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Page 1: The Langevin Equation Coffey_Kalmykov_Waldron

1 World Scientific Series in Contemporary Chemical Physics - Vol. 14

The Langevin Equation With Applications to Stochastic

Problems in Physics, Chemistry and

Electrical Engineering

Second Edition

W. T. Coffey Yu. P. Kalmykov J. T. Waldron

World Scientific

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The Langevin Equation

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SERIES IN CONTEMPORARY CHEMICAL PHYSICS

Editor-in-Chief: M. W. Evans (A1AS, Institute of Physics, Budapest, Hungary)

Associate Editors: S Jeffers (York University, Toronto) D Leporini (University of Pisa, Italy) J Moscicki (Jagellonian University, Poland) L Pozhar (The Ukrainian Academy of Sciences) S Roy (The Indian Statistical Institute)

Vol. 2 Beltrami Fields in Chiral Media by A. Lakhtakia

Vol. 3 Quantum Mechanical Irreversibility and Measurement by P. Grigolini

Vol. 4 The Photomagneton and Quantum Field Theory: Quantum Chemistry, Vol. 1 by A. A. Hasanein and M. W. Evans

Vol. 5 Computational Methods in Quantum Chemistry: Quantum Chemistry, Vol. 2 by A. A. Hasanein and M. W. Evans

Vol. 6 Transport Theory of Inhomogeneous Fluids by L. A. Pozhar

Vol. 7 Dynamic Kerr Effect: The Use and Limits of the Smoluchowski Equation and Nonlinear Inertial Responses by J.-L. Dejardin

Vol. 8 Dielectric Relaxation and Dynamics of Polar Molecules by V. I. Gaiduk

Vol. 9 Water in Biology, Chemistry and Physics: Experimental Overviews and Computational Methodologies by G. W. Robinson, S. B. Zhu, S. Singh and M. W. Evans

Vol. 10 The Langevin Equation: With Applications in Physics, Chemistry and Electrical Engineering by W. T. Coffey, Yu P. Kalmykov and J. T. Waldron

Vol. 11 Structure and Properties in Organised Polymeric Materials eds. E. Chiellini, M. Giordano and D. Leporini

Vol. 12 Proceedings of the Euroconference on Non-Equilibrium Phenomena in Supercooled Fluids, Glasses and Amorphous Materials eds. M. Giordano, D. Leporini and M. P. Tosi

Vol. 13 Electronic Structure and Chemical Bonding by J.-R. Lalanne

Vol. 14 The Langevin Equation: With Applications to Stochastic Problems in Physics, Chemistry and Electrical Engineering, 2nd Edition by W. T. Coffey, Yu. P. Kalmykov and J. T. Waldron

Vol. 15 Phase in Optics by V. Perinova, A. Luks and J. Perina

Vol. 16 Extended Electromagnetic Theory: Space Charge in Vacuo and the Rest Mass of the Photon by S. Roy and B. Lehnert

Vol. 17 Optical Spectroscopies of Electronic Absorption by J.-R. Lalanne, F. Carmona and L. Servant

Vol. 18 Classical and Quantum Electrodynamics and the B(3) Field by M. W. Evans and L. B. Crowell

Vol. 19 Modified Maxwell Equations in Quantum Electrodynamics by H. F. Harmuth, T. W. Barrett and B. Meffert

Vol. 20 Towards a Nonlinear Quantum Physics by J. R. Croca

Vol. 21 Advanced Electromagnetism and Vacuum Physics by P. Cornille

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World Scientific Series in Contemporary Chemical Physics - Vol. 14

The Langevin Equation With Applications to Stochastic

Problems in Physics, Chemistry and

Electrical Engineering

Second Edition

W. T. Coffey Trinity College, Dublin, Ireland

Yu. P. Kalmykov Universite de Perpignan, France

J. T. Waldron Trinity College, Dublin, Ireland

\fc World Scientific NEW JERSEY • LONDON • SINGAPORE • BEIJING • SHANGHAI • HONGKONG • TAIPEI • CHENNAI

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Published by

World Scientific Publishing Co. Pte. Ltd.

5 Toh Tuck Link, Singapore 596224

USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601

UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE

British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library.

First published 2004 Reprinted 2005

THE LANGEVIN EQUATION: WITH APPLICATIONS TO STOCHASTIC PROBLEMS EM PHYSICS, CHEMISTRY AND ELECTRICAL ENGINEERING, 2nd Edition

Copyright © 2004 by World Scientific Publishing Co. Pte. Ltd.

All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to photocopy is not required from the publisher.

ISBN 981-238-462-6

Printed in Singapore by World Scientific Printers (S) Pte Ltd

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Dedicated to the memory of our dear friends

Herbert Frohlich

and

Hannes Risken

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Preface to the Second Edition

Two principle considerations impelled us to write a new edition of "The Langevin Equation". First, more than 7 years have elapsed since the publication of the first edition and after a variety of suggestions and comments of our colleagues and interested readers of the first edition, it became increasingly evident that the presentation of the material in the book could be greatly improved. Secondly, in that period many new and exciting developments have occurred in the application of the Langevin equation to Brownian motion. In particular, one should mention the extension of the theory to include relaxation of molecules of an arbitrary shape, the extension of the Kramers treatment of the escape particles over potential barriers to non-axially symmetric potentials in the context of magnetic relaxation, the verification of the Kramers theory by measurements of the superparamagnetic relaxation time, stochastic resonance, anomalous relaxation, etc. In order to accommodate all the new developments, the book has been extensively rewritten; the chapters have been reordered so as to give a more logical presentation of the material. Moreover, a very large amount of new material has been added.

In particular, Chapter 1 has been expanded so that a complete account of the Kramers theory of escape of particles over potential barriers could be given along with an account of the extension of that theory to the decay of metastable states of multi-degree of freedom systems originally due to James Langer. This extension is particularly important in view of the application of Langer's method in the theory of superparamagnetism. In addition, a simplified account of the depletion effect of a bias field on the shallower of the two wells of the bistable potential has been given. This involves no other properties of the system save a knowledge of the partition function and the definition of the relaxation time of the correlation function and the overbarrier relaxation time. The relatively new topics of stochastic resonance and anomalous relaxation are also referred to in this chapter. In the latter context, it is shown how the hitherto empirical Cole-Cole formula may be derived

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Vlll The Langevin Equation

from a fractional diffusion equation based on the continuous time random walk proposed by the late Elliot Montroll and George Weiss. This chapter, because it is more elementary than the others, could reasonably be used for an introductory course on the theory of stochastic processes.

In Chapter 2, the sections dealing with Statonovich and Ito calculi have been rewritten and enlarged in order to present a comprehensive account of this subject. Moreover, the section dealing with matrix continued fraction methods of solution of recurrence relations has been revised and a general method of determining both the smallest nonvanishing eigenvalue of the Fokker-Planck equation and the correlation time by the continued fraction method has been given. Yet another significant addition to this chapter is a simple formula for the complex susceptibility of a system governed by a multistable potential. It is demonstrated that a knowledge of the overbarrier relaxation time, the effective relaxation time, and the integral relaxation time is sufficient to yield a simple two-mode relaxation formula for the complex susceptibility. The modes in question are the overbarrier relaxation mode and a single fast relaxation mode comprising all the fast relaxation modes in the wells of the potential. This formula is very useful in the interpretation of susceptibility measurements. Yet another new topic included in this chapter is the derivation of an exact integral formula for the integral relaxation time (the area under the curve of the relevant decay function). Such a formula is very useful for the generation of asymptotic expansions of the decay time and as a check of numerical results obtained by continued fraction methods. The integral relaxation time is also important in the context of the nonlinear response.

Chapter 3 is essentially unchanged from the first edition. Chapter 4 of the first edition which dealt with the itinerant oscillator model, more properly belongs to the realm of inertia effects and so has been moved in revised form to Chapter 10 of the present edition. Chapter 5 of the first edition dealing with rotational Brownian motion about a fixed axis in a potential excluding inertial effects, becomes Chapter 4 of this edition. Revisions incorporated in this chapter include the application of the continued fraction method for the determination of the smallest nonvanishing eigenvalue, developed in Chapter 2, to the Brownian motion in a potential. Another revision is the application of the two-mode susceptibility formula to the relaxation in a cosine potential.

The former Chapter 6 treating Brownian motion in a tilted cosine potential now becomes Chapter 5 and is substantially unchanged. We have merely simplified the presentation and corrected misprints. Chapter 6 of the present edition is completely new. It is concerned with the

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Preface IX

translational Brownian motion in a double-well (2-4) potential. The most interesting aspect of this potential is that the underlying recurrence relation for the statistical moments leads to divergent continued fractions. Nevertheless, by conversion of the continued fraction into an integral, it is possible to obtain a formula for the integral relaxation time, etc. Moreover, by application of the method developed in Chapter 2, it is again possible to use the two-mode relaxation formula to yield a simple expression for the position correlation function.

Chapter 7, which treats the noninertial rotation Brownian motion in space, has been substantially expanded. Here, a more general approach than that used in the previous edition has been given by regarding the Langevin equation for rotational diffusion in a potential as a Stratonovich equation. Moreover by means of an expansion of the averaged stochastic evolution equation for an arbitrary function in spherical harmonics, it is shown that a general recurrence relation for the statistical moments may be obtained in terms of the Clebsch-Gordan coefficients. The recurrence relation is valid for any potential that may be expanded in spherical harmonics. This general approach based essentially on the theory of angular momentum is then specialised to solve various problems arising in dielectric and magnetic relaxation, e.g., dielectric relaxation in nematic liquid crystals and in liquids subjected to a strong external dc electric field. The other substantial extension of this chapter is the solution for anisotropic rotational diffusion of an asymmetric top using the Euler-Langevin equation (again regarded as a Stratonovich equation), the properties of Wigner's D functions and the Clebsch-Gordan coefficients. This procedure leads to a completely general set of moment equations for the time evolution of the averages of the D functions. Examples of the application of the general theory include the retrieval of Perrin's result for the linear dielectric relaxation of a general ellipsoid and the nonlinear dielectric relaxation of a rod-like molecule in a weak ac field superimposed in a strong dc bias field.

Chapter 8 of this edition is concerned with the application of matrix continued fraction methods to the solution of various problems of rotational diffusion in axially symmetric potentials (excluding inertial effects). The material of the first edition has been augmented by giving a comprehensive account of the dynamical processes occurring in superparamagnetic particles, where solutions obtained by the effective eigenvalue method described in Chapter 2 are compared with the exact solution using the continued fraction method, thus establishing the parameter ranges in which approximate analytic solutions are valid. Another new topic introduced in this chapter is the nonlinear ac and

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X The Langevin Equation

transient response of polar molecules in dielectric and Kerr effect relaxation. These responses are evaluated exactly for the first time using the matrix continued fraction method, thereby enabling one to judge the accuracy of previously available analytic solutions obtained by perturbation theory. Yet another new result of Chapter 8 stemming from the recurrence relations for the time evolution of the averages of Wigner's D functions, which have been established in Chapter 7, is both exact and effective eigenvalue solutions for the complex susceptibility and relaxation times of nematic liquid crystals. These are written in terms of the order parameter of a nematic liquid crystal composed of molecules with the dipole moment not collinear with the long molecular axis. Important consequences of this investigation are simple formulae for the retardation factors and complex susceptibility which may be easily compared with experiments. The final new topic added in this chapter is stochastic resonance of superparamagnetic particles in the simplest uniaxial potential of the magnetocrystalline anisotropy. This problem is chosen as a detailed example of the stochastic resonance phenomenon.

Chapter 9 of the first edition was concerned with magnetic relaxation in non-axially symmetric potentials of the magnetocrystalline anisotropy. However, the material presented there does not represent a practical method of solution of the multi-term recurrence relation arising from the order and degree of the spherical harmonics, since all that was accomplished was to convert the solution of the multi-term recurrence relation into the diagonalisation of a set of linear equations. A much more efficient way of calculating the statistical averages is to formulate the problem in terms of matrix continued fractions. Thus, magnetic relaxation and ferromagnetic resonance in a strong magnetic bias field applied at an angle to the easy axis of magnetisation in unixial anisotropy and magnetic relaxation in a cubic anisotropy potential may be solved exactly. An important consequence of this calculation is the verification of the Kramers theory of escape of particles over potential barriers as applied to magnetic relaxation. The most important effect occurring in a nonaxially symmetric potential is the coupling between the longitudinal and transverse modes of the magnetisation which strongly influences both the ferromagnetic resonance and the overbarrier relaxation.

Chapter 10 is concerned with inertial effects in rotational Brownian motion. At the time of preparation of the first edition, the matrix continued fraction method had not been well developed as far as applications to inertial rotation were concerned. In this chapter, the matrix continued fraction method is applied to determine the exact solution for inertial orientational relaxation of a fixed rotator in a cosine

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Preface XI

potential. It is shown that the combination of inertial effects and an applied field produces strong absorption in the far-infrared range of frequencies. Referring to orientational relaxation in the absence of an external potential, exact solutions for orientational relaxation of rigid rotators and symmetric top molecules are presented using the Langevin equation method. Furthermore, the itinerant oscillator model considered in Chapter 4 of the first edition is treated in this chapter. There, an exact solution for the complex susceptibility without the assumption of small oscillations is obtained, again using the continued fraction method. This method allows one to judge the accuracy of previously available formulae for the susceptibility obtained using the small oscillation approximation.

Chapter 11 is completely new as it is concerned with anomalous relaxation, a topic which has risen to the fore in the period since the first edition was prepared. It is shown in this chapter how the fractional diffusion equation for a random walk with finite jump length variance and a random distribution of the waiting times may be used to justify the anomalous relaxation behaviour in the noninertial limit. The effect of the anomalous relaxation behaviour on system parameters such as the Kramers escape rate is considered. Moreover, it is shown how inertial effects and an external potential may be incorporated into anomalous diffusion by solving the fractional equivalent of the Klein-Kramers equation in phase space. It is also shown how the anomalous relaxation may be set in the context of the generalised Langevin equation of Mori.

We would like to thank a number of individuals and organizations who have greatly helped us directly or indirectly in preparation of this edition. It would be difficult to list all of them: in particular, we thank David Burns, Declan O'Connor, Derrick Crothers, Pierre-Michel Dejardin, the late Jean-Louis Dormann, Paul Fannin, Dimitri Garanin, Lawrence Geoghegan, Olivier Henri-Rousseau, Hamid Kachkachi, the late Rolf Landauer, Dave McCarthy, Bernard Mulligan, Stuart Rice, Wolfgang Wernsdorfer. We are indebted to Yuri Raikher and Victor Stepanov for their contribution on stochastic resonance in Section 8.3.3 as well as for useful comments and suggestions. Special thanks are due to Jean-Louis Dejardin and Sergey Titov who have carefully read the entire manuscript and proposed a number of corrections and improvements in the presentation. We would also like to thank the Trinity College Dublin Trust, the Royal Irish Academy, the Royal Society, the Enterprise Ireland International Collaboration Fund, INTAS and the United States Air Force Research Laboratory, European Office of Aerospace Research and Development for financial support. In particular, W.T.C. wishes to thank Trinity College Dublin for the award of a Berkeley Fellowship for the

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Xll The Langevin Equation

Academic Year 2001-2002 which greatly facilitated the production of this edition. Finally, we thank our families who bore with us during the heavy work of preparing this edition.

Dublin William T. Coffey Perpignan Yuri P. Kalmykov April 2003 John T. Waldron

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Preface to the First Edition

This book may be said to have its origins in a remarkable conference on Brownian motion held at the School of Theoretical Physics, Dublin Institute for Advanced Studies in August 1976 during the course of which several of the problems we shall discuss were first posed.

One of our main objectives in writing the book is to demonstrate how the concept of the equation of motion of a Brownian particle - the Langevin equation - first formulated by Langevin in 1908 - so making him the founder of the subject of stochastic differential equations, may be radically extended to solve the nonlinear problems which arise in the theory of the Brownian motion in a potential. This approach to the subject enables one to completely dispense with the underlying probability density diffusion equation - the Fokker-Planck equation, with all its attendant mathematical complications. These are particularly pronounced in the theory of rotational Brownian motion on which heavy emphasis is laid throughout the book.

The basis of our treatment is to regard the Langevin equation for a set of random variables (the time average of which is the desired quantity) starting from a set of sharp values of them at a given time as an integral equation for their values at a later time. The time average of this equation, calculated in accordance with the Stratonovich rule (as is also used to calculate the drift coefficient in the corresponding Fokker-Planck equation) may then be expressed in terms of a deterministic equation of motion for the set of sharp starting values. The method may be applied repeatedly to generate the hierarchy of differential-recurrence relations governing the time evolution of the averages usually generated by the Fokker-Planck equation. Hence that equation is shown to be redundant and any desired average property of a system governed by a Langevin equation may be directly calculated from that equation so extending Langevin's treatment of the Brownian motion to nonlinear systems. The problem of calculating the averages ultimately reducing to the task of diagonalising the system matrix, just as in the state space approach to

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XIV The Langevin Equation

dynamical systems theory as the differential-recurrence relations may always be written as a first order linear matrix differential equation with constant coefficients. This representation also has the outstanding advantage that it avoids the difficulties one encounters (with special functions etc.) by regarding the calculation of the eigenvalues of the Fokker-Planck equation as a Sturm-Liouville problem. All that is required are simple properties of matrices. The matrix representation of the solution also yields a simple general formula for the correlation time (in linear response theory the area under the curve of the normalised decay function) in terms of the inverse of the system matrix and the initial value vector with that vector being in turn a function of equilibrium averages only.

Another important result which follows from our treatment is that on extending the Cresser-Risken algorithm (Chapter 6) for the solution of three term recurrence-relations it is possible to have exact solutions for the Laplace transform of the after-effect function, the correlation time (which appears in terms of special functions), etc. for a variety of relaxation problems. The representation of the exact solution for the correlation time in terms of sums of products of special functions (usually confluent hypergeometric (or Kummer) functions) has the advantage that it allows that quantity to be rendered in integral form, the method of steepest descents may then be easily applied to determine the asymptotic behaviour.

In addition, for relaxation problems involving a strong external field and a bistable potential (see Chapter 8), we demonstrate that the conventional assumption that a single decay mode dominates no longer invariably holds. This behaviour occurs for external fields less than the critical field at which the two minima structure of the potential disappears. This in turn means that the assumption that the correlation time may be closely approximated by the inverse of the smallest non-vanishing eigenvalue (for sufficiently high barriers effectively the inverse of the Kramers escape rate) is no longer absolutely true. Such behaviour arises because only in certain cases such as relatively weak external fields can the contribution of the other decay modes be neglected.

Our ideas owe much to the work of the late Hannes Risken and the important suggestion of a change of variable in the Langevin equation to the quantity the average value of which one wishes to calculate, made by Frood and Lai in 1975, the other central idea being Doob's interpretation in 1942 of the Langevin equation as an integral equation (the relevant papers and books are cited in the text). This is treated in detail in Chapters 5-8.

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Preface xv

The book is arranged in a manner such that much of the material may be accessible to a beginning graduate or advanced undergraduate student. A synergetic approach in the spirit of Haken and the principles laid down by the Editors of the series Advances in Chemical Physics being followed as far as is possible with examples being taken from fields as diverse as laser physics and paleomagnetism. In particular, large parts of the material presented in Chapters 1 and 3 have been given as a final year undergraduate course in the School of Engineering, Trinity College Dublin. Chapter 1 being designed inter alia to serve as an expanded guide to the classical literature e.g. the Dover publication Selected Papers on Noise and Stochastic Processes, edited by Nelson Wax. Chapter 2 is devoted to the detailed development of our Langevin method for the calculation of averages described above. Chapters 4 to 10 on the other hand are concerned with particular applications of the theory, to dielectric relaxation, superparamagnetism, ring laser gyros to name but a few.

There are a number of organisations and individuals who have greatly helped us which we would like to thank, in particular. Derrick Crothers, James McConnell, Brendan Scaife, (with whom the tradition of dielectric studies at Trinity College Dublin and the Dublin Institute for Advanced Studies began) Myron Evans (Series Editor) James Calderwood, Roy Chantrell, Francis Farley, Akio Morita, Jagdish Vij, Peter Corcoran, Estomih Massawe, Paul Fannin, Lawrence Geoghegan, Patrick Cregg, Kevin Quinn, Vladimir Gaiduk, Jean Louis Dormann, Jean Louis Dejardin, Stuart Rice, Tony Wickstead, Wolffram Schroer, Maxi San Miguel and Eugene Walsh. We would also like to thank the Trinity College Dublin Trust, the Soros Foundation, the Royal Irish Academy, the Royal Society, the Nuffield Foundation, the British Council and Forbairt for travel grants etc. We thank our families who bore with us during the heavy work of writing the volume and last but not least Ms. Brenda McDonald for her magnificent typing efforts.

Trinity College Dublin 27 July 1995

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Contents

Preface to the Second Edition vii

Preface to the First Edition xiii

Contents xvii

Chapter 1 Historical Background and Introductory Concepts.... 1 1.1 Brownian Motion 1 1.2 Einstein's Explanation of the Brownian Movement 6 1.3 The Langevin Equation 11

1.3.1 Calculation of Avogadro's number 16 1.4 Einstein's Method 17 1.5 Necessary Concepts of Statistical Mechanics 23

1.5.1 Ensemble of systems 25 1.5.2 Phase space 25 1.5.3 Representative point 26 1.5.4 Ergodic hypothesis 26 1.5.5 Calculation of averages 27 1.5.6 Liouville equation 29 1.5.7 Reduction of the Liouville equation 32 1.5.8 Langevin equation for a system with one degree

of freedom 33 1.5.9 Effect of a heat bath. Intuitive derivation of the

Klein-Kramers equation 34 1.5.10 Conditions under which a Maxwellian

distribution in the velocities may be deemed to be attained 35

1.5.11 Very high damping regime 37 1.5.12 Low damping regime 40

1.6 Probability Theory 44 1.6.1 Random variables and probability distributions... 45 1.6.2 Properties of the Gaussian distribution 48 1.6.3 Moment generating functions 52 1.6.4 Central Limit Theorem 56

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XV111 The Langevin Equation

1.6.5 Random processes 57 1.6.6 Wiener-Khinchine theorem 59

1.7 Application to the Langevin Equation 60 1.8 Wiener Process 63

1.8.1 Variance of the Wiener process 64 1.8.2 Wiener integrals 66

1.9 The Fokker-Planck Equation 68 1.10 Drift and Diffusion Coefficients 76 1.11 Solution of the One-Dimensional Fokker-Planck

Equation 80 1.12 The Smoluchowski Equation 83 1.13 Escape of Particles over Potential Barriers — Kramers'

Escape Rate Theory .' 85 1.13.1 Escape rate in the IHD limit 91 1.13.2 Kramers' original calculation of the escape rate

for very low damping 95 1.13.3 Range of validity of the IHD and VLD formulae 99 1.13.4 Extension of Kramers' theory to many dimen

sions in the intermediate-to-high damping limit 102 1.13.5 Langer's treatment of the MD limit 104 1.13.6 Kramers' formula as a special case of Langer's

formula 109 1.14 Applications of the Theory of Brownian Movement

in a Potential 112 1.15 Rotational Brownian Motion — Application to

Dielectric Relaxation 113 1.15.1 Breakdown of the Debye theory at high

frequencies 118 1.16 Superparamagnetism — Magnetic After-Effect 121 1.17 Brown's Treatment of Neel Relaxation 128 1.18 Asymptotic Expressions for the Neel Relaxation Time... 133

1.18.1 Application of Kramers' method to axially sym-metric potentials of the magneto-crystalline anisotropy 133

1.18.2 IHD formula for magnetic spins 137 1.19 Ferrofluids 141 1.20 Depletion Effect in a Biased Bistable Potential 143 1.21 Stochastic Resonance 149 1.22 Anomalous Diffusion 152

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Contents xix

1.22.1 Empirical formulae for e(co) 156 1.22.2 Theoretical justification for anomalous

relaxation behaviour 157 1.22.3 Anomalous dielectric relaxation of an assembly

of fixed axis rotators 160 References 164

Chapter 2 Langevin Equations and Methods of Solution 169 2.1 Criticisms of the Langevin Equation 169 2.2 Doob's Interpretation of the Langevin Equation 171 2.3 Nonlinear Langevin Equation with a Multiplicative

Noise Term: Ito and Stratonovich Rules 172 2.4 Derivation of Differential-Recurrence Relations from

the One-Dimensional Langevin Equation 177 2.5 Nonlinear Langevin Equations in Several Dimensions.... 179 2.6 Average of the Multiplicative Noise Term in the

Langevin Equation for a Rotator 183 2.6.1 Multiplicative noise term for a three-

dimensional rotator 184 2.6.2 Multiplicative noise terms with / taken as zero

prior to averaging 186 2.6.3 Explicit average of the noise induced terms for

a planar rotator 188 2.7 Methods of Solution of Differential-Recurrence Relations

Arising from the Nonlinear Langevin Equation 190 2.7.1 Matrix diagonalisation method 191 2.7.2 Initial conditions 194 2.7.3 Matrix continued fraction solution of recurrence

equations 196 2.8 Linear Response Theory 201 2.9 Correlation Time 207 2.10 Linear Response Theory Results for Systems with

Dynamics Governed by One-Dimensional Fokker-Planck equations 210

2.11 Smallest Nonvanishing Eigenvalue: The Continued Fraction Approach 214 2.11.1 Evaluation of k\ from a scalar three-term

recurrence relation 215

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XX The Langevin Equation

2.11.2 Evaluation of X\ from a matrix three-term recurrence relation 219

2.12 Effective Eigenvalue 221 2.13 Evaluation of the Dynamic Susceptibility Using T , xef,

and A( 223 2.14 Nonlinear Response of a Brownian Particle Subjected to

a Strong External Field 226 2.14.1 Analytical solutions for the relaxation time of

one-dimensional systems 227 2.14.2 Nonlinear transient response in the rotational

Brownian motion 229 References 233

Chapter 3 Brownian Motion of a Free Particle and a Harmonic Oscillator 236

3.1 Ornstein-Uhlenbeck Theory of the Brownian Motion 236 3.2 Stationary Solution of the Langevin Equation —

The Wiener-Khinchine Theorem 238 3.3 Brownian Motion of a Harmonic Oscillator 241 3.4 Application to Dielectric Relaxation..., 243

3.4.1 Theorem about Gaussian random variables 244 3.5 Torsional Oscillator Model: Example of the Use of the

Wiener Integral 247 References 251

Chapter 4 Two-Dimensional Rotational Brownian Motion in N-Fold Cosine Potentials 252

4.1 Introduction 252 4.2 Langevin Equation for Rotation in Two Dimensions 253 4.3 Longitudinal and Transverse Effective Relaxation

Times in the Noninertial Limit 256 4.4 Polarisabilities and Dielectric Relaxation Times of a

Fixed Axis Rotator with Two Equivalent Sites 261 4.4.1 Introduction 261 4.4.2 Matrix solution 263 4.4.3 Longitudinal polarisability and relaxation times 266 4.4.4 Transverse polarisability and relaxation times... 274

4.5 Comparison of the Longitudinal Relaxation Time with the Results of the Kramers Theory 278

References 280

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Contents xxi

Chapter 5 Brownian Motion in a Tilted Cosine Potential: Application to the Josephson Tunnelling Junction 282

5.1 Introduction 282 5.2 Josephson Junction: Dynamic Model 283 5.3 Reduction of the Averaged Langevin Equation for the

Junction to a Set of Differential-Recurrence Relations.... 285 5.4 DC Current-Voltage Characteristics 287 5.5 Linear Response to an Applied Alternating Current 290 5.6 Effective Eigenvalues for the Josephson Junction 293 5.7 Linear Response Using the Effective Eigenvalues 298 5.8 Spectrum of the Josephson Radiation 302 References 307

Chapter 6 Translational Brownian Motion in a Double-Well Potential 309

6.1 Introduction 309 6.2 Relaxation Time of the Position Correlation Function.... 310 6.3 Comparison of Characteristic Times and Evaluation

of the Position Correlation Function 317 References 323

Chapter 7 Three-Dimentional Rotational Brownian Motion in an External Potential: Application to the Theory of Dielectric and Magnetic Relaxation 325

7.1 Introduction 325 7.2 Rotational Diffusion in an External Potential:

The Langevin Equation Approach 326 7.3 Gilbert's Equation Augmented by a Random Field Term 335

7.3.1 Langevin equation approach 337 7.3.2 Fokker-Planck equation approach 343

7.4 Brownian Rotation in the Uniaxial Potential 347 7.4.1 Longitudinal relaxation 347 7.4.2 Susceptibility and relaxation times 349 7.4.3 Integral form and asymptotic expansions 355 7.4.4 Transverse response 358 7.4.5 Complex susceptibilities 361

7.5 Brownian Rotation in a Uniform DC External Field 367 7.5.1 Introduction 367 7.5.2 Longitudinal response 368 7.5.3 Transverse response 372 7.5.4 Comparison with experimental data 375

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xxii The Langevin Equation

7.6 Anisotropic Noninertial Rotational Diffusion of an Asymmetric Top in an External Potential 378 7.6.1 Solution of the Euler-Langevin equation for an

asymmetric top in the noninertial limit 378 7.6.2 Linear response of an assembly of asymmetric

tops 388 7.6.3 Response in superimposed ac and strong dc bias

fields: perturbation solution 390 References 394

Chapter 8 Rotational Brownian Motion in Axially Symmetric Potentials: Matrix Continued Fraction Solutions 397

8.1 Introduction 397 8.2 Application to the Single Axis Rotator 398

8.2.1 Longitudinal response 398 8.2.2 Transverse response 403 8.2.3 Relaxation times 405

8.3 Rotation in Three Dimensions: Longitudinal Response 407 8.3.1 Uniaxial particle in an external field 407 8.3.2 Characteristic times and magnetic susceptibility 413 8.3.3 Magnetic stochastic resonance 421

8.4 Transverse Response of Uniaxial Particles 427 8.4.1 Matrix continued fraction solution 427 8.4.2 Transverse complex magnetic susceptibility 429

8.5 Nonlinear Transient Responses in Dielectric and Ken-Effect Relaxation 436

8.6 Nonlinear Dielectric Relaxation of Polar Molecules in a Strong AC Electric Field: Steady State Response 443

8.7 Dielectric Relaxation and Rotational Brownian Motion in Nematic Liquid Crystals 450

References 465

Chapter 9 Rotational Brownian Motion in Non-Axially Symmetric Potentials 468

9.1 Introduction 468 9.2 Uniaxial Superparamagnetic Particles in an Oblique

Field 469 9.2.1 Recurrence equations 469 9.2.2 Matrix continued fraction solution 473

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Contents xxm

9.2.3 Smallest nonvanishing eigenvalue, the relaxation time, and the complex susceptibility 478

9.3 Cubic Anisotropy 490 9.3.1 Recurrence equations 490 9.3.2 Matrix continued fraction solution 493 9.3.3 Complex susceptibility and relaxation times 499

References 505

Chapter 10 Inertial Langevin Equations: Application to Orientational Relaxation in Liquids 507

10.1 Introduction 507 10.2 Step-On Solution for Noninertial Rotation about a

Fixed Axis 508 10.3 Inertial Rotation about a Fixed Axis 512

10.3.1 Inertial effects and nonlinear response 512 10.3.2 Matrix continued fraction solution 519

10.4 Inertial Rotational Brownian Motion of a Thin Rod in Space 530 10.4.1 Derivation of recurrence equations 530 10.4.2 Evaluation of Q 536 10.4.3 Evaluation of C2 539 10.4.4 Evaluation of C, for an arbitrary / 540

10.5 Rotational Brownian Motion of a Symmetrical Top 544 10.5.1 Derivation of recurrence equations 544 10.5.2 Evaluation of Q and C2 548

10.6 Itinerant Oscillator Model of Rotational Motion in Liquids 557 10.6.1 Introduction 557 10.6.2 Generalisation of the Onsager model — Relation

to the cage model 558 10.6.3 Dipole correlation function 563 10.6.4 Exact solution for the complex susceptibility

using matrix continued fractions 566 10.6.5 Results and comparison with experimental

data 570 10.7 Application of the Cage to Ferrofluids 576 Appendix A: Statistical Averages of the Hermite Polynomials

of the Angular Velocity Components for Linear Molecules 589

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xxiv The Langevin Equation

Appendix B: Averages of the Angular Velocities Components 590 Appendix C: Evaluation of cos0(e) in the Low

Damping Limit 594 Appendix D: Sack's Continued Fraction Solution for the

Sphere 596 References 597

Chapter 11 Anomalous Diffusion 600 11.1 Discrete and Continuous Time Random Walks 600 11.2 A Fractional Diffusion Equation for the Continuous

Time Random Walk Model 603 11.2.1 Solution of fractional diffusion equations in

configuration space 612 11.2.2 Anomalous diffusion of a planar rotator in a

mean field potential 617 11.3 Divergence of Global Characteristic Times in

Anomalous Diffusion 621 11.3.1 First passage time for normal diffusion 622 11.3.2 First passage time distribution for anomalous

diffusion 626 11.4 Inertial Effects in Anomalous Relaxation 631

11.4.1 Slow transport process governed by trapping 632 11.4.2 Calculation of the complex susceptibility 634 11.4.3 Comment on the use of the telegraph equation

as an approximate description of the configuration space distribution function including inertial effects 639

11.5 Barkai and Silbey' s Form of the Fractional Klein-Kramers Equation 641 11.5.1 Complex susceptibility 644 11.5.2 Fractional kinetic equation for the needle model 648

11.6 Anomalous Diffusion in a Periodic Potential 654 11.6.1 Calculation of the spectra 660

11.7 Fractional Langevin Equation 665 Appendix: Fractal Dimension, Anomalous Exponents and

Random Walks 670 References 672

Index 675

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Chapter 1

Historical Background and Introductory Concepts

1.1 Brownian Motion

The first detailed account of Brownian motion was given by the eminent botanist Robert Brown in 1827 [1] while studying the plant life of the South Seas. In this study, he dealt with the transfer of pollen into the ovulum of a plant. He examined aqueous suspensions of pollen grains of several species under a microscope and found that in all cases the pollen grains were in rapid oscillatory motion.

Initially, he thought that the movement was not only "vital" (in the sense of not being due to a physical cause), but peculiar to the male sexual cells of plants. He quickly disembarrassed himself of this explanation on observing that the motion was exhibited by grains, which he called irritable particles, of both organic and inorganic matter in suspension. We describe the evolution of Brown's reasoning in his own words [1]:

"Having found as I believed a peculiar character in the motion of the particles of pollen in water it occurred to me to appeal to this peculiarity as a test in certain cryptogamous plants, namely Mosses and the genus Equisetum in which the existence of sex organs had not been universally admitted ... But I at the same time observed, that in bruising the ovula or seeds of Equisetum which at first happened accidentally, I so greatly increased the number of moving particles, that the source of the added quantity could not be doubted. I found also on bruising first the floral leaves of Mosses and then all other parts of those plants, that I readily obtained similar particles not in equal quantity indeed, but equally in motion. My supposed test of the male organ was therefore necessarily abandoned".

1

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2 The Langevin Equation

He proceeds:

"Reflecting on all the facts with which I had now become acquainted, I was disposed to believe that the minute particles or molecules of apparently uniform size, were in reality the supposed constituent or elementary molecules of organic bodies, first so considered by Buff on and Needham ..."

Brown investigated whether the motion was limited to organic bodies:

"A minute portion of silicified wood which exhibited the structure of Coniferae, was bruised and spherical particles or molecules in all respects like those so frequently mentioned were readily obtained from it: in such quantity, however, that the whole substance of the petrification seemed to be formed of them. From hence I inferred that these molecules were not limited to organic bodies, nor even to their products".

Later, he writes:

"Rocks of all ages, including those in which organic remains have never been found yielded the molecules in abundance. Their existence was ascertained in each of the constituent minerals of granite, a fragment of the Sphinx being one of the specimens observed".

Brown finally described the motion as [1]:

"Matter is composed of small particles which he called active molecules that exhibit a rapid irregular motion having its origin in the particles themselves and not in the surrounding fluid".

Following Brown's work there were many years of speculation [1,2] as to the cause of the phenomenon before Einstein made conclusive mathematical predictions of a diffusive effect arising from the random thermal motions of particles in suspension. Most of the hypotheses advanced in the nineteenth century could be dismissed by considering an experiment described by Brown in which a drop of water of microscopic size, immersed in oil and containing just one particle, unceasingly exhibited the motion. According to Nelson [1], the first investigator to express a notion close to the modern theory of Brownian movement (i.e., that the perpetual motion is caused by bombardment of the Brownian particle by the particles of the surrounding medium) was C. Weiner in 1863 [2].

We mention the very detailed experimental investigation made by Gouy, which greatly supported the kinetic-theory explanation. Gouy's conclusions may be summarised by the following seven points [1].

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Chapter 1. Historical Background and Introductory Concepts 3

Figure 1.1.1. Trajectory of a Brownian particle.

1. The motion is very irregular, composed of translations and rotations, and the trajectory appears to have no tangent.

2. Two particles appear to move independently, even when they approach one another to within a distance less than their diameter.

3. The smaller the particles, the more active the motion. 4. The composition and density of the particles have no effect on

the motion. 5. The less viscous the fluid, the more active the motion. 6. The higher the temperature, the more active the motion. 7. The motion never ceases.

Point 1 is of profound interest in view of the later work of N. Wiener [1], who proved in 1923 that the sample points of the Brownian-motion trajectory are almost everywhere continuous, but nowhere differentiable. Despite these careful observations in favour of kinetic theory, however, several arguments always seemed to militate against it. We give below two of the most prominent.

An early attempt to explain Brownian motion in terms of collisions was made by von Nageli. We consider the conservation of momentum during an atomic collision with a macroscopic Brownian particle of mass M and velocity V. If the surrounding molecules each have mass m and velocity v, the velocity change Av of the molecule on a single impact would be (m/M) v. If v is calculated from the kinetic-theory equation

±m(v2) = -kT

(k= 1.38 x 10~23 JKT1 is the Boltzmann constant and T is the absolute temperature) and then the principle of conservation of momentum is

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4 The Langevin Equation

applied, Av for a typical Brownian particle (10~6 m in diameter) in water at 300 °K is about 5 x 10"8 ms"1. The observed Brownian movement for this system, however, is greater than this by two orders of magnitude. Von Nageli was aware of this discrepancy; however, he could not explain it in terms of collisions because he assumed that these would produce zero net effect. Thus, he effectively calculated only the velocity change as a result of a single collision. His error lay in regarding the random collisions as occurring in regularly alternating directions that would keep bringing the target molecule back to its starting position. This assumption is invalid, because if n random collisions occur (see the discussion of the random-walk problem in Ref. [9]), the displacement (root-mean-square value) will be proportional to n m. Now, if the time interval between successive observations of the particle is T, n will be proportional to T. Thus the root-mean-square value of the displacement is proportional to Tm, and not zero as assumed by von Nageli.

Many investigators assumed (correctly) that the macroscopic Brownian particle could be treated simply as an enormous "atom" of mass M. This would also allow a test of the kinetic theory, because the law of equipartition of energy implied that the kinetic energy of translation of a Brownian particle and of a molecule should be equal. Thus, the speed s of a Brownian particle should be given by that of a molecule:

f<i2K*r, d.i.1) where s = ds/dt. For the system described above, s predicted by Eq. (1.1.1) is much greater than the visually observed value. The explanation is that the equipartition formula above holds only when the time between observations is of the order of the time between collisions. In practice, we cannot make observations to such a. fine degree.

To aid our argument let us consider (s ) more closely. Suppose we observe for 3 min at 30 s intervals the motion of a Brownian particle and plot its two-dimensional "random walk". Its trajectory looks like that shown in Fig. 1.1.2. Suppose the same random walk had been observed at intervals rof 10 s. There would then be 3 times as manypoints, and the overall impression would be that the particle is moving V3 times as fast. During a time interval rthe particle undergoes millions of collisions, so as the time between observations is decreased still further, the apparent velocity continues to increase, and only when ris of the order of the time between collisions will Eq. (1.1.1) hold true. We note that the trajectory that we will observe is in no sense the actual path of the particle (cf. the following remarks of Fowler [5,18]):

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Chapter 1. Historical Background and Introductory Concepts 5

Figure 1.1.2. Trajectory of a Brownian particle: (a) observed at intervals of 30 s and (b) at intervals of 10 s.

"We can never follow the details of the movement of the grain (Brownian particle) which has a kink at every molecular collision -about 1012 times a second in an ordinary liquid. What we observe in the way of displacements are of the nature of residual fluctuations about a mean value zero and have little direct connection with the actual detailed path of the grain, to our senses (pushed to their farthest limit in the form of the best cine camera taking pictures at 105 per sec) the details of the path are impossibly fine. The path may be fairly compared in a crude way to the graph of a continuous function with no derivative." (We shall say more of this later).

The above remarks of Fowler have been beautifully reinforced and added to by Schroeder [83], see his Fig. 1, p. 142, where the Brownian motion is cited as the supreme example of a random fractal phenomenon. Fractals are, in general (as defined by Mandelbrot [84]), exceedingly fine grained structures that exhibit self-similarity with respect to multiplicative changes in scale. In other words, a self-similar object appears [83] unchanged after increasing or decreasing its size. Self-similar objects typify many laws of nature which are independent or nearly so of a scaling factor. Examples of scale factors being [83] Planck's constant or the speed of light. Perhaps a useful working definition for physicists of a fractal is that attributed to Mandelbrot by Feder [85]:

"A fractal is a shape made of parts similar to the whole in some way ".

For the purposes of this book, fractals may be considered [85] as sets of points embedded in a space. (A detailed account of Brownian paths considered as fractals is available in the book of Mazo [100].)

In the context of a record of the Brownian motion as a random fractal object, the statistically self-similar nature of the observations

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6 The Langevin Equation

(realisations) of a Brownian motion (that is, the Brownian record looks "the same" [85]) constitutes a scale invariance or symmetry of the Brownian record. In Brownian motion, the range of lengths over which statistical self-similarity prevails may be from macroscopic sizes almost down to the mean free path of a molecule.

1.2 Einstein's Explanation of the Brownian Movement

It was left to Einstein in 1905 to explain Brownian movement essentially by combining (in the sense that the velocity distribution is in equilibrium but not that of the displacements), the elementary stochastic process known as the random walk with the Maxwell-Boltzmann distribution [2]. His ideas may be summarised thus: If a particle in a fluid without friction receives a blow due to a collision with a molecule, then the velocity of the particle changes. However, if the fluid is very viscous, the change in velocity is quickly dissipated and the net result of an impact is a change in the displacement of the particle. Thus, Einstein assumed that the cumulative effect of collisions is to produce random jumps in the position of a Brownian particle; that is, the particle performs a kind of random walk. Taking the jumps in the walk as small, he obtained a partial differential equation for the probability density distribution of the displacement in one dimension [2]. This equation is a diffusion equation similar to that for unsteady heat conduction. It is the simplest case of a class of equations (probability density diffusion equations) that have become known as the Fokker-Planck equations. Einstein obtained its solution, from which he was able to show that the mean-square displacement of a Brownian particle should increase linearly with time. By using the fact that at equilibrium the Maxwellian distribution of velocities must hold, he was able to express the constants in the solution in terms of the temperature and the viscosity of the fluid. Einstein's formula for the mean-square displacement was verified experimentally by Perrin in 1908 [1,18]. He obtained from Einstein's formula a value of Avogadro's number that agreed to within 19% with the accepted value. This provided powerful evidence for the molecular structure of matter.

It is interesting to recall that Einstein formulated his theory without having observed Brownian movement, but predicted that such a movement should occur from the standpoint of the kinetic theory of matter. We quote from his 1905 paper [2]:

"According to the molecular kinetic theory of heat, bodies of microscopically-visible size suspended in a liquid will perform movements of such magnitude that they can be easily observed in a microscope, on account of the molecular motions of heat. It is

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Chapter 1. Historical Background and Introductory Concepts 7

possible that the movements to be discussed here are identical with the so-called "Brownian molecular motion": however, the information available to me regarding the latter is so lacking in precision, that I can form no judgement in the matter... If the movement discussed here can actually be observed (together with the laws relating it that one would expect to find), then classical thermodynamics can no longer be looked upon as applicable with precision to bodies even of dimensions distinguishable in a microscope: an exact determination of actual atomic dimensions is then possible. On the other hand, had the prediction of the argument proved to be incorrect a weighty argument would be proved against the molecular-kinetic conception of heat".

And later:

"From the standpoint of the molecular kinetic theory of heat a dissolved molecule is differentiated from a suspended body solely by its dimensions and it is not apparent why a number of suspended particles should not produce the same osmotic pressure as the same number of molecules. We must assume that the suspended particles perform an irregular movement - even if a very slow one - in the liquid on account of the molecular movement of the liquid".

We remark by way of historical background that Laplace [3] in 1812 obtained a partial differential equation similar to the Fokker-Planck equation in a discussion of the mixing of balls drawn and replaced at random from two urns and that a similar equation was obtained by Rayleigh (1880, 1894), who wished to find the probability distribution function of a sum of n sinusoidal motions all having the same period and amplitude with a random distribution of phases. For n —> °°, Rayleigh [4] obtained a diffusion equation similar to that of Einstein. Both of the previous examples are forms of the random walk problem which is discussed at length in [9,12,21,27,79] in the context of the Brownian movement. The first clear statement of the random walk problem seems to have been made by Karl Pearson in 1905 [5,9].

"A man starts from a point O and walks / yards in a straight line; he then turns through any angle whatever and walks another / yards in a second straight line. He repeats this process n times. I require the probability that after these n stretches he is at a distance between r and r + dr from his starting point zero".

The phrase "turns through any angle whatever and walks another / yards in a second straight line" constitutes a stosszahlansatz or mechanism of the random walk process.

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8 The Langevin Equation

Bachelier in 1900 made a mathematical model of the French Stock Exchange and obtained a diffusion equation similar to that of Einstein [5,22,101]. Later [5] (1911, 1912), he studied the related problem of the Gambler's Ruin, which is in effect a type of random walk problem. He showed that when the sequence of bets placed by the gambler is large, it is simpler to formulate a continuous model of the process. He was again led to a type of Fokker-Planck equation. More examples are given by Gardiner [21] and further applications to financial problems are given in [90,101]. A rudimentary, nonetheless useful intuitive derivation of the specialised form of the Fokker-Planck equation known as the Smoluchowski equation may be given as follows.

Let us suppose that we have / Brownian grains per unit volume (we shall also term these representative points or realisations of the random variable describing the trajectory of a Brownian particle) suspended in a liquid per unit volume between r and r + dx at time t and let us further suppose that these particles are subject to an external force which is the negative gradient of a potential V(r) so that

K=-gradV(r) . (1.2.1)

Now consider a volume v in the liquid bounded by a closed surface S and let us calculate the (drift) current of particles crossing S due to the action of K. We have by Gauss's divergence theorem

jJvf(r,t)dv = -jsJd ndS = -jvdivJddv, (1.2.2)

where Jd is the current density of particles and n is the unit normal to S. Thus, we have the continuity equation

f- + divj r f=0, (1.2.3) at

which is the law of conservation of representative points. Now, the drift current is

Jd=f*. (1-2.4) Here v is the drift velocity of a particle moving in the liquid. On supposing that

K-<Tv = 0, where -£v is the viscous drag and £"is the drag coefficient of a particle, the drift current is

J r f = - ( / / O g r a d V . (1.2.5) The analysis so far takes no account of the thermal agitation of

the particles (the Brownian movement). In order to take account of this let us now add to Jd a diffusive term

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Chapter 1. Historical Background and Introductory Concepts 9

idiff=-Dgmdf, (1.2.6)

whereD is the diffusion coefficient, in this instance, kTI £. The addition of this diffusive term makes the distribution/(r, t) more nearly uniform. The continuity equation [Eq. (1.2.3)] then becomes

%- = Ddiv [grad/ + /gradV lkT), (1.2.7) at

which is the specialised form of the Fokker-Planck equation known as the Smoluchowski equation, which describes approximately the evolution o f / i n configuration space. (For a detailed account of Smoluchowski's method of derivation of this equation, which was based on a specific detailed kinetic model, namely, collisions of hard spheres, see Mazo [100]). Equation (1.2.7) was first [5] given by Einstein in 1905 for the special case of V=0. In general direct justification of inclusion of the thermal agitation by adding the diffusive term of Eq. (1.2.6) is difficult, nevertheless, it yields the same results as the detailed methods of derivation of the Fokker-Planck equation presented below with very much less labour. The stationary solution of the Fokker-Planck equation is the solution with / = 0. The diffusion coefficient D is found by requiring that in general the stationary solution should be the Maxwell-Boltzmann distribution (with certain restrictions as in the case of a particle moving in a tilted cosine potential, which are discussed in [13], see Chapter 5). The approximate Smoluchowski equation assumes that the velocity distribution has reached statistical equilibrium that is it has the Maxwellian distribution.

Another form of the Fokker-Planck equation which we shall refer to is the Klein-Kramers equation [5] (originally derived [9] by Klein in 1921) which describes the evolution of the density / ( r , v, t) of representative points in phase space (r, v), viz

— + v • grad r / gradv/ • gradrV = — at m m

<*KM+—vlf m

(1.2.8)

The Fokker-Planck equation is a partial differential equation of parabolic type. In the mathematical literature, it is called a forward Kolmogorov equation. It may be described in general terms as a diffusion equation with an additional first order derivative with respect to position (i.e., a convective or hydrodynamical derivative). The rigorous derivation of the Fokker Planck equation is given in Section 1.9.

The detailed derivation of Einstein's formula for the mean square displacement of a Brownian particle and the associated Fokker-Planck equation will be given in Section 1.4 below. We shall, however, first give

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10 The Langevin Equation

the calculation of the mean square displacement using the method based on the equation of motion of the random variable proposed by Langevin [6] in 1908 as it is the extension of this method to treat nonlinear dynamical systems without recourse to the Fokker-Planck equation which is the principal concern of this book.

More specifically the method of calculation of average properties of a dynamical system (e.g., mean electric dipole moment, mean square displacement, velocity correlation function, etc.) which has been hitherto used is to construct the Fokker-Planck equation in phase space from the underlying Langevin equation for the random variables representing say the position and velocity of a Brownian particle. The distribution function is then expanded where possible into a product of a set of orthogonal functions in the position and an orthogonal set in the velocities corresponding directly to the averages of the dynamical quantities which one wishes to calculate (e.g., for rotational Brownian motion about a fixed axis one will obtain a Fourier-Hermite series, see Chapter 10). The above procedure leads to a set of differential-recurrence relations for the coefficients of the generalised Fourier series, which govern the time behaviour of the averages of the dynamical quantities, which are desired. The same is true for the Smoluchowski equation, which supposes that equilibrium of the velocities has been attained, (there the configuration space distribution is expanded in a set of orthogonal functions in the configuration variables) and which is approximately valid if inertial effects are small. Our main objective is to show how (by a suitable transformation of and averaging of the Langevin equation interpreted as an integral equation according to the Stratonovich rule, Chapter 2, Section 2.3) the set of differential-recurrence relations may be generated directly from the Langevin equation. The Fokker-Planck equation is then bypassed entirely.

The advantages of this formulation of the theory are that: (a) In general, Langevin's method is far easier to comprehend

than that of the Fokker-Planck equation as it is based directly on the concept of the time evolution of the random variable describing the process rather than on the time evolution of the underlying probability distribution. Indeed to quote Wang and Uhlenbeck [12] the Langevin equation is "the real basis of the theory of the Brownian motion", since the drift and diffusion coefficients in the Fokker-Planck equation must be calculated from it a priori.

(b) The need to construct the Fokker-Planck equation from the Langevin equation is dispensed with.

(c) It is often very difficult to separate the variables in the Fokker-Planck equation by expanding in the orthogonal sets of functions

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Chapter 1. Historical Background and Introductory Concepts 11

corresponding to the desired averages, particularly in problems involving rotational Brownian motion (cf. Chapters 7, 9, and 10), Langevin's method avoids such a procedure, all that is required is the interpretation of his equation as an integral equation (see Section 1.10 and Chapter 2).

For ease of comparison for the reader with the work of the early investigators, we shall adhere to their notation as far as possible in Sections 1.3 - 1.6, which constitute a detailed account of the work of Einstein and Langevin including a brief summary of those parts of equilibrium statistical mechanics which are essential to our discussion. Thus, in these sections, no distinction is made between a random variable £(t) and one of its realisations xt) as is often done in theoretical physics. The notation fx,t) for probability density function in configuration space used by Einstein rather than W(x, t) (which is used in the rest of the book) or p (q, p, t) (when treating the Liouville equation in Section 1.5) will also be used in these sections. Section 1.6 constitutes an introduction to those parts of the theory of probability which are needed, hence in accordance with standard textbooks in statistics, the letter %(t) is used to denote a random variable and lower case roman ones, e.g., x (t) are realisations. In the remainder of the book unless it is evident from the context, we shall always use £, (t) to denote a random (stochastic) variable.

1.3 The Langevin Equation

The theory of the Brownian movement as formulated by Einstein [2] and Smoluchowski [9] although in agreement with experiment seemed far removed from the Newtonian dynamics of particles [1] as it appeared to rely entirely on the concept of the underlying probability density distribution of Brownian particles and the Fokker-Planck equation for the time evolution of that distribution. Langevin in 1908 by in effect introducing the concept of the equation of motion of a random variable (in this case the position of a Brownian particle), to quote Nelson [1], "initiated a new train of thought culminating in a truly dynamical theory of Brownian motion". In addition, by his formulation of the theory, he was the founder of the subject of stochastic differential equations. Notwithstanding this, it must be emphasised that the problem of reducing the Brownian motion to Newtonian particle dynamics is still incomplete as the problem or one formulation of it [1] is to deduce each of the following theories from the one below it:

Einstein-Smoluchowski Ornstein-Uhlenbeck Maxwell-Boltzmann

Hamilton-Jacobi

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12 The Langevin Equation

The reader is referred to Section 4 of [9], Chapter VI of [17], [5], and [27] for a further discussion of this problem. The Ornstein-Uhlenbeck theory, which with its subsequent additions, is itself a rigorous formulation of Langevin's ideas is discussed in detail later.

Langevin began by simply writing down the equation of motion of the Brownian particle according to Newton's laws under the assumptions that the Brownian particle experiences two forces, namely:

(i) A systematic force (viscous drag) -£x(t), which represents a dynamical friction experienced by the particle, x is the displacement and ^is the coefficient of friction.

(ii) A rapidly fluctuating force Fi), which is again due to the impacts of the molecules of the liquid on the particle, now called white noise. This is the residual force exerted by the surroundings or heat bath when the frictional force has been subtracted.

Thus, his equation of motion, according to Newton's second law of motion, is for a particle of mass m

md^1=_cdm+m ( 1 3 1 ) dt dt

The friction term C,x is assumed to be governed by Stokes' law which states that the frictional force decelerating a spherical particle of radius a is

£x = 6rn]ax, (1.3.2) where TJ is the viscosity of the surrounding fluid. The following assumptions are made about the fluctuating part F(t) [5]:

(i) F (t) is independent of x. (ii) F(t) varies extremely rapidly compared to the variation of x(t).

It is also assumed that the average F(0 = 0, (1.3.3)

since F(t) is so irregular. The overbar in Eq. (1.3.3) means the statistical average over an ensemble of particles, each particle in the ensemble starting with the same (sharp) initial conditions. Assumption (ii) above implies that each collision is practically instantaneous. This rapid variation can be expressed by [5]

F(t)F(t') = 2CkTS(t-t'), (1.3.4) where S(t) is the Dirac delta function, t and t' are distinct times and 2t,kT is called the spectral density. Moreover, if the force is passed through a filter, the spectral density is not a function of the angular frequency 0), that is we have white noise (Section 1.7 and Chapter 3). Formally speaking

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Chapter 1. Historical Background and Introductory Concepts 13

, T'

F(t)F(t') = F(t)F(t + T)= l im— J F(t)F(t + T)dT, r->-r J

0 which [21] is the time average of a two time product over an arbitrary range time T which is allowed to become infinite. In reality the autocorrelation function, Eq. (1.3.4), starting from 2£kT drops very rapidly to zero and as a result the spectral density <$F(a)) does not have the constant value 2£k T at very high frequencies (coloured noise). The situation where <J>f(&») is constant for all co (white noise), corresponding in the time domain to the delta function autocorrelation function of Eq. (1.3.4), is the limiting case of a purely random process (that is a process without a memory) which will never occur in practice. It will become apparent later (see Section 1.6.3) that F(t) is a centred Gaussian random variable and that F(t) obeys Isserlis's theorem [12,14,15], i.e., for 2n Fs,

F,F2...F2n = F(tl)F(t2)...F(t2n) = X J ! *"('*, Wtj). d-3-5)

where the sum is over all distinct products of expectation value pairs, each of which is formed by selecting n pairs of subscripts from 2n subscripts. For example, when n = 2, we have

F^F.F^WIWA+WIWA+WAWI-In general, from the theory of permutations and combinations, there will

be (2«)!/(2"n!) such distinct pairs. We also have for an odd number of observations

F(tl)F(t2)...F(t2n+l) = F1F2...F2n+l=0. Langevin [5], using his equation which is the first example of a

stochastic differential equation, derived the formula for the mean-square displacement of the Brownian particle in the following way. On multiplying Eq. (1.3.1) by x (t), we get

mx(t)x(t) = -£x(t)x(t) + F(t)x(t). On noting that

(1.3.6)

\dxL

XX =

2 dt and

.. 1 d xx =

2dt

fdx2^

dt -x

we have

m d

~2~dt

rdx2(t)^

\ dt

- mx (?) = • £ dx2(t)

2 dt + F(t)x(t). (1.3.7)

Equation (1.3.7) refers to only one selected Brownian particle. The complete solution of the motion of a macroscopic system would consist

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14 The Langevin Equation

of solving all the microscopic equations of the system. Because we cannot do this, we use, instead, the averaged equation of motion, i.e.,

m d

~2~dt

dx2

dt •mx2 =-?-—+ Fx. (1.3.8)

2 dt

In Eq. (1.3.8), it is assumed that Fx vanishes because of the irregular variation of the force F. In other words, the random force F and the displacement x are completely uncorrelated. From statistical mechanics it is known that when the velocity process has reached its equilibrium value, the Maxwellian distribution [5] can be assumed to hold so that the mean kinetic energy of the Brownian particle reaches an equilibrium value, viz.,

-mx2=-kT. (1.3.9) 2 2

Thus, Eq. (1.3.8) becomes

m d

~2~dt

f .—T7\ ^lUM= f f , ( 1 . 3 . , 0 , dt 2 dt

Let us write dx2 ldt = u. Equation (1.3.10) then reads mdu £

: — V — U - K T . (1.3.11) 2 dt 2

The solution ofEq. (1.3.11) is u = Ce-°/m+2kT/C, (1.3.12)

where C is a constant of integration. For very large times t compared to m/£, which is of the order of 1CT8 s, only the first term of the right hand side of Eq. (1.3.12) is of practical significance so that

u = 2kT/C- (1.3.13) We remark that neglecting the exponential term in Eq. (1.3.12) implies ignoring the effect of the inertia of the Brownian particle.

If we integrate Eq. (1.3.13) from t = 0, we get

x2-x2=(2kT/C)t.

Let XQ = 0 and write (Ax) instead of x

(Ax)2=(2kT/C)t, (1.3.14)

which is the formula of Einstein [2] as derived by Langevin. (Ax)2 has the following meaning, as has been remarkably well described by F. W. Sears [62]:

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Chapter 1. Historical Background and Introductory Concepts 15

"We observe a Brownian grain at time 0 and at time /. During the time interval (0, r) it has undergone a displacement Ay, whose projection on the x-axis is Ax. The same grain is observed at later times 2t,3t,..., and Ax is determined for each interval. These values are squared and their mean value is calculated; which is (Ax)2 . We emphasise that these displacements which we observe are not in any sense the detailed path of the grain, nor is Ax /1 its velocity. For example, in a time interval t = 1 sec. the particle makes millions of collisions and what we see are simply its initial and final positions, which we connect by a straight line, while the true path is a confused zigzag of linear segments. What we observe is already a greatly simplified "path" an example of which is given in Fig 1.1.2. If one imagines each of the linear elements in this figure to be composed of millions of straight lines one will begin to approximate the actual path. It is impossible to analyse this complicated motion in all of its details and we must therefore be satisfied to observe, in a certain time interval t, the corresponding magnitude of (Ax)1 which is only loosely related to the true path".

What we observe is the result of millions of collisions (cf. our discussion of the central limit theorem and the Wiener process given later in this chapter).

Finally we note that the noise force F(t) in the Langevin equation may be related to the drag coefficient ^ a s follows. We have, writing t' = t + r

J F(t)F(t + z)dz = - J F(t)F(t + T)dT 0 -°o

(1.3.15)

= CkTJ S(t)dT = £kT,

hence i °°

C = — \F(t)F(t + T)dT. (1.3.16) kI o

Equation (1.3.15) follows because F(t)F(t + z) is an even function of z. Equation (1.13.16) is thus a relation between the systematic frictional force and the random force and is a form of the fluctuation dissipation theorem.

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16 The Langevin Equation

1.3.1 Calculation ofAvogadro's number

We have seen that according to Stokes' law the friction coefficient £"of a spherical particle moving in a viscous liquid is

£=6xr/a, where a is the radius of the particle and rj is the viscosity of the liquid. If we combine this formula with the Einstein Eq. (1.3.14), we get

kTt (AxT =

or

where

J (A*r=.

3^770

RTt '3mjaN

N = R/k

(1.3.1.1)

(1.3.1.2)

w^ 1 v^«-¥*

Figure 1.3.1.1 Brownian fluctuation of a very light mirror suspended upon a fine quartz

fibre of torsion constant A. (a) Pressure 4 x 10 mm Hg. (b) Atmospheric pressure, (c) -4

Same system as curve b, except that the pressure is 10 mm Hg. The motion of the system is characterised by the angle <j> through which the mirror has rotated from its

position of equilibrium. One may expect that the system will perform Brownian motion of

such magnitude that A<p2 l2 = kT / 2 . Measurements of A and <j>2 permit one [E. Kappler,

Ann. Phys. 11, 233 (1931)] to obtain a value of the Boltzmann constant k as well as that

of the Avogadro number N. At low pressures, the motion approaches the sinusoidal

natural mode of oscillation of the system and tends to lose its random character. All of the

curves, in spite of the difference in their forms, yield identical values of <j>2 . (Taken from

R. Barnes and S. Silverman [26]; see detailed discussion in [1]).

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Chapter 1. Historical Background and Introductory Concepts 17

is the Avogadro number and R is the gas constant. If the gas constant R is known (8.314 J K mol"1), all the other variables in Eq. (1.3.1.1) except N may be determined in a suitable experiment. Thus one may use Einstein's formula to estimate the Avogadro number N. In 1908, Perrin computed the Avogadro number from observations of the Brownian movement, obtaining ./V= 6.85 x 1023 mol-1 [5]. He also confirmed the relation between t, rj, and T predicted by the Einstein equation. He was awarded the Nobel prize for his work in 1926. A detailed account of his work is given by Fowler [18] who has succinctly summarised the work of the early investigators:

(1) "We can see the manifestations of the molecular motions going on before our eyes. (2) We can check the assumptions of statistical mechanics in a rather detailed way by proving that the characteristics of the Brownian movement agree with the demands of the theory. (3) We obtain a direct, though not very accurate method of measuring molecular magnitudes".

A very detailed summary of the early experiments is given by Chandrasekhar [9] (see also [100,101]).

1.4 Einstein's Method

Einstein derived his expression for the mean-square displacement of a Brownian particle by means of a diffusion (Fokker-Planck) equation, which he constructed, using the following assumptions [2]:

(i) Each individual particle executes a motion, which is independent of the motion of all other particles in the system.

(ii) The motion of a particle at one particular instant is independent of the motion of that particle at any other instant provided the time interval is large enough. (This is essentially another way of stating the assumption that Fx-0 in the Langevin equation).

Einstein's method may be described as follows: we introduce a time interval t, which is large enough so that according to assumption (ii) above, the motion of a particle at time t is independent of its motion at time t ± T, but small compared to the time intervals between observations. Let us suppose that there are /particles in a liquid per unit volume between x and x + dx at time t (we shall consider only the one-dimensional case since there is no loss of generality in doing so). After a time T has elapsed we consider a volume element of the same size at point x. Each particle has a different (positive or negative) value. We

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18 The Langevin Equation

suppose that [5] the probability of a particle entering from a neighbouring element to x is a function of A = x- x and r (a brief description of the relevant probability theory is given in Section 1.6). We denote this probability by 0(A, r). Since the particle must come from some volume element, the density at time ris [5]

oo

f(x,t + T)= J f(x + A,t)0(&,T)dA. (1.4.1) —oo

Now positive and negative displacements are equiprobable. We have an unbiased random walk. Thus the function <p (A, r ) satisfies

<t>A,x) = </>-A,T). (1.4.2)

We now suppose that ris very small so that we can expand the left hand side of Eq. (1.4.1) in powers of T, i.e.,

f(x,t + T) = f(x,t) + T^ + o(T2). (1.4.3)

Furthermore, we develop/(x + A, t) in the right hand side of Eq. (1.4.1) in powers of the small displacement A so obtaining

A x x/ ^ A ^f(x,t) A2 d2f(x,t) f(x + A,t) = f(x,t) + A J \ ;+— J \ ' '+.... (1.4.4)

dx 2! dx The integral equation (1.4.1) therefore becomes

f + T^T = f \ 0dA + - \ A0rfA + - 4 j —<l>dA + ... (1.4.5) dt i dx^ 3 x 2 i 2!

Since <f)A,t) is a probability density function and 0 ( A , T ) =</>(-A,T), we must have

oo

J 0(A,T)dA = l, (1.4.6)

oo

j A^(A,-r)rfA = A = 0, (1.4.7) —oo

and oo

J A2</>(A,r)dA = A2 , (1.4.8) —oo

where the overbar denotes the mean over displacements for each particle.

We suppose that all higher order terms such as A4 on the right hand side

of Eq. (1.4.5) are at least of order T2. Hence, Eq. (1.4.5) simplifies to

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Chapter 1. Historical Background and Introductory Concepts 19

^ = _^lA2 ( 1 4 9 )

dt 2dx2

If we set

D =—A2 (1.4.10) 2T

then Eq. (1.4.9) can be written as

^ = D ^ . (1.4.11) dt dx2

Equation (1.4.11) is a diffusion equation in one dimension derived for small A from the integral equation (1.4.1) with D the translational diffusion coefficient. The solution of Eq. (1.4.11) may be found by assuming that all the Brownian particles are initially placed near the point x = 0 at time t = 0 (sharp initial conditions) [5]. This corresponds to finding the point-source solution of the diffusion equation that is the Green's function or the transition probability. The conditions to be imposed on the solution are

f(x,0) = S(x), (1.4.12) where S(x) is the Dirac delta function. Since / is a probability density function, we also have (see Section 1.6)

oo

j" f(x,t)dx = l. (1.4.13) —oo

The solution of Eq. (1.4.11) subject to these conditions is given by [5]

f(x,t)= , 1 e~x2^Dt, -°°<x<°°. (1.4.14) yjAjtDt

Equation (1.4.14) allows one to calculate the root mean square displacement of the particle in the x direction. We have

\x2=JlDt. (1.4.15) We emphasise (with Wang and Uhlenbeck [12] and Kac [27], see

also [5]) that one may only obtain the diffusion Eq. (1.4.11) when in small times T, the space coordinate A can only change by a small amount. In the general case, the process will always be governed by an integro-differential equation which is of the same type as the Boltzmann equation of the kinetic theory of gases [13] of which Eq. (1.4.1) is an example with 0(A,T) = 0(-A,T) as the stosszahlansatz. In addition, if r is the total displacement of the particle, i.e., [2]

r2 = x2 + y2 + z2 so that r2 = 3x2 . Einstein determined the diffusion coefficient D in terms of

molecular quantities as follows. Let us suppose that the Brownian

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20 The Langevin Equation

particles are in a field of force K(x) (e.g., the gravitational field of the earth); then the Maxwell-Boltzmann distribution for the configuration of the particles must eventually set in. This is

f = f Q e - V K k T ) . (1.4.16) If the force is constant, as would be true when gravity acts on the Brownian particles, the potential energy V is

V=-Kx, (1.4.17) where we suppose that the particles are so few that their mutual interactions may be neglected, (we have an "atmosphere" of Brownian particles). We may imagine the Maxwell-Boltzmann distribution to be set up as a result of the motion of the Brownian particles due to the force together with a diffusion current that seeks to satisfy Eq. (1.4.11). Now, the velocity of a spherical particle that is in equilibrium under the action of the applied force and viscosity is given by Stokes' law [cf. Eq. (1.3.2)]:

v = K l67tr]a). The number of particles crossing unit area in unit time (current density of particles) is then

J = vf = fK/(6xj]a), however, in order to preserve equilibrium, a diffusion current of equal strength flows in the opposite direction [cf. Eqs. (1.2.5) and (1.2.6)]

D ^ = J ^ . (1.4.18) ox 6K7]a

This is the mathematical statement of the fact that at equilibrium the rate of diffusion under the concentration gradient must just balance the directed effect of the field of force. Now we have from Eqs. (1.4.16) and (1.4.17)

! # = * - . (1.4.19) fdx kT

Thus, on comparing Eqs. (1.4.18) and (1.4.19) we obtain

D \ ^ A2 kT kT or D = — = ^ ^ = ^ , (1.4.20) kT 67U]a 2 T 67nja Q

which is Einstein's formula for the translational diffusion coefficient. The arguments used to establish the formulae for the mean-

square displacement may be extended to the rotational Brownian motion about a diameter of a particle in suspension. If 6 is an angular coordinate and if 61 is the mean-square displacement in time rdue to molecular agitation, then [cf. Eq. (1.4.11)]

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Chapter 1. Historical Background and Introductory Concepts 21

_ dt 3 0 2 ' where D = 62l2t). Suppose that an external torque of potential V6) acts on the body. At equilibrium we would then have

/ = fa-™™ and thus

/ dO ~ dOkT' For a particular particle under a torque X = -dV Idd, we have a steady angular velocity

rd0___§y_

dt~ W where the drag coefficient £"for a sphere rotating about a fixed axis in a viscous liquid is [33]

C = 8xa3?].

The number of particles passing by diffusion across a given value 6 of the coordinate in unit time is again

J £de de and thus

D = 02/(2T) = kT/C. Therefore, we have for rotational displacements

02=kTT/(47ra37i). (1.4.21) This is the rotational analogue of Eq. (1.4.20).

Einstein's formula for the displacement cannot be applied for any arbitrarily small time. We give the original argument of Einstein that illustrates this [2]. The mean rate of change of 0 as a result of thermal agitation is

id111 = jlkT /(t£) . (1.4.22)

This becomes infinitely great for indefinitely small intervals of time t. This is impossible, since each suspended particle would move with an infinitely great instantaneous angular velocity. The reason for this difficulty is that we have implicitly assumed in our development that events occurring during the interval t are completely independent of events in the time immediately preceding it. This is not true if t is chosen small enough because inertial effects will come into play.

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22 The Langevin Equation

Einstein establishes a range of validity for his formula using the following argument. Suppose that the instantaneous rate of change of 0at an initial time t0 = 0 is

Let us further suppose that the angular velocity CO (t) at some later time t is not affected by the irregular thermal processes that occur in the time interval (t0, t), but that the change in CO is determined solely by the viscous drag £co. Then we have

I6) = -Cco; (1.4.23) / is defined by the condition that I CO212 must be the rotational kinetic energy corresponding to the angular velocity cot). (The quantity / is termed // by Einstein). It is evidently the moment of inertia of the sphere about a diameter. By integration of Eq. (1.4.23), we get

e(t) = coit) = co0e~°'r. This is negligible only when t » the friction time 1/'£ that is, when the time interval between observations is large compared with / / £ If this condition is not satisfied, inertial effects must be taken into account. Einstein calculates that for bodies of 1 urn diameter and unit density in water at roomtemperature (300 °K) the lower limit of applicability of the formula for 62 is of the order of 10"7 s; this lower limit for the interval between observations increases in proportion to the square of the radius of the body. The same considerations hold for the translational as for the rotational motion of the grain. For practical purposes, the inertial effects will only start to come into prominence when Brownian movement is used to model high-frequency relaxation processes such as dielectric relaxation and Kerr-effect relaxation.

Einstein also showed how his theory may be applied to conduction processes in a conductor. The charge carriers are regarded as charged Brownian particles; thus, if f is replaced by the electrical resistance R of the conductor and the charge q replaces the displacement x of the Brownian particle, then Einstein's formula gives, for the mean square charge that has flowed across a section of the conductor at time t,

q2 = (2kT/R)t. (1.4.24) De Haas-Lorentz, in her book Die Brownsche-Bewegung, published in 1913 [9] (which contains a very thorough account of the history of the phenomenon to that time), lists six electrical systems in which fluctuations are treated by means of the Brownian movement. In the above example the autocorrelation function of the voltage e (t) across the conductor due to thermal fluctuations is

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Chapter 1. Historical Background and Introductory Concepts 23

e(t1)e(t2) = 2RkTSt1-t2). (1.4.25)

Equation (1.4.25), is closely related to the work of J. B. Johnson [Phys. Rev. 26, 71 (1925)] and H. Nyquist [Phys. Rev. 32, 110 (1928)] (see also Barnes and Silverman [26]). In particular, the result for the spectral density, namely,

<5>eco) = 2RkT, (1.4.26)

is known as Nyquist's theorem. 3>e(fi>) has the following meaning. Suppose using a filter we measure e(a))Aco, which is the voltage across the conductor in the angular frequency range (O),0) + Aco), where e(a>) is the Fourier transform of e (t) then

®e(0))Aa)/R = 2kTAa) (1.4.27) is the mean power distributed in the conductor in the angular frequency band Aco.

In order to facilitate the reader the following Sections 1.5 and 1.6 will illustrate the basic concepts of statistical mechanics and probability theory, which will be needed throughout the rest of the book. The reader familiar with these concepts may skip these sections if so desired.

1.5 Necessary Concepts of Statistical Mechanics

The development [3] of statistical mechanics and the introduction of stochastic processes into physics began in the nineteenth century, when physicists were attempting to show that heat in a medium is due to the random motion of the constituent molecules. Excellent historical summaries are given by Jeans [23], in the first and second editions of Chapman and Cowling [24] and by Born [25]. We briefly summarise the work of Maxwell who in 1859 [3,23] considered gases as if they were made up of small rigid spheres distributed randomly but with uniform average density in a vessel [5]. In his model, the molecules are supposed to have random velocities and to collide in a perfectly random fashion with each other and with the walls of the vessel. The process is also supposed to have lasted a long time so that equilibrium conditions will have been attained. The position of the molecule is represented by Cartesian coordinates x, y, z and its velocity by coordinates u, v, w so that

k-u, y = v, z = w. Maxwell asked what is the steady state probability f(u, v,w)du dv dw that the velocity components lie in small ranges between u and u + du, v and v + dv, and w and w + dwl His original argument, although now not regarded as completely satisfactory is of interest both for its simplicity and for its historical importance.

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24 The Langevin Equation

The derivation we give here is essentially that of Maxwell [5], with a few slight changes in nomenclature.

"Let N0 be the whole number of particles. Let u, v, and w be the components of the velocity of each particle in three rectangular directions, and let the number of particles for which u lies between u and u + du be N0f(u) du, where f(u) is a function of u to be determined. The number of particles for which v lies between v and v + dv will be N0f(v) dv, and the number for which w lies between w and w + dw will be N0f(w) dw, where / always stands for the same function. Now the existence of the velocity u does not in any way affect that of the velocities v or w, since these are all at right angles to each other and independent, so that the number of particles whose velocity lies between u and u + du, and also between v and v+dv and also between w and w + dw is

N O / ( K ) / ( V ) / ( W ) du dv dw.

If we suppose the A^ particles to start from the origin at the same instant, then this will be the number in the element of volume dudvdw after unit of time, and the number referred to unit of volume will be

N0f(u)f(v)f(w). But the directions of the coordinates are perfectly arbitrary, and therefore this number must depend on the distance from the origin alone, that is

/ ( " ) / (v) / (w) = ^(M2 + v2 + w2).

Solving this functional equation, we find

f(u) = CeA»\ ^ ( M2

+ v 2+ w 2 ) ^ C 3 / ( " 2 + v W ) . "

This proof, although attractive because of its simplicity, is deemed unsatisfactory because it assumes the three velocity components to be independent. The distribution may, however, be justified from rigorous considerations. The constant C3 is a normalising factor chosen so as to make the total probability unity:

C3 = oo oo oo

A(U2+V2+W2) j j j e dudvdw

—oo —oo —oo

The constant A is-ml (2k T), where m is the mass of a gas molecule. Boltzmann, in several papers beginning in 1868 [23,24],

generalised Maxwell's results by supposing that the molecules are also subjected to a conservative field of force

K(x, y, z) = -grad V(x, y, z), where V(x, y, z) is the potential energy corresponding to this force so that the total energy E of a molecule is

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Chapter 1. Historical Background and Introductory Concepts 25

E(x,y,z,u,v,w) = V(x,y,z) + m\u2 +v2 + vv2 J/2 .

He then found that

fx,y,z,u,v,w) = C'e-E(^-z^w)KkT), (1.5.1)

where C is a constant. Such a gas is said to have the Maxwell-Boltzmann probability distribution.

It follows from the Maxwell-Boltzmann distribution that the mean kinetic energy for each degree of freedom of the gas molecules is the same, that is,

—mu2 =—m(u2J = —m(v2)-—m(w2) = —kT .

This property, which applies also when there are several different kinds of molecule in the gas, is known as the equipartition of energy. If we interpret the mean kinetic energy as temperature, the equipartition theorem implies that gases in contact reach a common temperature, in agreement with experiment.

1.5.1 Ensemble of systems

In Sections 1.5.1 - 1.5.4, we shall follow closely the discussion of Tolman [17]. In the classical mechanics, we consider the behaviour of any given mechanical system of interest as it changes in time from one precisely defined state to another. In statistical mechanics, we have some knowledge of the system but not enough for a complete specification of the precise state. For this purpose, we shall consider the average behaviour of a collection of systems of the same structure as the one of actual interest but distributed over a range of different possible states. We speak of such a collection as an ensemble of systems. In the context of the present work, it is convenient to picture an ensemble as consisting [8] of the system in question and a very large number of copies of it with which it is in thermal equilibrium. By ensemble average, we understand an average over the ensemble at a given instant in time and denote such averages by angular braces ( ).

7.5.2 Phase space

In order to investigate the behaviour of such ensembles of systems, it is convenient to have a quasi-geometric language [17,89] which can be used in specifying the state of each system in the ensemble and in describing condition of the ensemble as a whole. Thus corresponding to

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26 The Langevin Equation

( Qj ^ ^

ZJm Figure 1.5.2.1. Phase plane trajectories x(t), v(t) = x(t)) of a damped harmonic

oscillator with the equation of motion x(t) + yxt) + tt^Jc(?) = 0.

any system of N degrees of freedom, we construct a conceptual Euclidian space CloflN dimensions with 2 N rectangular axes one for each of the coordinates q = (q^,..., qN) and one for each of the momenta p = (pi,... ,PN), where these values would determine the instantaneous state or phase (q, p) of the system. We speak [92] of such a conceptual space as a phase space for the system under consideration. For example, the phase space of the system treated in Section 1.5 is (x, y, z, u, v, w).

1.5.3 Representative point

The instantaneous state of any system in an ensemble can be regarded as specified by the position of a representative point in the phase space, and the condition of the ensemble as a whole can be described by a "cloud" of density p[<l(t),j>(t),t] of such representative points one for each system in the ensemble. The behaviour of the ensemble as time proceeds can then be associated with the "streaming" motion of the representative points as they describe trajectories in the phase space in accordance with the laws of mechanics. The representative points for the different systems are often spoken of as phase points. In Section 1.5 the coordinates of a representative point are (x, y, z, u, v, w). Excellent accounts of phase space and ensembles are given in Refs. 17, 89 and 92.

1.5.4 Ergodic hypothesis

Maxwell and Boltzmann hoped to justify the methods of statistical mechanics by showing that the time average of any quantity pertaining to any single system of interest would actually agree with the ensemble average for that quantity calculated by the methods of statistical mechanics. The postulate leading to this conclusion was called by Boltzmann the ergodic hypothesis and by Maxwell the assumption of

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Chapter 1. Historical Background and Introductory Concepts 27

continuity in phase. It states that the phase point for any isolated system would pass in succession through every point compatible with the energy of the system before finally returning to its original position in phase space. This is (see the discussion in pages 63-70 of [17]) not strictly true in the form postulated by the founders of statistical mechanics, thus in calculating average values one has in general to distinguish between an ensemble average and a time average. However, for a stationary process (i.e., where all time dependent averages are functions only of time differences, in other words the averages are invariant under time transformations) these two methods of averaging will always give the same result and one can therefore use either of them. For example, we have defined [21] the autocorrelation function as the time average of a two time product over an arbitrary range time T, viz.,

! r •Cx(z) = x(t)x(t + z)= lim — \ x(t)x(t + T)dt, (1.5.4.1)

where in all cases z for negative values is to be interpreted as |r|. Ergodicity means that for a stationary process, where

x(t)x(t + z) = x(t)x(t-z), (1.5.4.2) we may also consider ensemble averages in which [21] we repeat the same measurement always and calculate (ensemble) averages giving the same result as using Eq. (1.5.4.1), i.e.,

(x(t)x(t + z)) = x(t)x(t + z). (1.5.4.3)

1.5.5 Calculation of averages

It is convenient to summarise how averages of dynamical quantities may be calculated. Following Gibbs [5,17,89], we will describe the state of a given system in terms of coordinates q = (ql,q2,...,qN) and conjugate momenta P = (PI,P2,---,PN) (rather than velocities), the number of each being equal to the number of degrees of freedom of the system. Then, in the notation of Gibbs [5] the Maxwell-Boltzmann law (i.e., the probability of finding the state of the system in the range dQ. = dq\ ... dq^dpx ... dp^) is

pdQ. = Ce-mp'"PN'Ch-qN)dqldpv..dqNdpN, where E is the Hamiltonian or total energy of the system and J3 = (kT)~l. We suppose throughout that the system is in thermodynamic equilibrium at the temperature T. As before, the coefficient C is chosen so that

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28 The Langevin Equation

where the integration extends over all possible values of the variables. Since there is a continual interchange of energy between the system and its surroundings, the total energy content E at a given temperature does not have a definite value. In calculating the resulting averages for a system, it is convenient to introduce the partition function Z, defined by

Z = jae~PEdqidpi-dqNdpN =la e'^da •

The average value of any function of the p's and q's, A (p, q), is then given by

(A(P,q)) = i r ^ - / ^ a .

We shall use overbars and angle brackets to denote averages; overbars will denote time averages, and angle brackets, ensemble averages.

It is a fundamental tenet of statistical mechanics (cf. Section 1.5.4) that for a stationary process the time-average behaviour of a system is equal to the ensemble-average behaviour. In particular, the average total energy is given by

(E) = Z-l\ Ee-PEd£l = -—\nZ.

The average value of any coordinate is

Let us suppose that the coordinates are so chosen that the kinetic energy is expressed as a sum of squares of the momenta with constant

coefficients, that is, ^ . p] /(2m,), and the Hamiltonian is thus

i 2mi Then in calculating Z, or the average of any function of the coordinates only, the integrals that occur are the products of integrals over the position and momentum coordinates, respectively. For example, in evaluating (E)

!_/ 2\ _L J pfe-fipf'^dp.

2m; v ' I 2m: f 0-PPV^,

Now

e-PPi <*nidpi

OO OO -j

f e-apldp = 47tla~, f ple-apldp = —4itla~. J J 9/7 2a

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Chapter 1. Historical Background and Introductory Concepts 29

Hence, the mean value of pf I2mi is given by

Thus, we have the equipartition theorem described above: the mean value of the kinetic energy in any coordinate (one degree of freedom) is k T12. The same is true for potential energy terms, which are simply of the form

Ktqt 12, assuming that qt does not enter the Hamiltonian in any other form. If the coordinate qt enters the Hamiltonian simply as (a harmonic oscillator)

however, the particle cannot come into thermal equilibrium with the remainder of the system; thus there must be some kind of interaction between it and the rest of the system. This is generally provided by some mechanism such as the collision of gas molecules with the particle, which constitutes the oscillator. In general, if the interaction energy depends solely on the coordinates, and not on the momenta, then the equipartition theorem holds.

According to the theorem, associated with each degree of freedom is a mean kinetic energy kT/2. Hence one may then calculate the fluctuations due to this thermal energy in perfect generality from the laws of statistical mechanics. This is possible because the average energy of these random motions will be exactly the same for all systems at the same temperature (so long as each is in thermodynamic equilibrium with its surroundings), and will be entirely independent of the nature of the systems and the mechanisms that produce them. The energy distribution will be a function of the particular system in question. Barnes and Silverman [26] show how the equipartition theorem may be used to set a natural limit to the ultimate sensitivity of all measuring devices.

1.5.6 Liouville equation

In Section 1.2 above, we have very briefly alluded to the Klein-Kramers equation for the evolution of the probability density function W in the phase space (x, v), which unlike the (approximate) Smoluchowski equation, Eq. (1.2.7), used by Einstein includes exactly the effect of the inertia of the Brownian particles. We remark that the Smoluchowski equation is an approximate equation for the evolution of the density function in configuration space x, which assumes that equilibrium of the velocities has been attained. In the Langevin picture, this assumption is

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30 The Langevin Equation

contained in the steps in going from the exact Eq. (1.3.8) to the approximate Eq. (1.3.13). The Smoluchowski equation consequently describes the evolution of the density function in configuration space in the very high friction or small inertial effects limit. Returning again to the Klein-Kramers equation and the question of a plausible derivation of that equation along the lines that presented for the Smoluchowski equation in Eqs. (1.2.1-1.2.7) above, it is first necessary to refer to a purely dynamical theorem due to Liouville [17]. This theorem provides an ideal basis for the discussion of the effect of a heat bath on a dynamical system. For convenience, we will use Hamilton's canonical variables, namely, position q and momentum p. In this section, we shall also follow [17] and use the symbol p for the density of representative points in the phase space (q, p).

The dynamical evolution of a conservative system is, in general, described by the Liouville equation (see Tolman [17]), which, for a system of N particles, which have 3N degrees of freedom, with Hamiltonian

3W 2

H = Z ^ + V(qi,q2,...,q3N) (1.5.6.1) «=i 2mi

is

^ = 0, (1.5.6.2) Dt

where Dl Dt is the total or hydrodynamic derivative operator defined by - ^ - s A + u-grad (1.5.6.3) Dt dt

and u = q(t),p(t) = (ql,q2,---,q3N,pl,---,p3N) is the 67V-dimensional vector, which is the flow vector in phase space. A very detailed account of this equation is given by Tolman [17]. The hydrodynamic derivative means the derivative evaluated at a moving phase point q(f)>P(0 • The statement Dp/Dt = 0 means that there is no tendency for phase points to "crowd" into any particular region of phase space, i.e., phase space behaves like an incompressible fluid whose representative point is q(0»P(0J. Put in yet another way, the density in the neighbourhood of any selected moving representative point is constant along the trajectory of that point. This principle is known as the principle of conservation of density in phase, in other words, phase points stream [89]. In mathematical terms, the principle is

/>[qOb),P(fo).'o] = P[<l(0,P(0,f]- (1.5.6.4)

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Chapter 1. Historical Background and Introductory Concepts 31

The second important property arising from the incompressible nature of phase space is the principle of conservation of extension ("volume") in phase space

dq(t)dp(t) = dq(t0)dp(t0), (1.5.6.5) so that, even though the shape of the region Av in phase space may alter with the course of time, its volume does not [89].

We give here a brief outline of the derivation of the Liouville equation for one degree of freedom. The position coordinate now being denoted by x, and the momentum by p. These constitute a set of canonical variables satisfying Hamilton's equations [92]:

dH . dH P = -—, x = —-. (1.5.6.6)

ox op In the absence of the creation or annihilation of particles, the rate of decrease of particles in a volume Av of phase space must be balanced by the flow of particles out of the volume. Hence, if AS is the surface bounding the volume Av, we have

-\ — jj p(x, p,t)dpdx = - J pu-dS = -jjdiv(pu)dpdx, (1.5.6.7)

Av AS Av

where u is the velocity of a particle in phase space at the point (x,p). Here, the divergence theorem is used. Now since the volume element Av is arbitrary, Eq. (1.5.6.7) reduces to

^ + div(/?u) = 0, (1.5.6.8) at

which is the continuity equation of fluid mechanics. Now

div(pu) = —(px)-ox

and, using Eqs. (1.5.6.6),

.. , , 3 , ., d , . , .dp .dp f^ ^ div(/>u) = — (px) +—(pp) = x^- + p^-+p

ox dp ox op dx dp — + —

ydx dp j

(1.5.6.9)

dx dp d (dH)+d_(dH_

dp\ dx = 0 (1.5.6.10)

dx dp dx y dp from the equality of the mixed second order partial derivatives. Furthermore,

x^-+p^- = u-gmdp (1.5.6.11) ox op

and the continuity Eq. (1.5.6.8) yields:

^ + ugrad/7 = = 0. (1.5.6.12) dt Dt

Equation (1.5.6.12) is known as the Liouville equation, which for N particles moving in three dimensions (so that we have a 6N dimensional

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32 The Langevin Equation

phase space or 127V dimensional if the rotational degrees of freedom are added) is:

dp

dt

3JV

i=l

dH dp dH dp = 0.

i J It is often written as

Dp dp

Dt dt where p, H is the Poisson bracket

+ p,H = 0,

3N

P,H = Z i=i

dH dp dH dp

(1.5.6.13)

(1.5.6.14)

(1.5.6.15) dptdqt dqtdpt

The derivation for one degree of freedom, first given in 1838, is sufficient because, one must recall that Liouville' s theorem is a purely dynamical theorem, which is entirely equivalent to Hamilton's equations. The theorem will also apply for non additive Hamiltonians provided canonical variables can be defined as is so in micromagnetics as we shall see later.

1.5.7 Reduction of the Liouville equation

The Liouville equation is an equation with a number of variables of order 1023 and so is not tractable. In order to discuss the average dynamical behaviour of a particle or system embedded in a heat bath, it is necessary [9] to modify the Liouville equation by both reducing it and generalising it, reducing it by limiting the degrees of freedom to a small but representative set with a well defined potential and generalising by the addition of terms on the right hand side of Eq. (1.5.6.15) to account for the mean interaction between this set and the remaining degrees of freedom [the background or heat bath]. The first and best known such reduction and generalisation of the Liouville equation is due to Boltzmann (this is an integro-differential equation for the one particle distribution function [24]). Boltzmann, in his attempt to demonstrate that the effect of molecular collisions, whatever the initial positions and velocities of the molecules of the gas, would be to bring about a Maxwell-Boltzmann distribution of positions and velocities, formulated his famous equation [23,24]. This equation describes the time evolution of the density of molecules in phase space provided that only encounters [24] between two molecules (i.e., two-body interactions) are ever of any importance.

The Boltzmann equation, which is a closed equation [23,24] for the single particle distribution function, is now the fundamental equation that allows one to describe the bath itself in a microscopic way. The

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Chapter 1. Historical Background and Introductory Concepts 33

particular law of binary collisions (stosszahlansatz) describes the interactions between the molecules of the bath. The binary collision assumption amounts to stating that encounters with other molecules occupy only a very small part of the lifetime of a molecule. To put this statement in another way [24], it states that encounters in which more than two molecules take part are neglected in both number and their effect in comparison with binary encounters. Furthermore, in considering binary encounters between molecules having velocities within assigned ranges, it is assumed that both sets of molecules are distributed at random and without any correlation between velocity and position in the neighbourhood of the point where the collision takes place. This is the "molecular chaos" assumption of Boltzmann [23]. The theory of the Brownian movement instigated by Einstein, Smoluchowski and Langevin and by Bachelier [22] in 1900 for financial systems is essentially a particular case of this reduction whereby, in a collision, the positions are unchanged and their velocities are altered by such small amounts that they can be treated as infinitesimal, so that the Boltzmann equation reduces to a linear partial differential equation in phase space which, for the particular case of mechanical particles, which always have a separable and additive Hamiltonian, is now known as the Klein-Kramers equation. The Klein-Kramers equation is intimately connected to the Langevin equation, which, for the sake of completeness, we will again refer to at this juncture.

1.5.8 Langevin equation for a system with one degree of freedom

We have seen that Langevin treated the Brownian motion of a free particle embedded in a heat bath by simply writing down the Newtonian equation of motion of the particle accounting for the interaction of the particle with the bath by adding to the Newtonian equation a systematic retarding force proportional to the velocity of the particle superimposed on which is a rapidly fluctuating force which we now call white noise. We shall slightly generalise Langevin's treatment by supposing that the particle moves in a potential V (x), thus the Langevin equations are

m = £@-, P«) = -4-V[x(t)]-&Q- + F(t). (1.5.8.1) m dx m

In Kramers' paper of 1940, the force -£plm + F(t) [X(t) in his notation] is aptly termed "irregular force due to the medium". The white noise force F(t) has the following properties:

F(0 = 0, F(tl)F(t2) = 2kTCS(t1-t2). (1.5.8.2) The overbar means the statistical average over the realisations of F, i.e., the values it actually takes on. Since F (t) is a random variable, then pit)

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34 The Langevin Equation

and x(t) are also random variables. All averages over p and x are performed over a very small time interval r with p and x taking sharp initial values at the starting time t. The statistics of F(t), written down above, are, however, insufficient to describe the problem fully as neither the Klein-Kramers equation nor the set of statistical moments of the system generated by directly averaging the Langevin equation can be written down without supposing that F(t) is also Gaussian, i.e., F(t) must obey Isserlis's theorem (see Section 1.3).

7.5.9 Effect of a heat bath. Intuitive derivation of the Klein-Kramers equation

The intuitive derivation of the Kramers equation, which we shall now give follows that of Einstein [2] who included thermal agitation in the continuity equation for a particle subjected to a force, K = -dV Idx, by simply adding a diffusion term to the continuity equation for the number density or concentration of particles in configuration space. This enabled him to write down the Smoluchowski equation for the evolution of the number density in configuration space. We shall apply the same procedure to the Liouville equation, supposing that that equation is reduced to the Liouville equation for a single particle with the behaviour of the other particles (or bath) being represented by the drift and diffusion terms we shall add so that the hydrodynamical derivative DpiDt is no longer zero and so the phase points diffuse.

We shall first consider the behaviour of the system governed by (1.5.8.1) and (1.5.8.2) without the white noise term or fluctuating term (but including the damping term). Equation (1.5.6.9) becomes

div(/?u) = x—— ox

.dp (C dV)dp \m ax ¥+" (1.5.9.1)

dx m dp\dx ,

Since x = p/m and x are independent variables they play the role of generalised coordinates, and since dV Idx is independent of p, Eq. (1.5.9.1) reduces to

Pdp (cdv\dP <r div(/?u) = -

m ox P + — -2-p. (1.5.9.2)

m comes

p = 0. (1.5.9.3)

Km dx J dp m Hence, the equation of continuity, Eq. (1.5.6.8), becomes

dt m dx \m dx J dp m To take account of the white noise, we must now, following Einstein's method for the Smoluchowski equation, add on the diffusion term Dd2p/dp2, where the diffusion coefficient D is independent of p, and so Eq. (1.5.9.3) becomes

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Chapter 1. Historical Background and Introductory Concepts 35

dt m dx dx dp dpym dp, (1.5.9.4)

Now we insist, following Einstein, that the equilibrium solution (the Maxwell-Boltzmann distribution) be a solution of Eq. (1.5.9.4). Thus

D = £kT (1.5.9.5) and so Eq. (1.5.9.4) becomes

dp pdp dVdp £ d ( Irrdp^

dt m dx dx dp mdp\ dp (1.5.9.6)

Equation (1.5.9.6) is the Klein-Kramers equation for the evolution of the density p in phase space. The effect of having a nonzero right hand side of the Liouville equation is to cause a disturbance of the streaming motion of the representative points so that they diffuse onto other energy trajectories. The energy of the Brownian particle is no longer conserved as energy is interchanged between that particle and the bath. Equation (1.5.9.6) is clearly the same as Eq. (1.2.8) specialised to the phase space (x, v) if the replacement p —> W, p —> mv, is made.

In the analysis presented in this section and the sections immediately preceding it, we have adhered as far as possible to the original notation used by Kramers [19,67] in his discussion of the derivation of the Klein-Kramers equation. We shall now briefly outline how the Smoluchowski equation may heuristically be derived from the Klein-Kramers equation. In order to accomplish this it is necessary to refer to the conditions under which a Maxwellian distribution of velocities may be assumed to prevail in the Klein-Kramers equation.

1.5.10 Conditions under which a Maxwellian distribution in the velocities may be deemed to be attained

Having obtained the Klein-Kramers equation for the time evolution of the distribution function in the phase space (x,p), Kramers [19,67] proceeded to examine the conditions under which equilibrium in the velocities may be assumed to have been attained, the displacement having not yet attained its equilibrium value. The importance of such an investigation is that it allows one for sufficiently high values of the friction parameter, to write an approximate partial differential equation for the time evolution of the distribution function in configuration space only. This approximate equation is known as the Smoluchowski equation [cf. Eq. (1.2.7)]. In order to explain the reasoning of Kramers on this

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36 The Langevin Equation

subject it will be useful to recall Einstein's 1905 result for the mean square displacement of a Brownian particle namely (t = \t\ in all cases)

( A x ) 2 ) ~ f . (1.5.10.1)

This equation, as we saw in Section 1.4, was derived by constructing the partial differential equation for the time evolution in configuration space only and was later rederived by Langevin, as we saw in Section 1.3 by considering times well in excess of the frictional relaxation time m I £ which allowed him to postulate an approximate Maxwellian distribution for the velocities. Equation (1.5.10.1) has the flaw that it is not root mean square differentiable at very small times. In 1930, Uhlenbeck and Ornstein [15] showed using the Langevin equation without the assumption of Eq. (1.3.9) above namely that equilibrium of the velocities has been attained, that the exact solution for the mean square displacement of a free Brownian particle is (a detailed derivation of the result is given in Chapter 3)

((Ax)2) = ^(/3t-l + e-P!), (1.5.10.2) \ / mp

which is differentiable at short times. Here j3 = £/m is the friction coefficient per unit mass and not the thermal energy. Einstein's result Eq. (1.5.10.1) is regained if t » /?" ' ; The mean square displacement, which we shall write as x2 for brevity, is thus governed by the two characteristic times, viz.,

2kT' *W=^X (1.5.10.3)

(the diffusion time) and Tf=/3~l (1.5.10.4)

(the frictional time). The ratio of these times is

(1.5.10.5) rf _ 2kT

Tdiff x2m/32

Now diffusion effects, where a Maxwellian distribution of velocities approximately holds, will predominate over inertial effects if the left-hand side of this equation is much less than 1, which by transposition means

x«j3~lylkT/m (1.5.10.6)

so that the quantity p-^kTIm (1.5.10.7)

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Chapter 1. Historical Background and Introductory Concepts 37

defined by Kramers is a characteristic diffusion length which crucially determines whether inertial or diffusion effects will predominate.

As far as Brownian motion under the influence of a potential V(x) is concerned, Kramers applies the above reasoning to this problem by supposing that the force K = -dV/dx does not vary greatly over distances of the order of the diffusion length Eq. (1.5.10.7). So one would expect that starting from an arbitrary initial distribution p(x,p,0) a Maxwellian distribution of p would be reached after time intervals At» J3~l that allows him to postulate that

p(x,p,t)~fx,t)e-p2/(2mkT\ (1.5.10.8)

The reasoning of this section is of crucial importance in the study of dielectric relaxation where the omission of inertial effects in the Debye theory of dielectric relaxation leads to the phenomenon of "black water" or infinite integral absorption at high frequencies. The above considerations may now be used in the heuristic derivation of the Smoluchowski equation from the Klein-Kramers equation. The range of the friction or dissipative parameter in which the Smoluchowski equation provides an accurate description of the configuration space distribution function is termed by Kramers [19, 67] the very high damping regime. This terminology distinguishes the region of validity of the Smoluchowski equation from the intermediate to high damping (IHD) and low damping (LD) regimes where inertial effects are important so that the complete phase space description provided by the Klein-Kramers equation must be used. The distinctions between the various damping regimes are of vital importance in the application of the theory of the Brownian movement to reaction rate theory [19, 67] as we shall describe in detail in Section 1.13.

1.5.11. Very high damping regime

This is a limiting case of the IHD regime, where it is supposed that the damping is so large that equilibrium of the velocity distribution has been sensibly attained. In this situation, it is possible to obtain a partial differential equation for the evolution of the distribution function in configuration space only. This approximate diffusion equation is called the Smoluchowski equation. First, we recall what we mean by large viscosity. By large viscosity we mean that the effect of the Brownian forces on the velocity of the particle is much larger than that of the external force K(x). The Smoluchowski equation may be derived, according to Kramers, by assuming that K does not vary sensibly over a distance of the order of the diffusion length, Eq. (1.5.10.7). We expect

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38 The Langevin Equation

that, irrespective of the initial pdistribution, the distribution (with m=\ following Kramers [19, 67])

p(x,p,t)~f(x,t)e-p2,2kT) (1.5.11.1) (i.e., a Maxwell velocity distribution) will hold after a very short time lapse (~/?~ ) (cf. the arguments used by Langevin and Einstein to neglect inertial effects.) The high barrier then ensures that a slow diffusion of particles over the barrier will take place, which may be expected to satisfy the Smoluchowski equation for the density/(x, t) in configuration space:

dt~ dx\pf p dx (1.5.11.2)

Kramers then examines the approximate validity of Eq. (1.5.11.1). He remarks that as long as no perfect temperature equilibrium is attained, Eq. (1.5.11.1) will hold only approximately, even when the external force is equal to 0 [cf. the approximate Eq. (1.5.10.1)]. He claims that in that case, while the Maxwell velocity distribution will hold exactly for each particle, it will not hold exactly at each value of x, since otherwise there would be no diffusion current. Thus the behaviour is unlike that described by a single space variable Fokker-Planck equation such as that for the magnetisation relaxation in axially symmetric potentials of the magnetocrystalline anisotropy (to be treated later), which is an exact equation.

In order to derive Eq. (1.5.11.2) from the Klein-Kramers Eq. (1.5.9.6) in heuristic fashion we first rewrite Eq. (1.5.9.6) in the form

dp J d i a V ,dp K kTdp) d(K kTdp — = B pp + kT— p + — — p — dt ydp fidxjy dp PH P dx) dxpy p dx

(1.5.11.3) This can be checked most easily by directly rewriting Eq. (1.5.11.3) in the form of Eq. (1.5.9.6). We now integrate both sides of this equation (with respect to the momentum) along a straight line in phase space

x + pl P = x0 (constant). The integration extends over all possible momentum values fromp = -°° to p = + oo. Note

3 _ dx0 d _ d , d _ dx0 d _ 1 d dx dx dx0 dx0 dp dp dx0 P dx0

whence dldp- P~ldldx is the zero operator along this line. If we denote the integral of palong this line by f(x0,t), we obtain:

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Chapter I. Historical Background and Introductory Concepts 39

~J^**\pp~p**. In Eq. (1.5.11.4), note that the line integral is strictly

dt dp

f3P pdx ds,

(1.5.11.4)

(1.5.11.5) x+plp=xa

where ds is the element of arc length along the line in question. However, since ds =dx +dp and dpi'dx = -J3 or dp = -J3dx and also, since we let jB —> oo, i.e., we take the high friction limit, we can approximate ds ~ dp. Thus; the position coordinate has the value x ~ x0 along the line x + p IP = x0. We also use the fact that

I £*=•* J pds = -f(x0,t). (1.5.11.6) J at x+p//B=XQ x+p/fl=xQ

The right hand side of Eq. (1.5.11.4) can be simplified to yield

+-

- f x+p//3=x0

kT d d

d_ dx

dp ~ a K(x0)

P dxo dxo X+Pip=x0

Thus, we have

J pdp»-dxn

dx0

K(x0) f(x0,t)

J pds x+p//3=x0

kTdf(x0,t)

dt f(x0,t) = -

dxn

K(x0) f(x0,t)-

fi J— p

kTdf(x0,t)^ dxn

dxn

(1.5.11.7) . P ' " P - 0

which is a diffusion equation in configuration space and is the Smoluchowski equation.

The approximate validity of Eq. (1.5.11.7) is a consequence of the approximate validity of Eq. (1.5.11.1) if it is also assumed that in the region of values of p that dominate the integral (that is, \p\<-Jkf), the variation of x (which is of the order of Jkf I /?) is small compared to distances over which the force K and the density in configuration space/ undergo marked variations. These are, however, the conditions which a priori have to be imposed in order to ensure the applicability of Eq. (1.5.11.2). We remark that, since we integrate along the line pip = 3 - x and since both x and x0 are of the order of the diffusion length -JkT I p, we expect p to be of the order of JkT , i.e., the thermal value.

We have given above a heuristic derivation of the approximate equation for the distribution function in configuration space known as the Smoluchowski equation from the Klein-Kramers equation. The first

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40 The Langevin Equation

rigorous treatment of the problem was given by Brinkman [H. C. Brinkman, Physica 22, 29 (1956)]. He showed that the solution of the Klein-Kramers equation could be obtained by expanding the distribution function in appropriate sets of orthogonal functions, and in the small inertial limit, the results predicted by the Smoluchowski equation could be obtained. Further discussions of the problem, which are based on the Chapman-Enskog method [24] in the kinetic theory of gases are well summarised by van Kampen [10]. The overriding advantage of considering the very high damping limit discussed above is that it is possible to derive a diffusion equation in configuration space only. Moreover, this equation is in a single coordinate. Yet another example where this may be done (which is of overwhelming importance is the application of the theory of the Brownian motion to reaction rate theory, see Section 1.13, below) is the very low damping limit. It is impossible for an arbitrary damping, however, to derive a diffusion equation in a single variable from the Klein-Kramers equation.

1.5.12 Low damping regime

We again refer to the original derivation of the Klein-Kramers equation given by Kramers in 1940 [19] where a Brownian particle moves in the potential well generated by the force K(x) [see Fig. (1.13.1)]. In this instance Kramers restricts the discussion to the situation where the particle would perform an oscillatory motion in the well if it were not for the presence of Brownian forces. Small viscosity means that the Brownian forces cause only a tiny perturbation in the undamped energy during one oscillation in the well meaning that the Brownian forces will cause gradual changes in the distribution of the ensemble over the different energy values.

We now write the original Klein-Kramers equation in the canonical variables (x, p) as a diffusion equation in the energy (£) and phase (w). We can do this, since for small damping the energy is a slowly varying quantity and the phase a fast-varying quantity. Thus we will be able to average the density over the fast phase variable and get a diffusion equation in the slow (almost conserved) energy variable. We define the time average along a trajectory corresponding to the saddle point (now a one dimensional maximum) of the potential energy

- 1 r

p(E,t) = — j p(E,w,t)dw, (1.5.12.1) T o

where 7" is the time required to execute one cycle of the almost periodic motion at the saddle point energy E. (We have assumed that the average is taken along the energy trajectory so that

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Chapter 1. Historical Background and Introductory Concepts 41

dt = dw (1.5.12.2) ai = aw \I.D.IA.Z.

along a trajectory). If we define the action, /, at energy E by the equation 1(E) = (p pdq (1.5.12.3)

E=conxt

and allow the energy to vary by an amount dE over a thin ring of thickness dl, we can account for the slow diffusion of energy. (The assumption that the damping has negligible effect over one period means that the energy loss per cycle of the motion at the saddle point energy «kT, see Section 1.13 below.) We assume that the motion of the particles in the well would always be librational, i.e., have closed trajectories, in the absence of the Brownian forces (that is if it were not for the slow diffusion of energy). This condition indicates that the trajectories of the motion are almost closed, (i.e., almost periodic) except for a leisurely spiralling of the particles towards the minimum of the energy, due to the energy loss d E per cycle. The simplest example of a librational motion arises from the energy function for the harmonic oscillator:

2 2

E = -^ + 2m 2

Thus, the trajectories in phase space are (closed) ellipses. Now the Klein-Kramers Eq. (1.5.9.6) is

dp__dV_dp _p_dp_ R d f

dt dx dp m dx dp pp + mkT—

dp

(1.5.12.4)

(1.5.12.5)

If there were no dissipation of energy, we would have by Liouville's theorem,

dp _dV dp p dp

dt dx dp m dx So the remaining (diffusive) part of Eq. (1.5.12.5), viz.,

(1.5.12.6)

pMpP + mkTdjf °P\ op J

(1.5.12.7)

assuming that the dissipation of energy is very slow, describes the dissipation of energy. We now transform Eq. (1.5.12.5) into an equation in the energy and phase variables by using the Hamiltonian

2

E = l— + V(x). 2m

We have

(1.5.12.8)

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42 The Langevin Equation

which, on taking the positive sign and integrating the resulting differential equation between points xy = x(0) and x, yields:

X j /

f , . . =t + w, (1.5.12.10)

where the constant of integration w defines the phase. Now Eq. (1.5.12.2) implies that

vv = l (1.5.12.11) and since the variation in energy is very slow we have

£ = 0, (1.5.12.12) i.e., almost a conservative system. On noting that

dV P = — dx

we have by the chain rule dp _dp_dE dpdw_ dp__dpdE dp dw

dx dE dx dw dx dp dE dp dw dp where

dE__dV_ ^ _ j _ _ ^ 3 £ _ p dx dx dx x p dp m '

[using Eq. (1.5.12. 10)]. Thus d__dV_j)_ m_d_ dx dx dE p dw

and

JL=zA dp m dE

so that Eq. (1.5.12. 5) becomes dp dp Pp d ( ,dp dt dw m dEVF F dE

dp

(1.5.12.13)

(1.5.12.14)

(1.5.12.15)

Now, defining the average over one cycle of the almost periodic motion at the energy E as

- 1 r

p = — J p(E,w,t)dw, (1.5.12.16)

we have from Eq. (1.5.12.15)

dp = dp | fip d

dt dw m dE pp + pkT^

V dE (1.5.12.17)

Now

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Chapter 1. Historical Background and Introductory Concepts 43

r dp=TI f ^ . ^ 4 ^ J ^ = f • (L5-12-18) dt dtr dt

Also T •

?P- = U?£dw = Udp = 0 (1.5.12.19) dw T'[dw T' H

since the integral is taken over one complete cycle of the motion, p(T) = p(0). Note that Eq. (1.5.12.19) holds only approximately since

dE dw However, Eis slowly varying, so that in one cycle dE~0 and so

dp « -^-dw. aw

Eq. (1.5.12.17) then becomes

dp_j5(Jldp _ , dp dp dp 2lrrd2P p -r— + pp^r~+ pkT~——+ p kT—hr F dE FFdE F dEdE F dE2 dt m

which, using Eq. (1.5.12.14), simplifies to

f-4'2 dt

m dE dE m dE2

On using also the fact that 7" dp i Kdp , I d K , d _

-r- = —\ rr-dw = —z— pdw = -—p dE T' J0 dE T'dE dE'

and assuming that

p2p = p2p, we find that Eq. (1.5.12.21) becomes

dt

— \

m dE dE m dE2

Now, Eq. (1.5.12.16) yields

Pl=—,\pldw.

(1.5.12.20)

(1.5.12.21)

(1.5.12.22)

(1.5.12.23)

(1.5.12.24)

(1.5.12.25)

Furthermore, since we are taking the average along a trajectory dw - dt and the periodic time in the well is 27rlcoE), where co = a)(E) is the angular frequency of oscillation, so that

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44 The Langevin Equation

.. ijtlco , , inla : — pat-— pmxdt = (b pdx = / , ^ Jo 2;r J

0K 2 * ^ 2;r

(1.5.12.26) com r , <ym

I p at =— i rnnxat- (b ndx = l7t \ 2n

where / is the action over one period. Thus Eq. (1.5.12.24) becomes:

<- . 3 ^ dt

- ,„dp 1 CO d

dE 2n dE p + kT

dE

P 1 + Q)l__d_

27T dE F BE

(1.5.12.27)

Since

0 dE CO = 2K—,

dl Eq. (1.5.12.27) becomes (writingpfor p):

dt Pdl Ip + kT

2nldp CO dl

This corresponds to a diffusion along the /- or diffusion term proper is

H an dE) r di ^U^krP'^

codl

(1.5.12.28)

(1.5.12.29)

coordinate; the

(1.5.12.30)

and corresponds to a diffusion coefficient D given by D = 27Tj3kTI/co. (1.5.12.31)

The above paragraph constitutes an extended version of Kramers' original method of arriving at the energy diffusion equation. Further discussion of this will be found in van Kampen [10] and in Coffey et al. [67]. The energy controlled diffusion equation (1.5.12.29) is of vital importance in reaction rate theory in the context of the contribution of Kramers to that theory as we briefly describe in Section 1.13.

1.6 Probability Theory

This section reviews the fundamental concepts of probability theory which is the real basis of the theoretical treatment of the Brownian motion. Again readers who are familiar with these concepts may conveniently skip this section. By way of introduction, we cannot improve on the elegant description of the emergence of probability theory as a fundamental tool in physics, given by Born [25] in his Natural Philosophy of Cause and Chance.

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Chapter 1. Historical Background and Introductory Concepts 45

"The new turn in physics was the introduction of atomistics and statistics. To follow up the history of atomistics into the remote past is not in the plan of this lecture. We can take it for granted that since the days of Democritus the hypothesis of matter being composed of ultimate and indivisible particles was familiar to every educated man. It was reviewed when the time was ripe. Lord Kelvin quotes frequently a Father Boscovich as one of the first to use atomistic considerations to solve physical problems; he lived in the eighteenth century, and there may have been others, of whom I know nothing, thinking on the same lines. The first systematic use of atomistics was made in chemistry, where it allowed the reduction of innumerable substances to a relatively small stock of elements. Physics followed considerably later because atomistics as such was of no great use without another fundamental idea, namely that the observable properties of matter are not intrinsic properties of its smallest parts but averages over distributions governed by the laws of chance... The theory of probability itself, which expresses these laws, is much older; it sprang not from the needs of natural science, but from gambling and other, more or less disreputable, human activities".

The most important concept of that theory that we shall need is the random variable. This is a quantity that may take on any of the values of a specified set with a specified relative frequency or probability. The random variable could be a vector molecular property such as centre of mass velocity, angular momentum, or dipole orientation; or a tensor, such as polarisability. It is regarded as defined not only by a set of permissible values such as an ordinary mathematical variable has, but by an associated probability function expressing the relative frequency of occurrence of these values in the situation under discussion.

1.6.1 Random variables and probability distributions

We may formalise the foregoing concepts as follows. Let Q. be a set called the sample space of an outcome of a random experiment or occurrence [7]. Each subset A c f l is called an event. We wish to formalise the idea of the chance of obtaining an outcome lying in a specified subset A c Q. into a definition of probability. We shall need a function that assigns a unique number or measure (called probability) to each such A.

Definition 1: We call P (A) a probability function if to each event AczQ.,P(A) assigns a number P (A) to each A such that

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46 The Langevin Equation

0<P(A)<1, /»(«)= 1,

and if Au A2,... are events with Ai nAj=0 (i * j), where 0 denotes

the empty set, then

* IK =Z^K)-,n=i ; n=\

For two events A and B, the conditional probability is given by [7]

P ( A I B ) = f i , (1.6.u) P(B)

where P(AnB) is the probability function of both A and B occurring and P(A\B) means the probability function of event A occurring if event B occurs. If the two events A and B are statistically independent then

P (A I B)=P (A) so that

P(AnB) = P(A)P(B). (1.6.1.2) Consider the experiment of obtaining an event £e Q.. The experiment can be described by a set of functions £(C)- The £ ( 0 are called random variables because the independent variable £ cannot be predicted. It only takes on values (its realisations) according to the underlying probability law. More formally a random variable may be defined as follows [7].

Definition 2: A random variable £is a real-valued function with domain Q., i.e., for each f e i l , £ ( 0 e 91. For an n-dimensional random variable \ = (^u...,^n), we have ^(f) e 9T. 9? denotes the real line.

Here we give a brief outline of some properties of a random variable £,, which are relevant in the present context. The notation of Gnedenko [28] for random variables and realisations is adhered to as far as possible.

Definition 3: If £ is a random variable, its distribution function is defined as

Ff(x) = P[£<x],

where x is a real number. The x's are the realisations (values which it actually takes on) of the random variable £

If it is possible to write the distribution function as [8] X

Fs(x)= J ff(x)dx, (1.6.1.3)

meaning that the probability that an observed value of £ lies in the interval (x, x + dx) is fgdx, then <f is said to be a continuous random

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Chapter 1. Historical Background and Introductory Concepts 47

variable and /£ (x) is its probability density function. A discrete random variable £will have a probability mass function P^(A) such that

xeA

The mean value of a random variable £ denoted by (§ is defined as CO

(£)= J x/^(x)<ix for £ continuous (1.6.1.4) —oo

and

(4) = X • xv^# (*./) f° r £ discrete. (1.6.1.5)

Using Eqs. (1.6.1.4) and (1.6.1.5), it is obvious that

(ct) = c4), < £ + £ ) = <£ + <&) d.6.1.6)

(c being a constant). When (£) vanishes, £is said to be a centred random variable [8].

Likewise for any function g (£) of a random variable £ we have oo

(#(£))= J g(x)f^(x)dx for £ continuous (1.6.1.7) - c o

(*(£)) = £,- * ( * ; ) M * ; ) for ^discrete. (1.6.1.8)

The variance Var [cf] of £ denoted by cr2(£), is defined as [7,8] the

mean of (£ - (£» 2 , that is,

^ ( a = ( ( ^ - ( ^ ) ) 2 = ( # 2 - 2 ^ ) + (^)2) = ( ^ ) - ^ ) 2 . (1.6.1.9)

The positive square root of the variance denoted by <J(%) is called the standard deviation of £ The variance is a measure of the spread or dispersion of the values about the mean, which the random variable £can assume. It is in other words the. fluctuation of £

The covariance Cov(£j,£2) of any two random variables ^ and <f2 is defined as [7]

Cov(fi^2) = ( ( f i - (# 1 ) ) (^ 2 - ( fe») = ( ^ 2 ) - ( # 1 ) ( f e ) . (1.6.1.10)

Cov(£[,£2) is a measure of how the random variables £ and cf2 are interrelated. When ^ and £2 are independent then

Cov(<f„&) = 0 . (1.6.1.11) One defines the correlation coefficient of two random variables £i and £2

as [7]

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48 The Langevin Equation

Cov (£ ,£ , ) P= IPT(IY (1.6.1.12)

/? is a measure of the dependence between <f, and £2.

7.6.2 Properties of the Gaussian distribution

The distribution function of a continuous random variable, which is of great importance to the theory of Brownian motion is the normal distribution, also known as the Gaussian distribution:

fe(x) = —j==e ' - ° ° < ^ < 0 0 , <r>0, (1.6.2.1) (TV2^

which has the following properties (i) as x -> ±oo, /^ -» 0;

(ii) (£) = //; Var[a = <T2;

(iii) /^ has a maximum at £ = //.

It is denoted by N(^),<72). It is easily shown that the distribution of c £ is N(c(^),c2a2) and if we substitute z for £-(<f), the distribution of £-<#>, is 7V(0,<72). Likewise the distribution of %-%))!o is TV(0,1) called the standard normal distribution.

We must now introduce the concept of multidimensional random variables. Let ^ and £2 be two continuous random variables then we define their joint probability density function f^g (x,y) by the equation

f^2(x, y)dxdy = P(x<^<x + dx,y < £2 < y + dy).

We also define the corresponding join? distribution function by

F^2(x,y) = P& < x,%2 < y) = J J f^x,y')dxdy , —oo —oo

oo oo

Thus

J J F^2(x,y)dxdy^l —oo —oo

and if F? = is differentiable,

f^(X>y)= dxdy •

We also define the marginal distribution function Fg (x) of £t as

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Chapter 1. Historical Background and Introductory Concepts 49

°° X X

F(i (x) = P (£ < x, £ < oo) = \ \ f^2 (x', y')dx'dy' = J ^ (x')dx', —oo —oo —oo

where F^ (x')dx' is called the marginal distribution function of E,\

As before we may define the average value of a function g (<fi, £2) by means of the equation

oo oo

(«(6.&))=J J g(x,y)f^2(x,y)dxdy. —oo —oo

Clearly (£+£) = <£) + (&). We remark that

oo oo

(6&) = J J *y/$ 42 (*. y)dxdy • —oo —oo

In certain cases, this may be written as oo oo

(6&) = j" x /v, (*)<** J >"?& (y)dy, —oo —oo

where h and # are the marginal probability density distribution functions of £i and £2, respectively, thus

<^2H6)&>. ^ and £2 are t n e n said to be independent random variables and by Eq. (1.6.1.10) are uncorrected. We remark however that two uncorrelated random variables are not necessarily independent with the exception of Gaussian random variables. We have so far confined ourselves to 9?2, all these results may be carried over to 9T. Generally, if £ is a vector valued random variable and the ^ are all independent then

In general, if £ is a random variable with n components %y, &., ..., £„, its distribution function Fg g (x1,...,xn) is called the joint probability distribution of the n variables &,..., £n. We are often interested in the distribution of a subset having s < n variables £i,..., <£. The probability that they have certain values between (xl,xl +dxY) and (x2,x2 + dx2), etc. irrespective of the remaining variables £,+i,..., £, is

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50 The Langevin Equation

F^...^xU—'Xs)~ J - - J fzl..4nxl' — ixs>xs+l>~->xn)dxs+\—dXn-

—oo —oo

This is the most general definition of the marginal distribution function for the subset. The most important practical example of a multidimensional distribution is the n dimensional Gaussian distribution defined as follows [7,12,99]: let (£,...,£„) be n (not necessarily independent) random variables with means equal to zero. Let/(jci ... xn) be their joint probability density function; then we say (£ ... £„) are normally distributed in n dimensions if (we drop the suffixes from f* *

for convenience)

/ ( x ) = * e-» rM-'«/2 J (2;^y/2^/deTM

where M = l//yl is the matrix of the second moments

ttj =(&]> (i,j = l,2,...,n),

M is the inverse matrix; det (M) is the determinant of the matrix M; x is the column vector with components (xi,...,x„), xT is its transpose, and the quadratic form is

i n n

d e t M ^ % ,J ' J

(My is the cofactor of fly in the matrix M). The marginal distributions

/ ( x , ,x, ...x, ) for the multidimensional normal distribution may be

determined from oo oo

/ K •••*,,) = J ••• f / (xh .-.^,\+l ,-,\ )d\+1 ...dxiN .

—oo —oo

These are r-dimensional normal distributions. If the £ are independent then (%%) = 0 (j •*• j) and M becomes a pure diagonal matrix. Here

f(xl...xn) = fxl)fx2)...fxn),

where each fxi) is a one-dimensional normal distribution with mean zero and variance //,;.

We may easily evaluate the quadratic form explicitly, for example in the two dimensional case [7,99] with the probability density function, f(xi,x2),

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Chapter 1. Historical Background and Introductory Concepts 51

M = fthi fhi^

KMn /hi. The vectors x and xT are

, M_1=(detM)_1 M22 Mn

~Mn Mi ) , detM = / / n /^ 2 - / / 1 2 .

x = , x —\X\X2) K2J

so that

x r M 'x = (detM) in22x\ ~'2-fh2xix2+fh\x\\

We now introduce the standard notation

M1 = o j . M2 = °l. M2 = ^ ^ . where ox and (J are the standard deviations of £1 and £,2 and /? is the

correlation coefficient. Our probability density function then assumes the standardform

f&42 (xi>x2): 1 2(1-/?2)

*? 2l*l*2 , *2

2 ^ , < r ( l - /> 2 / / 2

^ a*a*2. ^2

x\ *2

We have considered for simplicity the case of zero means. If the means are not zero the multidimensional normal distribution is

/ (x) = -1 -i(x-n)' R-1(X-M;

(2^)"/2^/deTR

where R is now the covariance matrix with elements (£,£,) - (£,)(£,) and (£,). For example, our two-dimensional Gaussian distribution now becomes

(xi-Mxt) 2p(jc,-//^ )(x2-Mx2) r [xi-Mxj,

1 •4,6 (-Xi'^)-"

2(l-/»") «* <

(-°o < JCJ , jt2 <°°). In order to show that / (x ) is indeed a probability density function, we have to prove that [7]

00 00

j" f(x)dx= $ ...j f(xl,...xn)dxl...dxn=l. —00 — 00 —00

In order to accomplish this we note that R must be positive definite thus [7] there exists a non singular real matrix Q such that R = QQ r . We now write x = Qy + \i. The Jacobian of this transformation is

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52 The Langevin Equation

= |detQ|.

Thus, the quadratic form is reduced to a sum of squares and so n

DO - oo oo - ^ y ; / 2

1.6.3 Moment generating functions

We now introduce the concept of a moment generating function, which is an integral transform of a distribution function. In general [30]

oo

m^(u)= \ euxf^(x)dx. —oo

In particular, consider the function oo

fy(u)=\ eiuxf^x)dx = e^). —oo

This function is termed the characteristic function of the random variable £ Evidently, two random variables have the same characteristic functions if and only if they have the same probability density function. As an example of computing characteristic functions, we evaluate <j>$u) for E, with the normal distribution (which in the centred case has the remarkable reproductive property that it is itself Gaussian)

ffW = -1 -(x-n)1 l(2o2)

-e

We have by completing the square in the argument of the exponential

^ i V2;r Further, if £ is a centred Gaussian random variable (i.e., ju = 0), then

<j)^(u) = e

so that

In particular, if u = 1,

( « * ) = r ^ ' 2 , (1.6.3.D

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Chapter 1. Historical Background and Introductory Concepts 53

which is a most useful relation. We remark that, by Fourier's integral theorem [81]

1 °° f^x) = —\ e-iux<f>?(u)du,

2K _ so that a knowledge of the characteristic function of a random variable is equivalent to a knowledge of its probability density function.

We now establish a fundamental property of characteristic functions, namely the nth derivative of <j>$(u) when evaluated at u = 0 gives the nth moment about zero of the random variable £,. We have

**-<•*>• "-T = < • < £ > •

d20t(u)

u=0 du -wr •

«=o and so on. For example, for the normal distribution N(ju, a ) one has

dfy(u) fy(u) = e

iufi-a2u2/2

du = iju = i(£),

«=o and so on. One may continue to show that for this distribution,

(<f2) = //2+cr2 , ( £ 3 ) = 3 / /CT 2 +/ / 3 , (£4) = 3a2 + 6 /A7 2 +/ / 4 .

Further, if (£,) = ju = 0 , we have by successive differentiations

(<f2"+1) = 0, (£2") = (2n-l)!!(<f2)",

which is a most important (reproductive) property of N (0, cr2). The characteristic function for the n dimensional Gaussian

distribution is

tfc(u)= j e,aT*f(x)dx = -1 J 0 m r x-(x-^) 7 'R- I (x-n) /2

(2^r)n/2VdetR

By elementary matrix multiplication, we have

/ x - - ( x - f i ) r R " 1 ( x - j i )

dx.

m

Hence

= m r ^ - - u r R u - - ( x - n - / R u ) r R ~ ' ( x - f i - z R u ) .

(1.6.3.2)

In particular, for the two-dimensional case with (£ ) = (<f2) = 0> we have

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54 The Langevin Equation

P&& \U],u2)-e

which will be widely used in what follows as we shall see that this characteristic function plays a fundamental role in the theory of Brownian movement. Several properties of Gaussian random variables may be deduced from it.

If £1 and £,i are independent random variables and we have the linear combination z = CC\E,\ + Oi £2, then

More generally if (£u... gn) are n independent random variables and we have the linear combination z = CC\& + ccz^z +...+ 0Cn^n then

n

^(")=n^(^M)-1=1

This may be used to establish an important property of independent Gaussian random variables. Let

Z-N^of), ^(u) = eiu»-°?u2'2,

What is the probability density function of the linear combination

Evidently n n

and thus

f z = N f n n ^

V*=i k=i

So a linear combination of independent normally distributed random variables is also a normally distributed random variable with

n mean /U = Z 0 * / ^ a n ^ variance cr2 = fecial

The result just given applies to a sum of independent Gaussian random variables. Let us now consider the analogous result for a linear

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Chapter 1. Historical Background and Introductory Concepts 55

combination of n not necessarily independent Gaussian random variables. For simplicity, let us consider the sum of two Gaussian random variables £,x + £2 and calculate the characteristic function of this sum, that is,

This may be evaluated by simply writing ux = u2 in our expression for the characteristic function of the two-dimensional Gaussian distribution above. We have

<j> (u) = eKUx] +Mx2 ) M ° * ' +2pa*iax2 +<7*2)u2/z

Thus, <fi + £2 is a Gaussian random variable with mean

MXl +MXl

and variance

In the general case for any real constants c and c2, the variables c £i and c2 £2 are Gaussian [8]. It follows that any linear combination of £1 + £2 is Gaussian. This may be extended to show that any linear combination of n Gaussian random variables is itself Gaussian. Furthermore, if ^ are all centred so too is the linear combination. This result may be used to prove Isserlis's theorem [14], which is of central importance in the theory of Brownian motion. We have demonstrated that if &, £2 ,•••,& are centred Gaussian random variables and c1,c2,...,c„ is a set of real numbers, then £ = <f,Cj+... + <fncn is also a centred Gaussian random variable. We now establish, following [8] Isserlis's theorem, namely (see Section 1.3)

1\<J\ <-<ln <h

(Although this result is often quoted in literature on Brownian motion and is crucial to the theory, the proof of it is not easily available.)

For £ defined above

( ^ ) = (2*<^>)"1/2 J e*e-*2l^dx = e-^12

and, therefore,

sf (r)-t^): r=0 ' • i=0 ^ J •

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56 The Langevin Equation

For £,r =(^lcl+... + ^ncn)r, the multiplier of cxc2...cn in (£r> is

8n rn\(gyg2...gn), while the multiplier of cxc2...c2s in (£2)'v is

j,<i2<...<i2i.

'l<Jl *2*<J2i

Thus, on comparing the coefficients of cc2...cn on both sides of Eq. (1.6.3.4), we have Isserlis's theorem. The second Eq. (1.6.3.3) may also be written in the form

&&-£>=zn(«A)' the summation being over all products of mean values of different pairs with decreasing suffixes.

1.6.4 Central Limit Theorem

The properties of characteristic functions are of considerable use in developing a fundamental statistical theorem known [101] as the Central Limit Theorem, which may be stated as follows.

Let £, be a sequence of independent random variables each having arbitrary distributions, then the sums,

€ = &+&+&...+€„)/fi, approach a normally distributed random variable as n approaches infinity. Further, if £ has mean zero and variance (£?) = of < °° then E, has mean zero and variance o2, where

1 n

^-2 1 V" T 2 c1

The theorem may be proved heuristically as follows. Let

( ^ ) = ( ^ ) = vf,i = l,2,...n

(the exact values of these higher order moments will be of no concern to us in the present investigation as long as they are uniformly bounded). The characteristic function (/>zu) may be written as

k=\ X '

Taking the logarithm of this gives the logarithmic characteristic function 71

fn^(") = Z l n f 1 1 - 3 3 4 4 \

' u ok irkur uv ' k=i In 6n4n 24n2

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Chapter I. Historical Background and Introductory Concepts 57

For n —> °o, the term on the right-hand side under the summation sign may be approximated by -u2al l(2n). Hence,

lim [ln^(w)J = —u2cr2

and thus

fy(u) = e-uV/2, where

a2 = lim (\ n ^

1 * - > 2

\nk=\ J

The probability density function, /^(x), of £ is then

/ ^ ) = J - f fy(u)e-i»*du = —^e-x2/(2a2\ (1.6.4.1)

which proves the theorem. It should be noted that a rigorous proof of the theorem requires justification of the various limiting processes involved in going to Eq. (1.6.4.1). This can be done by appealing to Lebesgue's dominated convergence theorem [29].

The most important concept of probability theory in relation to Brownian motion is the notion of a random process, which we now outline.

1.6.5 Random processes

Consider a random variable £ which depends on the time t, i.e., £ = g(t). A random process (also known as a stochastic process) is [11] a family of random variables £(t),te T], where t is some parameter, generally the time, defined on a set T. £(?) does not depend in a completely definite way on the independent variable t [12]. Instead one gets different functions y (t) in different observations. To describe the random process completely [11] we decompose the set Tinto instants t\ < t2 < ••• < tn < T and then approximate the family of random variables £(t) by %(t\), ^(t2), ••-. £(tn)- We may then use the following set of joint distribution functions: P\(yi, t\) dy\ is the probability of finding %(t) in the interval (yx, y\ + dy). P2(y\, h; y2, t2) dy\ dy2 is the joint probability of finding %t\) in the interval (y\, y\ + dy\) and <^(t2) in the interval (y2, y2 + dy2). Pi(yu h; y2, t2; y3, t3)dyldy2dy3 is the joint probability of finding %(t\) in the interval (y\,y\+ dy\), ^(t2) in the interval (y2,y2 + dy2) and %(t3) in the interval (y3, y3 + dy3). This may be continued up to Pn(yh tx\y2, t2; ...; y„, tn).

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58 The Langevin Equation

£0

Figure 1.6.5.1. A realisation of the random variable £ (t).

The process is stationary when the probability distribution underlying the process during a given interval of time depends only on the length of that interval and not on when the interval began. Another way of saying this is that the underlying mechanism that causes the set of random variables £(t) to fluctuate does not change with the course of time. This means that a shift of the time axis does not influence the functions Pn, and as a result our set becomes P\(y)dy, which is the probability of finding £, in (y, y + dy), P2(yi, y2, t) dy\dy2 which is the joint probability of finding a pair of values of £in ranges (yh y\ + dy\) and (j>2, yi + dy2) which are a time interval t apart from each other (where t = \ti -1$) and so on.

The functions Pn may now be determined [12] by experiment from a single record £ (t) taken over a sufficiently long time. One may then cut the record into pieces of length T with T long in comparison to all periods contained in the process. The different pieces may then be considered as the different records of an ensemble or collection of observations. One has in general to distinguish between an ensemble average and a time average. The two methods of averaging for a stationary process will nevertheless always give the same result [12].

A random process is said to be purely random [12] if the successive values of £ are statistically independent. This kind of process is given by [11]

JiU.'i). Piyi>tiiy2>h) = Piyi>ti)piy2.h)> (1.6.5.1)

P3yi>h>y2't2'y3'h) = Fi(yi>ti)py2>ti)pyi'h)-A random process is called a Markov process if all the information about the process is contained in P2 [11, 12]. On recalling the definition of conditional probability, we write [12] P2(y2> hfyu h) dy2 f° r t n e

probability of )> being in the interval Cy2, y2 + dy2) at time t2 given that y\

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Chapter 1. Historical Background and Introductory Concepts 59

had occurred at time t\. Analogously this can be extended to higher order probabilities. Thus, a Markov process is a process such that [12] the conditional probability that v lies in the interval (yn, yn + dyn) at time tn, given that y\, y2,..., y„-\ had already occurred at times t\,t2,...,tn_\, depends only on the value of ;y at time tn_\, i.e.,

) = P2(yn, tn\y„-\, tn_i). (1.6.5.2) A property of a stochastic process of particular interest in connection with the Brownian motion is the correlation function [102].

The time-correlation function of two-time dependent random variables ^(f) and ^-(r) with zero average values is defined where we suppose that the physical system has obtained a steady state (thermal equilibrium) as [8] the ensemble average (^*(/,)^(f2)) (the symbol *

denotes the complex conjugate). & and ^ will in general depend on position and velocity variables or position and conjugate momentum variables of the system which we can denote by u (t). Thus [8]

<£('i)£/f2)>=J ^h)^(h)fh)du, where/(fj) is the probability density function of the system at time t\ in

the absence of an external field. If i * j , then (^\tx)^j(t2)) is called the

crosscorrelation function of £ and -. If i =j, ( ^ f^ )^ -^ ) ) is called the

autocorrelation function of £ It is a measure of the dependence of the same random variables at different times. The definition using the

complex conjugate ensures that (£2(0) must be real. The autocorrelation function of a stationary stochastic process is not affected by a shift in time, i.e.,

<£('l)<?,('2)> = <#( ' l - '2)£/0)>.

7.6.6' Wiener-Khinchine theorem

Let us now consider a stationary random process £(t) with zero average value where £(t) is now a real function of time and consider the Fourier transform of %(t) [10, 12, 13] namely

oo

!(»)= J e-m$(t)dt. (1.6.6.1) —oo

then we have the Wiener-Khinchine theorem [13] (proved in Chapter 3) 1 °°

(g(t)g(t + T)) =— J <^^(0))cos(OTd0), (1.6.6.2)

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60 The Langevin Equation

<*>£(©) = 2 j (% (t)£(t + T)) COS COTdT, (1.6.6.3) 0

where

<^«y) = lim ^£((0j'i = lim ^o),T)l*o),T')\

is called the spectral density of the random function ^t). Equation (1.6.6.3) can be rewritten (recalling that the autocorrelation function is an even function of r ) as

oo

$>4(6))=\ Zt)Zt + T))eiondT, (1.6.6.4) —oo

that is, for a stationary process, the spectral density is the Fourier transform of the autocorrelation function. In addition, by the ergodic theorem, we will have

1.7 Application to the Langevin Equation

As an illustration of the use of the concepts developed in Section 1.6 we consider how they may be used to evaluate the mean square value of the velocity from the Langevin equation (1.3.1). This also will serve as an introduction to the Ornstein-Uhlenbeck [1,15] theory of the Brownian movement which is discussed in detail in Chapter 3.

We write the Langevin Eq. (1.3.1) in the phase space (x, v) as xt) = vt), vt) = -/3vt) + F(t)lm (1.7.1)

or ^

\VJ

'0 1 ^ ' j ^ 0) F(t)

KVJ (1.7.2)

,0 -P) where J3 is the friction coefficient per unit mass. We may write this as a matrix differential equation

X(t) = AX(t) + BF(t)/m. (1.7.3) We want a solution of Eq. (1.7.3) with the initial condition describing a sharp value at t = 0. This is given by

X(t) = eAtX0+ — f exl,-nBF(t')dt', (1.7.4)

where we have assumed that the particle started off at the sharp (corresponding to delta function initial conditions in the associated Fokker-Planck equation) phase point X0 = X(0) = (x0, v0).

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Chapter 1. Historical Background and Introductory Concepts 61

Next, by calculating e ' from the matrix equation

eA'=L-l(sI-Ayi,

where oo

LfO)=l e-"f(t)dt

(1.7.5)

(1.7.6)

is the Laplace transform of/(f) [16], I is the identity matrix, and L l is the inverse Laplace transform, we have

^ 1 F^l-e-*)

In component form we have

\ , s , f

voJ 0

Ff)dt'. (1.7.7) m

v(t) = Voe-e' + \ e-M>-'')mdt>. i rn

(1.7.8)

Taking averages and noting that Ft) = 0, we obtain from Eq. (1.7.8)

m = v0e-/" (1.7.9) and

1 'r v\t) = vyW+-A-l e-^FiOdt'j e-^-^Ft")df. (1.7.10)

Using the fact that m

F(tl)F(t2) = 2DS(tl-t2), we have

v2(0 = v 0 V 2 # + e - 2 ^ e^'^dit'-ndt'dt". (1.7.11) 0 0

From the properties of the Dirac-delta function [13]

we have

\ f(t)S(t-a)dt = f(a), p<a<q, (1.7.12)

J e^'+t"W-t")dt" = e2^

so that the mean square velocity is given by

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62 The Langevin Equation

v2(t) = v20e-2^+e-^' — \ e

2 ^ = V o V 2 * + - ^ - ( l - * - 2 n (1-7.13)

The variance is given by

o * = 7 w - ( ^ ) 2 = - * ( l - e - 2 * ) . (1.7.14)

The value of D is found from the long time limit of Eq. (1.7.13), we have

limv2a) = I>/(/?m2). (1.7.15) /—>oo

We can also assume that for long times the Maxwell-Boltzmann [1,5,15] distribution sets in, so that

limv2(0 = H 7 m . (1.7.16)

Comparing Eqs. (1.7.15) and (1.7.16), we see that D = mjBkT = (kT. (1.7.17)

The spectral density <&F(aJ) of the fluctuating force F (t) in the Langevin Eq. (1.7.1) is, according to the Wiener-Khinchine theorem, given by

oo oo

<DF(<y)=r eimFQi)Fr)dr = 2\ eionD8z) = 2D. (1.7.18)

—oo —oo

(from now on, we shall follow the notation of Wang and Uhlenbeck [12] and regard 2D as the spectral density; D is also used as a symbol for diffusion coefficient, however the difference will be apparent from the context wherever we speak of white noise). Hence <bFaf) is independent of the angular frequency CO, so F (t) is called a white noise force by analogy with white light where the spectral density is constant in the visible range of frequencies. If <3>f 0)) depends on a>, then F(t) is called a coloured noise force.

Nyquist's theorem may be proved using the above results by considering the Langevin equation for the series LR circuit, namely

L ^ - + Ri(t) = et), (1.7.19) dt

where L is the inductance, R is the resistance, i(t) is the current, and e(t) is the random e.m.f. with

e(0 = 0, etl)et2) = 2DStl-t2). By replacing ^by the resistance R in Eq. (1.7.18), we have

3>e(ai) = 2D = 2RkT (1.7.20) as the spectral density of the noise voltage. We emphasise that pure white noise cannot actually exist since the power dissipated in R in a frequency range Q)\, Oh is

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Chapter 1. Historical Background and Introductory Concepts 63

j\*ea>)d(o=2kTa\-atl),

which is infinite if we extend the integral to all frequencies. In practice both quantum and memory effects will come into play, which limits the "flatness" property of the spectrum. Although pure white noise does not occur as a physically realisable process, it is of fundamental importance as an idealisation of many real physical processes leading, inter alia, to the Stratonovich and Ito calculi (see Chapter 2, Section 2.3), which play such an important role in the interpretation of the Langevin equation. Quantum effects will come into play at frequencies of the order 1013 Hz. We shall consider neither of these effects in the present work.

The process described by the Langevin equation (Eq. 1.7.1) with ^-correlated fluctuating force F(t) is a Markov process [13] because the solution of the first order equation (Eq. 1.7.1) is uniquely determined by its initial conditions. Hence the conditional probability of the process at time tn depends only on the value x(?„_i). The ^-correlated force F(t) at time t < tn_\ cannot change the conditional probability at time t > tn_\. The Markov property ceases to be valid if Ft) is no longer ^-correlated.

1.8 Wiener Process

We now consider the special type of stochastic process known as the Wiener process [1] introduced by Wiener in 1923 [30] to provide a rigorous mathematical description of the statistical properties of the trajectory of a Brownian particle. The fundamental properties of the Wiener process are set out in this section.

Let £ (t) = X (t) which for the purpose of illustration denotes the displacement after a time t of a particle undergoing Brownian motion so that X (0) = 0 by definition. Consider a time interval (s, t) which is long compared with the time between impacts of the particles of the surrounding medium on the Brownian particle. This is to say that the Brownian particle has been "drummed" during (s, t). We make the following assumptions:

1. The displacement [X (t) - X (s)] of the Brownian particle over the time interval (s, t) is the sum

±[Xtk)-X(t„)]

of the small displacements X (tk) - X (tk-i) of the Brownian particle caused by the impacts of the particles of the surrounding medium.

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64 The Langevin Equation

2. The probability distribution of X(tk) depends only on X(tk„) and not on X (tk-2), X (tk^), etc., so that we have a Markov process, whence

[xao-xts)] ,[x(f2)-x(r,)] ,...,[x(o-xan-,)], are independent random variables. Formally, we say that the process has independent increments.

3.(X(tl)-X(s)) = Xtk)-X(tk_l)) = 0. 4. Since [X(t)-X(s)] is the sum of a large number of

independent random variables [X (tk) - X (tk-i)], each having arbitrary distributions, it follows from the central limit theorem that [X (t) -X(s )]/Var[X (t) - X(s)] approaches a Gaussian distribution as n —> °o . This is the same as saying that the characteristic function of [X(t)~ X(s)]is

Mu\ »2Var[X(0-X(j)]/2

Formally, we say that a stochastic process consisting of a family of random variables X (t), t = 0 is a Wiener process if

a. X (t), t = 0 has stationary independent increments. b. X (t) is normally distributed for t - 0. c. <X(f)) = 0forf = 0. d. X(0) = 0. A fine example of three simulated sample paths (realisations) of

the Wiener process is given in Fig. 3.5 of Gardiner [21], see also his Fig. 1.2 showing the motion of a point undergoing Brownian motion and Fig. l,p.l42ofSchroeder[83].

1.8.1 Variance of the Wiener process

Assuming that Var [X (t)] = f(t), we have

f(tl+t2) = (x2(tl+t2j) = ([X(ti+t2)-X(tl) + X(t1)-X(0)]2),

since X(0) = 0. On multiplying out, we have

/(*, +t2) = ([X(t1+t2)-X(tl)f) + ([X(tl)-X(0)]2), (1.8.1.1)

since

([*(*!+f2)-X(r,)][X(r1)-X(0)])

= (X(tl+t2)-X(tl))(X(tl)-X(0)) = 0,

because X(tl+t2)-X(tl) and X(^)-X(0) are independent random

variables and (X(t)) = 0. By stationarity, we may express Eq. (1.8.1.1) as

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Chapter I. Historical Background and Introductory Concepts 65

m+t2) = ([X(t2)-X(0)]2) + ([X(tl)-X(0)]2)

= (x2(t2)) + (x2(tl)) = f(t2) + f(tl).

Let t2 = t, h = -s; then f(t-s) = f(t) +f(-s) and the only function which satisfies this functional equation is

f(t-s) = c2(t-s), (1.8.1.2)

where c is a constant to be determined. By stationarity

Var[X(t)-X(sj\ = Var[X(t-s)-X(0)]=Vax[X(t-s)]

[since X(0) = 0]; so that from Eq. (1.8.1.2) Var[X(t)-X(s)] = c2\t-s\, (1.8.1.3)

in order to ensure a positive variance. Therefore, [X(t)-X(s)] is a Gaussian random variable with the probability density function

\2^t-s\r]Xe-[Xt)-X(S)fl^^. Now, we wish to evaluate the covariance K(s,t) given by

K(s,t) = Cov[X(s),X(t)] = ([X(S)-(X(s))] [ X ( 0 - ( X ( 0 > ] ) .

Since (X(s)) = (X(t)) = 0, we have K(s,t) = X(s)X(t)) = (X(s)[X(t) -X(s) + X(s)])

= (X(s)[X(t)-X(s)]) + (x2(s)) = (x2(s))

as X (s) and X (t) -X (s) are independent. Therefore

K(s,t) = (X2(s)) = Var[X(s)] = c2 min(s,0, (1.8.1.4)

where "min" denotes the minimum of s and t. On comparison with Eq. (1.3.14) since we have chosen X(t) as the displacement, it is obvious that c2 = 2kT/£ in this instance.

We now consider the differences <f (A) of the Wiener process as these will allow us to evaluate integrals involving the Wiener process. In what follows, the random variable X (t) above is replaced by the symbol B (t) and we shall no longer necessarily suppose that B (t) represents the displacement of a Brownian particle in which case the constant c2 will alter. For example, the white noise force F (t) in the Langevin equation is usually written as (strictly speaking a meaningless equation as B (t) is not differentiable)

F(t) = mB(t) so that

(F(h)F(t2)) = m2(B(h)B(t2)) = 2kTCS(tl~t2).

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66 The Langevin Equation

Figure 1.8.1. A crude picture of realisations of F (f) and of Bt) = f' F(t')dt'.

Hence c2 becomes c2 = 2kT£/m2. We remark that the Wiener process B (t) is not stationary since B (0) = 0; however, the increment B(t + At)-B(t) is stationary, therefore, dB(t) and so B(t) are stationary processes [8].

The white noise force F (t) used by Langevin is related to B (t) by the integral [13]

B(t) = \ Ft')dt' o

so that the Wiener process smoothes the white noise process.

1.8.2 Wiener integrals

We consider two overlapping time intervals

A = [tltt2] and A' = [t^t^] w i t n h <l'\ <h<t2

(1.8.1.5)

and

£(A) = [S(f2)-B(0], t(A') = [B(t)-B(t)]. Multiplying these two equations together and averaging with the aid of Eq. (1.8.1.4), we have

(^(A)^(A')) = c2 |?;-?2 | = c2 |AnA'| (1.8.2.1)

and the special case A =A'

(£2(A)) = c2IAI. (1.8.2.2)

We may interpret differentials such as d B (t) in terms of £ because dB(t) = [£(f + dt) - £(f)] = #(*) (1.8.2.3)

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Chapter 1. Historical Background and Introductory Concepts 67

so that oo oo

f(t)dBt)=\ f(t)£(dt) (1.8.2.4) —oo —oo

written as a Stieltjes integral [30] and termed a Wiener integral. This means [1] that even though the sample paths (realisations) of the Wiener process are continuous with probability one but not differentiable, nevertheless integrals of the form of Eq. (1.8.2.4) can be defined for any square integrable/since integration by parts is permitted.

We now wish to discuss in detail the Wiener integral oo

<f[/]=J f(t)§(dt). (1.8.2.5) —oo

Let us take for/(f) the step functions

fn(0 = clZAl(t) + c2ZA2(t) + ... + cnZAn(t), (1.8.2.6)

where A; = tt -1\ and %A (0 is m e indicator function of the interval A,

defined as

fl, if te A,

^ 'HcUf.cA, . Thus

^/J=Z^[fi(0-5«)]=Z^(A,.). 1=1 ( = 1

We now define the Wiener integral £ [ / ] by

<?[/]= lim <?[/„]. (1.8.2.7) n-*oo

(It suffices to define the integral for the step functions since we may approximate any continuous function f(t) on the interval (-°°, °°) by a series of step functions [29]).

We have on taking mean values

^[/]>=lim(^[/n])=limi:Ci.(^(A;.)) = 0.

Consider now

(^/j)=((i:/«(^(4)=(ii^(A^(Ai))

= c2±±\AinAj\cicj=S±c?\Ai\. i'=i j=i i=i

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68 The Langevin Equation

Whence

( ^ [ / ] ) = l i m ( f [ / n ] ) = c 2 j f\t)dt. (1.8.2.8) —oo

In a similar way, one may deduce that oo

%[f]Z[8]) = c2\ f(0g(t)dt. (1.8.2.9) —oo

Equations (1.8.2.8) and (1.8.2.9) are very useful formulae. For further detailed treatment, the reader is referred to the fundamental paper of Doob reprinted in the anthology edited by Wax [9]. More mathematical details are given by Nelson [1] and Doob [30]. The advantage of the formulation of the theory in terms of the Wiener integral is that it allows all the mathematical operations used in the theory to be precisely defined.

We are now equipped with the mathematical tools necessary for our treatment of the theory of the Brownian movement.

1.9 The Fokker-Planck Equation

The Fokker-Planck equation is an equation for the evolution of the distribution function (which is defined on the phase space for the problem) of fluctuating macroscopic variables [13]. It is essentially a specialised form of the Boltzmann integral equation [12], [13] with the stosszahlansatz of Brownian motion. The diffusion Eq. (1.4.11) for the distribution function of an assembly of free Brownian particles is a simple example of such an equation. The main use of the Fokker-Planck equation is as an approximate description for any Markov process £fct) in which the individual jumps are small [10]. We shall derive the Fokker-Planck equation following the exposition of Coffey etal. [11].

Consider a stochastic process £(/) in which we take a set of instants t\ < t2 < t3 where for the present we assume that y\ and t\ are fixed. We define the conditional probability P2(y2, htyh h) dy2

a s t n e

probability that g(t2) lies in the interval (y2, y2 + dy2) given that g(t) had a value y\ at time t\ and P^iyz, t3\y2, t2; y\, h)dy^ the probability that £(f3) lies in the interval (y3, y3 + dy3) given that £(r2) had a value y2 at time t2 and %t\) had a value yx at time ^.If we multiply P2 by P3 and integrate with respect to y2, the resulting probability density function will only depend on y\ and t\, i.e.,

oo

PAy^h\y\A)dy^=\ P2y2,t2\yl,h)Piy^h\y2,t2;yvtl)dy2dy^\.9.\)

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Chapter 1. Historical Background and Introductory Concepts 69

or oo

Piy^hhx,h)= J P2y2,h\y\'h)pAy^hh2,h;yl,h)dy2, (1.9.2) —00

which is called the Chapman-Kolmogorov equation. If we restrict ourselves to a Markov process, we will then have

Pi(y3,h\y2,t2\yutl) = P2(y3,t3\y2,t2) (1.9.3) and

Piy^hhx,h) = \°liP2y2,t2\yl,h)P2y^h\y2,t2)dy2,(\.9A)

which is the Chapman-Kolmogorov equation for a Markov process also known as the Smoluchowski integral equation, essentially due to Einstein [2]. In Eq. (1.9.4) let us write

p2 =w> yj = y> ^2 = z,y=x, t2=t, t3=t + At and suppress the t\ dependence so that (compare Eq. (1.4.1))

oo

W(y,t + At\x)= J W(z,t\x)W(y,t + At\z,t)dz. (1.9.5) —oo

In Eq. (1.9.5), we write for economy of notations W(y,t + At\z,t) = W(y,At\z) (1.9.6)

so that oo

Wy,At\x)= J W(z,t\x)W(y,At\z)dz. (1.9.7) —oo

We wish to derive a partial differential equation for the transition probability W(y, 11 x) from this integral equation under certain limiting conditions. We have to consider

J R(y)d-^^dy, (1.9.8)

where R(y) is an arbitrary function satisfying

lim R(y) = 0,md R(n\y) exists at y = ±<=o. (1.9.9) y—>i°°

In Eq. (1.9.9), R^(y) is the nth derivative of R(y) with respect to y. Using the definition of the partial derivative, we have

~W(y,t + At\x)-W(y,t\x)~ f fl(v)—rfv= J R(y) lim

J dt J A;->0 At dy. (1.9.10)

If we assume that we can interchange the order of the limit and integration, then Eq. (1.9.10) reads

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70

dw J R(y)—dy=]im f R(y) J At A/_ifl J

The Langevin Equation

W(y,t + At\x)-W(y,t\x)

dt A/-*0 At dy. (1.9.11)

We now substitute for W(y, t + At I x) in Eq. (1.9.11) using oo oo

J R(y)W(y,t\x)dy= j i?(z)W(z,?lx)rfz (1.9.12) —oo —oo

to obtain

dW I /?(y)^r-dy=lim — J At Af_^n A * a? At->0At

J W(z,flx)J fl(y)W(y,Aflz)rfydz

J R(z)W(z,t\x)dz

(1.9.13)

Let us expand R(y) in a Taylor series about the point y = z so that

/?(?) = fl(z) + (y-z)K'(z) + 2!

/?'(z) + ... . (1.9.14)

Therefore

I R(y)—-dy = hm — dt

^2

f W(z,rljc)f K(z)+(y-z)#(z)

+^-^-R"(z) + ..]W(y,At Iz)rf)*fc- j /?(z)W(z,f Ix)dz

(1.9.15)

InEq. (1.9.15), we have

J W(y,At\z)dy = l, (1.9.16)

since W(y, At I z) is a probability density function. Therefore

dW /4U/ 1

f R(y)—dy=]im — J dt A»->O At

j Wz,t\x)\ (y-z)R'(z)

+ (y-zf 2!

R\z) + ..)Wy,At\z)dydz

(1.9.17)

Let us write

an(z,At)= j (y-z)nW(y,At\z)dy. (1.9.18)

Then

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Chapter 1. Historical Background and Introductory Concepts 71

f R(y)^dy=]im±l W(za\x)alz,At)R'(z)+^^-R'Xz) + ..]dz. dt &i-^oAt J I 2!

(1.9.19) Interchanging limits with integration again, Eq. (1.9.19) reads

°° Aw °°

J R(y)—dy=j W(z,t\x) (1.9.20)

,. aAz,At) n„ N ,. a2(z,A?) n „ t x 1 , x h m - 1 ^ — - f l O O + h m - ^ - 2 — - R (z) + ... dz.

Ar->o At AJ->O 2! A/ We now suppose that [cf. Eqs. (1.9.35) and (1.9.36) below]

l i m a " ( z , A ? ) = 0 , forn>2. (1.9.21) Ar->0 A?

Thus

J R(y)?j-dy=] W(z,tIx)[D(l)(z,t)R'(z) + D2)(z,t)R"(z)]dz, (1.9.22) .aw —oo

where

• D ( i ) ( Z j 0 = l i m £ l ^ M j (1.9.23) Al->0 At

D ( 2 ) ( Z f 0 = l i m 5 a ( 5 ! M . (1.9.24) A<->O 2Af

We need to factor R(z) out of the right-hand side of Eq. (1.9.22). To do this we use integration by parts. Thus, we have

oo oo oo

J W(z,t\x)Dm(z,t)R'(z)dz= | udv = uvfm-j vdu, (1.9.25) —oo —oo —oo

where

u = W(z,t\x)Dm(z,t) and dv = R\z)dz (1.9.26) so that

du = —\w(z,t\x)Dm(z,t)\dz, and v = /?(z). (1.9.27)

Hence oo oo -»

j W(z,t\x)Dmz,t)R'(z)dz = - \ R(z)—[Dw(z,tW(z,t\x)]dz (1.9.28) —oo —oo ^

by Eq. (1.9.9). Similarly applying integration by parts twice to the last term of Eq. (1.9.22), we have

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72 The Langevin Equation

oo oo -\2

J W(z,t\x)Di2\z,t)R"(z)dz= J R(z)-^[rt1\z,t)Wz,t\x)\dz. (1.9.29) —oo —oo " ~

Substituting Eqs. (1.9.28) and (1.9.29) into Eq. (1.9.22), we have

I M ^ + i [ D < , ' w ] ~ ^ [ D ' 2 V ] ) * = 0 °-930) (since z is a dummy variable). Thus

M ^ = _ | . [ ^ ) ( y , 0 W ( y > H x ) ] + ^ [ ^ ( y , 0 W ( y , * l x ) ] . (1.9.31)

Equation (1.9.31) is the Fokker-Planck equation for a one-dimensional Markov process governed by the random variable £(t), D W is called the drift coefficient and D ^ the diffusion coefficient which are to be calculated from the Langevin equation. The condition that the Taylor series may be truncated at n - 2 will be justified if the driving stimulus is white noise in the underlying stochastic differential equation (Langevin equation). This is apparent from the properties of white noise, i.e.,

F(tl)F(t2) = 2DStl-t2) (1.9.32)

and Isserlis's theorem, Eq. (1.3.5), namely

F(tl)F(t2)...Ft2n) = ZY[FiA ' F(h)F(t2)...F(t2n+l) = 0.(1.9.33) •r>'s

For n = 2, for example,

F(tl)F(t2)F(t,)F(t4) = 4D2S(t1-t2)S(t2-t4) (1.9.34)

+Stl-ti)St2-t4)+S(tl-t4)St2-t3),

which gives rise to a\ of order (At)2 in Eq. (1.9.21). From Eq. (1.9.34) we see that a3, a5, ...,a2n+\ are all zero and

from Eqs. (1.9.32) and (1.9.33)

a2n~(At)n. (1.9.35)

Hence

lim— a2 = 0 , n>\. (1.9.36) Af->0A?

However, if the driving stimulus is not white noise, higher order terms must be included in the Kramers-Moyal expansion, Eq. (1.9.20), and one no longer has the Fokker-Planck equation.

We again emphasise with Wang and Uhlenbeck [12] in relation to Eq. (1.9.31) that Eqs. (1.9.21)-(1.9.24) and (1.9.32)-(1.9.33) are necessarily only approximations. The basic equation is always

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Chapter 1. Historical Background and Introductory Concepts 73

Boltzmann's integral equation [5,23]. The above equations hold when in each collision the velocity (v for example) of the particle can change very little so that the Boltzmann equation may be approximated by the diffusion (Fokker-Planck) equation (1.9.31).

Since in general we will be dealing with the multivariable form of the Fokker-Planck equation, it is necessary to quote the form of that equation for many dimensions characterised by a set of random variables £ = £[,•••£„. The multivariable form of the Fokker-Planck equation [13] is with W = W(y,t I x), y denoting a set of realisations of the random variables £:

^^=-Z#-K(y.0W(y,0]4z^-[i>8)(y.0W(y,0] & i 3v(

L J 2kJ dykdytL J

(1.9.37) For simplicity, let us suppose that the process is characterised by a state vector y having only two components (yi,^) (these, for example, could be the realisations of the position and velocity of a Brownian particle), and so the two variable Fokker-Planck equation written in full is

^ - s ^ K H + i i ^ K H 0-9,8) dt ,=1^y,L J 2kJdykdy,

or

In general, D$ = D$ so that

where the various drift and diffusion coefficients are

(1.9.39)

(1.9.40)

D ^ l i m ^ , D < 2 > = i i m - ^ - , ( i ,y=l,2). (1.9.41) A/->0 At A/-»0 At

We reiterate that we have assumed in writing down our Fokker-Planck equation [referring to Eq. (1.9.31) for convenience] that

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74 The Langevin Equation

D « ( y , 0 = l i m 5 l ^ 0 , D ( 2 ) ( y > 0 = l i m £ 2 ( 3 ^ 0 , (1.9.42) A(-»0 A? A;->0 At

and

D W ( y , 0 = l i m M Z ^ = 0 (1.9.43) Ai->0 At

for n > 2. This allows us to truncate the Kramers-Moyal expansion Eq. (1.9.20). In the Fokker-Planck equation, these quantities (which express the fact that in small times At in the process under consideration, the only alteration in the random variable £ is that due to the rapidly fluctuating Brownian force F(t), which is the central idea underlying the theory of the Brownian motion) are to be calculated from the Langevin equation. The procedure emphasises again that that equation is the basic equation of the theory of the Brownian movement. We remark that the time At is of such short duration that (taking as an example y as the position and momentum of a particle) the momentum does not significantly alter during the time At and neither does any external conservative force. Nevertheless, At is supposed sufficiently long that the chance that the rapidly fluctuating stochastic force F(t) takes on a given value at time t+At is independent of the value which that force possessed at time t. In other words, the Brownian force has no memory.

We shall now explicitly calculate the drift and diffusion coefficients in the Fokker-Planck equation for the simplest one-dimensional model, which is as follows. The Langevin equation for the process characterised by the one-dimensional random variable ^(t), which describes for example the velocity of a particle of mass m undergoing one-dimensional Brownian motion, is

$t) + ftt) = F(t)lm, (1.9.44) where

FJT) = 0, F(t)F(t + T) = 2fimkTS(T). If we integrate this equation over a short time At, we have the integral equation with £(t + At) being the solution of Eq. (1.9.44) which at time t has the sharp value v, so that

/+A/ , t+At

£(t + At)-y = - f P%t')dt' + — f F(t')dt'. (1.9.45) m -J

Thus (for a more detailed exposition, see our detailed calculations in Section 1.10), taking the statistical average of the realisations of E, in a small time At, we can readily calculate the drift coefficient Dm(y,t):

^M=^m±MzyM = -Py. (1.9.46) A;-»0 At

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Chapter 1. Historical Background and Introductory Concepts 75

In order to calculate the diffusion coefficient D(2\y,t), we square £(t + At)-y to obtain the integral equation

[(t + At)-y]2=12y2(At)2

, t+At * t+Att+At (1.9.47)

-2Atj3y— f F(t')dt'+ — f f F(t')F(t")dt'dt". m • m \ ,

The first term on the right-hand side is of the order (At)2. The middle term vanishes because F and the sharp initial value y are statistically independent. The last term is treated as follows. It is, on averaging,

t+Att+At

(2PkTlm) | | 8(t''-t")dt'dt'' = (2/3kTlm)At,

where we have noted the property of the Dirac delta function, Eq. (1.7.12). Hence

D™(y,t) = lim [fl* + A ° ~ y ] 2 =2/3kTlm. (1.9.48) A;-^0 At

The third Kramers-Moyal coefficient D^\y,t) is calculated as follows. We form

i t+At

[^(t + At)-y]3=-/32y\At)3+3y2j32(At)2- J F(t')dt' m t

-3yj3At ( , t+At \ 2 f j t+At

— [ F(t')dt' + — [ F(t')dt' (1.9.49)

The only term -At, which will contribute to the average in this equation, is the one involving the triple integral. However, this will vanish for a white noise driving force because by Isserlis's theorem, all odd values are zero. Thus, D(3\y,t) = 0. Likewise, we can prove that all the D(n)(y,t) = 0 for all n> 3.

Thus, the transition probability W satisfies the Fokker-Planck equation

at ay m dy corresponding to the Langevin equation (1.9.44). W must also satisfy, since it is a transition probability,

KmW(y,t\x) = S(y-x), limW(y,t\x) = W(y), (1.9.51)

where W(y) denotes the stationary solution.

- = j3^-(yW)+F V 2 (L 9-5°)

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76 The Langevin Equation

We shall now discuss how drift and diffusion coefficients may be evaluated in the most general case.

1.10 Drift and Diffusion Coefficients

The quantities Dm and D(2) for the nonlinear Langevin equation may be calculated in the following way [13]. The most general Langevin equation in one stochastic variable £has the form [13]

£(t) = h£(t),t) + gZ(t),t)F(t). (1.10.1)

If g is constant, Eq. (1.10.1) is called a Langevin equation with an additive noise term, while if g depends on £ Eq. (1.10.1) is called a Langevin equation with a multiplicative noise term. We shall consider only the multiplicative noise case since it is more general. We wish to evaluate [13]

Dw(y,t)= lim A/->0

£(f + A Q - y | At \m=y

and

D» (w)=I l lmM<i±Mz2L 2 A/->O At

\Z0)=y

(1.10.2)

(1.10.3)

where t;(t + At) is a solution of Eq. (1.10.1) which at time t has a sharp value y such that

W) = y- d-io.4) Following [13] we write the Langevin equation (1.10.1) in the

integral form t+At

£t + At)-y= J \hZt'),t') + gZt'),t')Ft')\dt'. (1.10.5)

m

Figure 1.10.1. Three typical realisations of a random variable %(t) starting from a sharp initial point £(h) = y.

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Chapter J. Historical Background and Introductory Concepts 77

We now expand h and g as a Taylor series about the sharp point £ = y so that on recalling that the increment during the interval (t, t') is ^t') - y we obtain

h(^0,t') = h(y,t') + [^t')-y]^^- + . dy

(1.10.6)

g^t'),t') = gy,t') + W)-y]^^- + ... , (1.10.7) dy

where

d

dm rM£(O,0 <?(')=? dy 9£(0 #w=>

3g(y,0 dy

Using these expansions in the integral equation (1.10.5), we have t+At t+At -\. , /s

' ' (1.10.8) t+At t+At 7) ( t'\

+ | g(y,t')F(t')dt'+ J [ £ ( 0 - y ] ^ ^ F ( O A / + . . . .

We may now iterate for <f (0-y in the integrand using Eq. (1.10.5) to get

J+A; t+At 9/ , ( v f\ f

£(t + At)-y= f h(y,t')dt' + f l ^ f h(y,t")dt'dt" t dy \

t+At

+ f ^ ^ J gy,t")Fndt'df + \';\(y,f)F(f)df

+ T M M f h(y,t")F(t')dt"dt' a^

+ J 3g(y,0

a^ J,

(1.10.9)

J g(y,t")Ft")F(t')dt"dt' + o(&)

so that the last term involves & product of noises. Now we recall that

(1.10.10) F(t) = 0, Ft')Ft") = 2DSt'-t") (D= (kT) and the property of the Dirac delta function [13]

f(t)S(t-a)dt = ^f(a). (1.10.11)

Thus, we have from Eq. (1.10.9)

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78 The Langevin Equation

&t + At)-y = J hy,t')dt'+2D J -*±ll-±\g(y,t")8t'-t")dt"dt'+...

, / ~ \ i ,3g(y,f + 0,Af) = hy,t + @lAt)At + Dg(y,t + &2At) 6W —2—!-At+o(At) (1.10.12)

dy (0 < 0, < 1). Here, we have taken into account Eq. (1.10.11), i.e.,

2D] gy,t")St'-t")dt" = Dgy,t'). (1.10.13) t

Thus, we obtain

Da) (y ,0=l imS±SEz = / K y ,0 + D ^ ! 0 g ( y , 0 . (uo.H) Ar->o At dy

Equation (1.10.14) may also be considered as an evolution equation for the sharp value y. This is the basis for the approach to the subject portrayed in this book: the sharp initial condition corresponding to the delta function initial distribution in the Fokker-Planck picture so that in effect we are calculating the time dependence of the components of the transition probability when we impose the sharp initial condition on the Langevin equation. We emphasise that ^(t) in Eq. (1.10.1) and y in Eq. (1.10.14) have different meanings. £(f) is a stochastic variable while y-^(t) is a sharp (definite) value at time t. We have distinguished the sharp values from the stochastic variables by deleting the time argument. The last term in Eq. (1.10.14) is known as the noise induced drift.

The other integrals in Eq. (1.10.9) have been ignored because [13,15] they will either give a contribution of the form (At)n for n = 2 if there are 2n Fs and by Eq. (1.9.36) they will vanish, or if there are (2n+l) F's they will vanish by Isserlis's theorem (1.9.33).

Similarly for D(2)(y, t), we have from Eq. (1.10.5) t+At t+At

[(t + At)-yf= f J h(€,t')h(£,t')dt'dt" t t

t+At t+At

+2 f h(£,t')dt'$ g£,t')Ft')dt' (1.10.15)

t+At t+At

+ J | g^t')g^,f)Ft')Ft")dt'dt'. t t

The first two terms of Eq. (1.10.15) will give contributions of the order (At)2 and they will vanish according to Eq. (1.9.36). Thus

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Chapter 1. Historical Background and Introductory Concepts 79

t+At t+At

[Z(t + At)-y]2=2D j j gZ,t')gZ,t")28t'-f)dfdt".\.\Q.\6) t t

Therefore

,„ 1 \E(t + At)-yf , D2\y,t) = -lim^ / n =Dg\y,t). (1.10.17)

2 A<->O At

Having illustrated the one-dimensional problem, we will now illustrate how the procedure is applied to obtain the drift and diffusion coefficients for the two-dimensional Fokker-Planck equation in phase space for a free Brownian particle. This equation is, as we have seen, often called the Kramers equation or Klein-Kramers equation [13]. In general, the Fokker-Planck equation in the context of a dynamical system, the motion of which in the absence of heat bath is governed by Hamilton's equations with a separable and additive Hamiltonian comprising the sum of the kinetic and potential energies, is known as the Klein-Kramers equation.

We have seen that the Langevin equation for a free Brownian particle may be represented as the system

x = v, v = -j3v + F(t)/m (1.10.18) with

F(Fj = 0, Ftx)F(t2) = 2mj3kT8tx -t2).

The corresponding Fokker-Planck equation for the transition probability density Win phase space with x = yx, v = y2 in Eq. (1.9.40) is

+ I ^ [ D ! ? ( , , v , w ] + I ^ [ D g < V ) w ] + i l [ z > ! S ( , , v W ] .

Since x = yx, Ax = Ay and proceeding as in Eq. (1.9.42)

D^=lim 4 t t= l im — = v. (1.10.20) A/->0 At Af->0 At

Now, the change in velocity in a small time At is i t+At

Av~-/3vAt + — f Ft')dt'.

Thus the drift coefficient D? is

m

i d )

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80 The Langevin Equation

Dl)=lim — = -/3v. (1.10.21) AJ->O A?

Likewise, the diffusion coefficients D[f(x,v) and D$(x,v) are

Z)g)(x,v) = lim ^ = lim ^ * > 1 = 0, (1.10.22) Af->0 At A/->0 A?

( 2) , , ,. AxAv vAfAv A,2 (*>v) = hm = lim

Ar->0 At A/->0 A?

:lim-/?v2Af + v f ^ U ' = 0 (1.10.23)

because F(f) = 0. In order to evaluate

D^(x,v) = lim (*0L.t (1.10.24) Ar->0 A?

consider

(Av) 2 =/?V(A0 2 -^^ f F(r>' + -V I j F(t')F(t")dt'dt". m ' m , ,

(1.10.25) The first term on the right-hand side of Eq. (1.10.25) is of order (At)2, the second term vanishes on averaging, and

t+At t+At t+At t+At

J j" Ft')F(t")dt'dt" = 2D j" j 8t'-t")dt'dt" = 2DAt, t t t t

(D = pk Tm), whence the diffusion coefficient is

D%(x,v) = 2kTj3/m. (1.10.26)

Thus, we obtain

dW $W _ „

dt dx 3v m dv2

which is the desired Fokker-Planck equation.

d(vW) kT 32W (1.10.27)

1.11 Solution of the One-Dimensional Fokker-Planck Equation

As a simple example of the solution of the Fokker-Planck equation, we consider the Brownian motion of a free particle in velocity space only. The Langevin equation for this problem is Eq. (1.10.18). The

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Chapter 1. Historical Background and Introductory Concepts 81

corresponding Fokker-Planck equation in velocity space v is Eq. (1.9.50), namely,

dW. '^(PvWv) + ^ ^ . (1.11.1)

dt dv m dv2

Equation (1.11.1) is evidently a particular case of Eq. (1.10.27) The easiest way to solve this equation is to construct the

characteristic function. We have on taking Fourier transforms oo

W(u,t)=j Wvv,t\v0,0)e-iuvdv (1.11.2) —oo

so that on integrating by parts

J dv au J dv —oo —oo

Hence, our original Fokker-Planck equation is transformed into the first order linear partial differential equation (dropping the suffix v)

dW /3kT 2~ dW —— = -— u W -pu—— at m au

or dW dW /3kT 2 -

1- pu = -— u W . (1.11.3) dt du m

We make a small digression here and consider the general solution of the first-order linear partial differential equation of the form

P(x,y,z)^ + Q(x,y,z)^ = R(x,y,z). (1.11.4) ox ay

This equation is satisfied by the function 0 defined by the equation 0(x,y, z) = 0

if [12]

dx ay oz To solve the equation, we form the subsidiary system

dx _ dy _ dz P(x,y,z) Q(x,y,z) R(x,y,z)

In the problem at hand,

P=l,Q=/3u, R = -(/3kTu2lm)W ,

Hence, our subsidiary system is

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82 The Langevin Equation

dt du dW

1 fiu [j3kTu2/m)w

and the general solution of this system is

W(u,t) = v(ue-P')e-kTu2/2m), (1.11.5)

where *P is an arbitrary function, the value of which for the present problem is found as follows. The initial distribution of velocities has the form W(v,0\v0,0) = S(v-v0) so that W(u,0) = exp(-iuv0). On setting ? = 0in Eq. (1.11.5), we have

eT'"v° =^(u)e-kTu2/2m) or ¥(u) = <r'"v°+*r"2/(2m). Hence

-fit \ _ -iv0ue~fi'+kTu2e~2/1' /(2m)

and

^f(ue-fit) = e

. . 2 , fi,_kTu_ll_e-2f!,\

W(u,t) = e 2m

The above equation is the characteristic function of a one-dimensional Gaussian random variable having mean

(v(0) = v o ^ r (1.11.6)

and variance

-2=([v(«)-(v«)>T = ( l - ^ ' ) . 0.11.7) (We have shown in Section 1.7 how these results can be obtained directly from the Langevin equation). Thus the conditional probability distribution (transition probability) of the velocities in the (Ornstein-Uhlenbeck) process is

(v(0-(v(0))2

W(v,flv0,0) = — ) = e ^ , (1.11.8) crv 2n

with a1 and jU = (v(t)) given by the two preceding equations. The stationary solution is found by taking UmW which by

inspection is I mv2

UmW(v,t\v0,0) = J-^—e~2kT

,_».. v °' > \27tkT which is independent of vo- Let us now suppose that VQ also has this distribution, then

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Chapter 1. Historical Background and Introductory Concepts 83

mvo

Wv0) = J-^-e 2kT ]2xkT

which is a one-dimensional Gaussian distribution. Thus, by using P(A I B)P(B) = P(AnB), we have the joint distribution

mlv +VQ-2v0ve~P' I

w ( v , ( , v 0 , o ) = — 2 _ _ _ . " " r ' ' -" '"i . d.11.9) 27TkT^\-e ~ipt

Since the process is stationary (that is the underlying mechanism does not depend on when the process began), we can write (shifting the time axis) v(tl) = vQ, v(t2) = v and ^ = |^2_?i|- Thus, W has the two-dimensional Gaussian distribution

m [v2(l)-2v(tlMh)P+v2«2)]

W(v(tl),v(h), t) = ^==e~W ^ ,(1.11.10) InkJ^l-p1

which depends on the time only through the time difference r, p is the correlation coefficient given by /?(T) = e~^M, which we shall obtain in Chapter 3 using the Wiener-Khinchine theorem and the Langevin equation. This procedure may be also applied for many dimensions as in [12]. Note that the same symbol J¥ is used for the various probabilities for notational convenience; the particular context being indicated by the arguments.

1.12 The Smoluchowski Equation

We have remarked that the Smoluchowski Equation is a special form of the Fokker-Planck equation first given by M. von Smoluchowski in 1906 [5,9,100] which approximately describes the time evolution of the concentration of Brownian particles if inertial effects are small. In particular, Smoluchowski considered the problem of the Brownian movement of a particle under the influence of an external force [5,13]. He showed that if an external force K (x, t) acts on the particle, then the distribution function W(x, t) in configuration space satisfies the approximate equation [5, 13] (see also Sections 1.2 and 1.9 above)

dW(x,t) _ 1

dt ~ C

Equation (1.12.1) is a one-dimensional Smoluchowski equation. It is a differential equation in configuration space because W does not explicitly depend on the velocity. A more general form of this equation for

r) r)2

-—K(x,t) + kT— ox dx

W(x,t) . (1.12.1)

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84 The Langevin Equation

= D^Y (-oo<x<oo), (1.12.3)

orientational relaxation under the influence of an external potential V(u, t), where u is a unit vector specifying the orientation of a rigid body in space has the form (see Section 1.15 above)

C ^ ^ - = kTW2W(u,t) + div(grad[V(u,t)]W(u,t)). (1.12.2) at

(See also Ref. [9] for the general form of the Smoluchowski equation for the translational Brownian motion under an external force K = -gradV ).

The Fokker-Planck equation derived by Einstein (see Section 1.4) is an example of the simplest form of the Smoluchowski equation

<^ = D — dt ~ dx2

where D = k T/ £ is the diffusion coefficient and for convenience we place the assembly of particles at the origin so that Xo = 0.

In mathematical terms, our problem is to solve this equation subject to the delta function initial condition

P(x,0\0) = S(x).

This is a way of stating that all the particles of the ensemble were definitely at x = 0 at t = 0 (sharp initial condition). The solution of this equation subject to the delta function initial condition is called the fundamental solution of the equation, and mathematically speaking this solution is the Green's function of Eq. (1.12.3), it is also the transition probability. The solution again is best effected using Fourier transforms. On taking the Fourier transform over the variable x we have the characteristic function

oo

P(u,t) = J P(x,t\0)eiuxdx, —oo

whence dP(u,t) n 2 K , s —^-2-L = -Du Pu,t)

dt so that

Pu,t) = Au)e-D"2' ( f>0). The initial condition is

P(«,0)= J S(x)eiuxdx = l, —oo

whence A (u) = 1, and we have

P(u,t) = e-Du2' ( f>0).

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Chapter 1. Historical Background and Introductory Concepts 85

In order to invert this transform, we use the formula

7 -a2y2 ^ , 4n -p2/a2 e y cos2pydy = e ' ,

o 2a

Thus

P(xt\0\- f r~Du2'r~iuxV/r -I I A , t U l 1 C C (MX

2nL (AnDt y/2

(« 2 >o)

-x2/(4Dt) C- (1.12.4)

The above equation is a one-dimensional Gaussian distribution with zero mean and variance (note that P tends to zero as x tends to infinity and to the delta function as t tends to zero), such that a2 = 2D \t\. Thus

x^2DW^m-jm, (M2.5) ' mp 3nrja

where r] is the viscosity of the suspension and a is the radius of the Brownian grain. This result is the same as that obtained in 1908 by Langevin by simply writing down the equation of motion of the Brownian particle and averaging it directly as in Section 1.3.

Equation (1.12.3) also serves to define the Wiener process the probability density function of which is the fundamental solution

P(^(0,/|jc(5),s) = c(2w|/- i |1 / 2)"1

e- [ j c ' )-* ( , ) ] 2 / ( 2 c 2 1 '^ (1.12.6)

of the diffusion equation dP = c2 d2P dt ~ 2 dx2

with c2 = 2kT' IC, in this case. We shall now summarise the last fundamental result of the early

investigations of the theory of the Brownian movement, namely, the famous transition state theory of Kramers [9,13,19] originally developed to explain the breaking of a chemical bond under the influence of thermal agitation.

1.13 Escape of Particles over Potential Barriers — Kramers' Escape Rate Theory

The origin of modern reaction rate theory which we must very briefly outline before describing the Kramers theory, stems from the 1880s when Arrhenius [10,20,67] proposed, from an analysis of the experimental data, that the rate coefficient in a chemical reaction should obey the law

r = v o e - A V / ( * r \ (1.13.1) where AV denotes the threshold energy for activation and v0 is a prefactor [67]. After very many developments summarised in [20], this equation

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86 The Langevin Equation

led to the concept of chemical reactions, as an assembly of particles situated at the bottom of a potential well. Rare members of this assembly will have enough energy to escape over the potential hill due to the shuttling action of thermal agitation and never return [20] (see Fig. 1.13.1), so constituting a model of a chemical reaction.

The escape over the potential barrier represents the breaking of a chemical bond [20]. The Arrhenius law for the escape rate T (reaction velocity in the case of chemical reactions) of particles that are initially trapped in a potential well at A, and that may subsequently, under the influence of thermal agitation, escape over a high (»kT) barrier of height AV at C and never return to A, may be written in the context of transition state theory (TST) [10,20] as

T = TTS=^e-AVIkT), (1.13.2) TS In

where the subscript TS stands for transition state. The attempt frequency, Q)A, is the angular frequency of a particle performing oscillatory motion at the bottom of a well. The barrier arises from the potential function of some external force, which may be electrical, magnetic, gravitational and so on. The formula has the form of an attempt frequency times a Boltzmann factor, which weighs the escape from the well.

A very unsatisfactory feature of the Arrhenius formula is that it appears to predict escape in the absence of coupling to a heat bath in contradiction to the fluctuation-dissipation theorem. This defect was remedied and reaction rate theory was firmly set in the context of non-equilibrium statistical mechanics by the pioneering work of Kramers [19].

Figure 1.13.1 Single well potential function as the simplest example of escape over a barrier. Particles are initially trapped in the well near the point A by a high potential barrier at the point C. They very rapidly thermalise in the well. Due to thermal agitation however very few may attain enough energy to escape over the barrier into region B, from which they never return (a sink of probability), i.e., the barrier C is assumed to be sufficiently large so that the rate of escape of particles is very small.

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Chapter 1. Historical Background and Introductory Concepts 87

He chose (in order to take into account non equilibrium effects in the barrier crossing process which manifest themselves as a frictional dependence (i.e., a coupling to the heat bath of the prefactor in the TST formula) as a microscopic model of a chemical reaction, a classical particle moving in a one-dimensional potential (see Fig. 1.13.1). The fact that a typical particle is embedded in a heat bath is modelled by the Brownian motion. This represents (essentially through a dissipation parameter) in the single particle distribution function, all the remaining degrees of freedom of the system consisting of the selected particle and the heat bath, which is in perpetual thermal equilibrium at temperature T. In Kramers' model [10,19], the particle coordinate x represents the reaction coordinate (i.e., the distance between two fragments of a dissociated molecule - a concept first introduced by Christiansen [10, 67] in 1936). The value of this coordinate, xA, at the first minimum of the potential represents the reaction state, the value, xB, significantly over the summit of the well at B (i.e., when the particle has crossed over the summit) represents the product state, and the value, xc, at the saddle point, represents the transition state. We remark that in his calculations of 1940, Kramers [19,67] assumed that the particles are initially trapped in a well near the minimum of the potential at the point A. They then receive energy from the surroundings and the Maxwell-Boltzmann distribution is rapidly attained in the well. Over a long period of time however rare particles gain energy in excess of the barrier height AV. Kramers then assumed that such particles escape over the barrier C (so that there is a perturbation of the Maxwell-Boltzmann distribution) and reach a minimum at B, which is of lower energy than A, and once there, never return. We list the assumptions of Kramers.

1) The particles are initially trapped in A (which is a source of probability).

2) The barrier heights are very large compared with k T (Kramers takes k to be 1).

3) In the well, the number of particles with energy between E and E + dE, is proportional to e~E dE, that is a Maxwell-Boltzmann distribution is attained extremely rapidly in the well.

4) Quantum effects are negligible. 5) The escape of particles over the barrier is very slow (i.e., is a

quasi-stationary process) so that the disturbance to the Maxwell-Boltzmann distribution (postulate 3) is almost negligible at all times.

6) Once a particle escapes over the barrier it practically never returns (i.e., B is a sink of probability).

7) A typical particle of the reacting system may be modelled by the theory of the Brownian motion, including the inertia of the particles.

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88 The Langevin Equation

It is worth mentioning here that assumption 5 above relies heavily on assumption 2. If the barrier is too low, the particles escape too quickly to allow a Boltzmann distribution to be set up in the well. If the barrier is high, on the other hand, before many particles can escape, the Boltzmann distribution is set up. As required by postulate 3, we assume, therefore, that the ratio AW (k T), is at least of the order, say, 5.

The model, which yields explicit formulas for the escape rate for very low and intermediate to high dissipative coupling to the bath (so including non equilibrium effects in the TST formula), is ubiquitous in any physical system in which there is noise-activated escape from a potential well. It has recently attained new importance in connection with fields as diverse as dielectric relaxation of nematic liquid crystals [37], magnetic relaxation of fine ferromagnetic particles [70], laser physics [63,64], and Josephson junctions [13].

Kramers' objective was to calculate the prefactor ju in the escape rate, namely,

r = ju^e-AVKkT) (1.13.3) In

from a microscopic model of the chemical reaction. The fact that a microscopic model of the reacting system (viz., an assembly of Brownian particles in a potential well) is taken account of in the calculation of the prefactor // means that the prefactor is closely associated both with the stochastic differential equation underlying the Brownian motion process, which is the Langevin equation for the evolution of the random variables (position and momentum) describing the process, and the associated probability density diffusion equation describing the time evolution of the density of the realisations of these random variables in phase space. This is the Fokker-Planck equation, which like the Boltzmann equation is a closed equation for the single particle or single system distribution function.

By supposing that p~0 (quasi-stationarity) in Eq. (1.5.9.6), Kramers discovered two asymptotic formulae for the escape rate out of a well for a system governed by the Langevin equation. The first is the intermediate-to-high damping (IHD) formula

2x (l + p2/(4co2

c)f2-/3/(2<»c) e - A v / ( * r ) j ( l A 3 4 )

where /? = C,im. In the IHD formula, the correction ju to the TST result in the prefactor of Eq. (1.13.3) is essentially the positive eigenvalue (characterising the unstable barrier crossing mode) of the Langevin equation (1.5.8.1) linearised about the saddle point of the potential V(x)

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Chapter 1. Historical Background and Introductory Concepts 89

(which in the case considered by Kramers is a one dimensional maximum). A further discussion of this is given later on.

Equation (1.13.4) formally holds [67], when the energy loss per cycle of the motion of a particle in the well with energy equal to the barrier energy Ec = AV, is very much greater than k T. The energy loss per cycle of the motion of a barrier crossing particle is J3I(EC), where Ec

is the energy contour through the saddle point of the potential and / is the action [67] evaluated at E = Ec. This criterion effectively follows from the Kramers very low damping result (see below). The IHD asymptotic formula is derived by supposing

(i) that the barrier is so high and the dissipative coupling to the bath so strong that a Maxwell-Boltzmann distribution always holds at the bottom of the well and (ii) that the Langevin equation may be linearised in the region very close to the summit of the potential well, meaning all the coefficients in the corresponding Klein-Kramers equation are linear in the positions and velocities.

If these simplifications can be made, then the Klein-Kramers equation, although it remains an equation in two phase variables (x,p), may be integrated by introducing an independent variable which is a linear combination of x and p so that it becomes an ordinary differential equation in a single variable.

A particular case of the IHD formula is very high damping (VHD), where T from Eq. (1.13.4) becomes

2xj3 Here the quasi-stationary solution, p~0, may be obtained in integral form by quadratures and the high barrier limit of the solution (which is appropriate to the escape rate) may be found by the method of steepest descents. It is now unnecessary to linearise the Langevin equation about the saddle point (here a one-dimensional maximum) as the solution may be obtained by means of the Smoluchowski equation. Thus, in the moderate and very high friction regimes, Kramers' problem may be solved by reduction to essentially one-dimensional problems.

For small friction J3 (such that J3I(EC) « kT, however, the IHD formula derived above fails predicting just as the TST formula, escape in the absence of coupling to the bath because [67] the tacit assumption that the particles approaching the barrier from the depths of the well are in thermal equilibrium (so that the stationary solution applies) is violated

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90 The Langevin Equation

(due to the smallness of the dissipation of energy to the bath). Thus, the spatial region of significant departure from the Maxwell-Boltzmann distribution extends far beyond the region where the potential may be sensibly approximated by an inverted parabola.

Kramers showed how the very low damping (VLD) case, where the energy loss per cycle f3l(Ec) is very much less than kT, may be solved by again reducing the Klein-Kramers equation to a partial differential equation in a single variable. This variable is the energy or equivalently the action. Here the energy trajectories form closed loops so that they do not differ significantly from those of the undamped librational motion in a well with energy corresponding to the saddle energy AV or EQ . Thus the only effect of escape is to produce a very slow spiralling of the closed energy trajectories towards the origin in the phase space (x, p). He solved the VLD problem (cf. Section 1.5.12) by writing the Klein-Kramers equation in angle-action (or angle-energy) variables (the angle is the phase or instantaneous state of the system along an energy trajectory) and taking a time average of the motion along a closed energy trajectory near the saddle energy trajectory. The average, being along a trajectory, is, of course, equivalent to an average over the fast phase variable, hence, a diffusion equation, Eq. (1.5.12.29), in the slow energy (or action) variable emerges. Thus, once again, the time derivative of p (when p is written as a function of the energy using the averaging procedure above) is exponentially small at the saddle point. Hence, the stationary solution in the energy variable may be used. This procedure, which will shortly be described in detail following the original approach of Kramers, yields the Kramers' VLD formula:

r = coAl3l(Ec)e_Ay/(kT)

In kT This formula holds when in Eq. (1.13.3) y. « 1, i.e., Pl(Ec) « kT and unlike the TST result vanishes when /3 —> 0, so that escape is impossible without coupling to the bath. Thus, in all cases, analytical formulae for the escape rate rest on the fact that, in the damping regimes under consideration, the Fokker-Planck (Klein-Kramers) equation may be reduced to an equation in a single coordinate.

The low damping formula is of particular significance in that it clearly demonstrates that escape is impossible in the absence of coupling to the bath. Similarly, if the coupling to the bath is very large, the escape rate becomes zero. Kramers, in his original paper made several estimates of the range of validity of both IHD and VLD formulae and the region in which the TST theory embodied in Eq. (1.13.1) holds with a high degree

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Chapter 1. Historical Background and Introductory Concepts 91

1.0

0.8

0.6

0.4

0.2

0.0 •

IN.

1 -1

—— IHD formula

PKEc)~kT ^ ^

ID, TST (JJ = 1)

\ \ \ \ HD formula

\ N.

N

^*^*^^^^ "* *-P~wc

HD(M~U0)

0.0 0.2 0.4 0.6 PK2(Oc)

0.8 1.0

Fig. 1.13.2. Diagram of damping regions for the prefactor |i in Eq. (1.13.3). There are three regions, namely, LD, intermediate damping (ID) (TST), and HD, and two crossovers between them. The ID case is difficult to realise since the IHD correction to it is of the order /?/ &fc, and the ID case requires very small values of jil coc and very low temperatures in order to render both IHD and LD corrections small.

of accuracy. He was, however, unable to give a formula in the intermediate region between IHD and VLD, as there /3lEc) ~ kT so that no small perturbation parameter now exists. A number of investigators in the period since Kramers' work have investigated this problem. We shall, merely quote the formula of Mel'nikov and Meshkov [J. Chem. Phys. 85, 1018 (1986)] who treated the problem by constructing an integral equation for the evolution of the energy distribution function which they solved using the Wiener-Hopf method [67] and obtained the formula:

r F = ——exp

2K

_1_

In j h 1- kT '

X2 +1/4) dX

A2+1/4 IHD' (1.13.7)

which is valid for all values of the friction, f3. The reader is referred to Ref. [67] for details.

1.13.1 Escape rate in the IHD limit

In order to calculate the reaction rate in what is called the intermediate to high damping (IHD) limit, where the damping forces are strong enough to ensure equilibrium in the system (except for a small region near the barrier top where the potential may be approximated by an inverted harmonic oscillator potential), we first remark that the Langevin equation may, in this limit, be linearised in the region of the maximum of the

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92 The Langevin Equation

potential at C. This corresponds in the Klein-Kramers equation to having coefficients which are linear in the momentum and displacement and such an equation is a "linearised" Klein-Kramers equation.

Just as small viscosity, the process is governed by diffusion in a single coordinate which, on this occasion, is a linear combination of the displacement and the momentum rather than the energy. The appropriate diffusion equation is:

dp _dV dp p dp Rd

dt dx dp m dx dp pp + mkT

dp , (1.13.1.1)

Equation (1.13.1.1) is the Klein-Kramers equation, which we have derived in Section 1.5.

We now assume that the function V is sufficiently well behaved to be able to expand it as a Taylor series about xc (the value of x, where the top of the barrier is located). We write taking the barrier top as the zero of the potential

V = -co2cx-xc)

212. (1.13.1.2) Considering the situation as quasi-stationary, i.e., very slow diffusion over the barrier, made possible by the condition AV » kT, where the barrier height

AV = Vxc)-V(xA), (1.13.1.3)

we find that Eq. (1.13.1.1) reduces to the stationary equation (taking m = 1)

where x' = x -

dp ox dpy dp J

xc. We now make the substitution

= 0,

p = C(x',p)e and Eq.(1.13.1.4) becomes:

-(p2-(4x'2)/(2kT)

dp dx dp dp

(1.13.1.4)

(1.13.1.5)

(1.13.1.6)

Here £x',p) is a crossover function, which varies rapidly near the barrier and takes on the value 1 in the well and 0 over the barrier.

We see immediately that £= constant is a solution of Eq. (1.13.1.6). However, this solution corresponds to thermal equilibrium, hence to a situation of no diffusion. This would yield, of course, the Maxwell-Boltzmann distribution. We can obtain, however, another solution if we assume that the crossover function satisfies the condition

C = C(u), (1.13.1.7)

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Chapter 1. Historical Background and Introductory Concepts 93

where u = p-ax and a is a constant. We remark that our solution must satisfy the following conditions:

(i) to the right of the barrier at C, the density must go to zero in order to reflect the fact that at the beginning of the process, practically no particles have reached the sink B. (ii) near the source point A, the Maxwell-Boltzmann distribution holds to a high degree of accuracy.

We take account of these two conditions by imposing the boundary conditions: ^—» 0 as x —» °o (i.e., f o r x » x c ) and £= 1 atx = 0(i.e., at A). (1.13.1.8) Equation (1.13.1.7) with Eq. (1.13.1.6) now leads to

[a-P)p-a%x~\(;' + 0kTC' = O. (1.13.1.9)

To solve this equation, we wish to write the coefficients £', £" in terms of the single variable u = p-ax' rather than x' and p. This can be achieved in a very neat way if we write

(a - /3)p - afcx' =(a- P)p - ax') = (a- f3)u, which imposes on the constant a the condition:

a>2c=a(a-p) or a = /3l2)±^jo)2

c +j32/4 . (1.13.1.10)

Equation (1.13.1.9) now takes the form of a conventional ordinary differential equation in u:

j3kTC" + (a-p°)uC' = 0, (1.13.1.11) which has as solution

„ (a-P)u'1

C = C']e ipkT du, (1.13.1.12)

where C" is a constant of integration. If we ignore the minus sign in Eq. (1.13.1.10) then

a-j3 = -j3/2 + ^co2+j32/4 = A+

will be positive and Eq. (1.13.1.12) will therefore represent a diffusion of particles over the barrier at C. The quantity A+ then corresponds to the unique positive eigenvalue of the Langevin equation (1.5.8.1) linearised about C with the noise term omitted and characterises the (unstable) barrier crossing mode. We may take - ° ° for the lower limit of the integral in Eq. (1.13.1.12) corresponding to the boundary condition

^ - > 0 a s M->-OO, (1.13.1.13) i.e., £"—» 0 for x » xc or far to the right of the barrier top. On the other hand, in the region x « xc far to the left of the barrier top, that is, in the

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94 The Langevin Equation

depths of the well, we may extend the upper limit of integration in Eq. (1.13.1.12) to +°o to get

£ = C'42xPkTI(a-P) (1.13.1.14)

since \ e ax dx = 4nloc. Furthermore, the density in phase space near —00

A (the minimum of the potential) will be: p(x,p) = C'j2xj3kT/(a-j3) gAv/(«-)e-(^+^)/(2*7-)j ( U 3 . U 5 )

where near the bottom of the well (x ~ xA = 0), the potential is approximated by V(x) = -AV + co2

Ax212. The number q of particles passing the barrier top in unit time,

i.e., the probability current, may be obtained [67] by integrating pp over p from -00 to +00 (q in this ID case strictly means the number of particles crossing unit area in unit time; the calculation of the current from the current density in more than one dimension is a complicated mathematical task [67]). By putting x' = 0 so that x = xc (i.e., the line of flow is through the saddle line), since pp is the current density, we obtain fromEqs. (1.13.1.5) and (1.13.1.12)

p1 p (a-P)p'2

q=\ ppdp = C'\pe 2kT J e 2pkT dp dp - C'kT 27t/3kT I a (1.13.1.16) —00 —00 —00

while the number of particles nA trapped near the minimum A is from Eq. (1.13.1.15)

The probability of escape is, therefore, the number crossing the saddle line in unit time divided by the number in the well, viz.,

r = S- = JT^fi7Ze-AV,ia'), (1.13.1.18) nA 2K

which together with the eigenvalue, Eq. (1.13.1.10), yields

r = - ^ ( J y f f 2 / 4 + fl£-£/2)e-AV/(tr). (1.13.1.19)

27t(ocy* 1 If we now take the limit of the right hand side of this equation, in the two cases of large and small ft, we find in the first instance /3» 2coc)

r = (JuMjB27-l)e-AV,ikT) ~^Le-hvm). (1.13.1.20)

A7CCOc\y 1 271/3

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Chapter 1. Historical Background and Introductory Concepts 95

r = fk 2K

The result embodied in Eq. (1.13.1.20) is, in effect the non-inertial limit where the dissipation in the system is so large that the inertia of the escaping particles has practically no effect (compare the original Einstein theory of the Brownian movement with the later inertia corrected version of Uhlenbeck and Ornstein, see Chapter 3).

Many authors take the limit in the second instance, i.e., weak damping ( y ? « 2 « c ) in Eq. (1.13.1.19):

^ + /32/(44)-fi/(2coc)yAV/^ ^e~hvl(-kT\ (1.13.1.21)

which is the Arrhenius result predicting escape in the absence of coupling to the bath, i.e., /?—>0. However, taking the limit as /?—>0 is not permitted as we shall now explain.

We first remark that the IHD solution Eq. (1.13.1.19) which we have described relies [67] on the assumption that the friction is large enough to ensure that the particles approaching the barrier from the depths of the well are in thermal equilibrium. If the friction coefficient becomes too small, this condition is violated and the IHD solution is no longer valid because the space interval in which the non-equilibrium behaviour prevails exceeds that where an inverted parabola approximation to the potential is valid. Hence, the need for a different treatment for very small values of the friction such that J3l(Ec) « kT and for crossover values of the friction, where J3I(EC) ~kT. Put in a more simple way, in the VLD case, the coupling to the bath is so weak that the assumption of a Maxwell-Boltzmann distribution in x and p in a relatively large region extending almost to the top of the barrier is not valid, because the damping is so small that the motion of an escaping particle is almost that of a librating particle with energy equal to the barrier energy and without dissipation.

Having treated in detail the IHD case, we now consider very low damping.

1.13.2 Kramers' original calculation of the escape rate for very low damping

We follow, as closely as possible, the original reasoning and phraseology of Kramers using the energy controlled diffusion equation of Section 1.5.12. As usual, a stationary state of diffusion, i.e., dp/dt = 0, with current density q corresponds to

q = -j3(lp + kTI^-) = -j3kTIe-E/kT)^~(peEIkT)) (1.13.2.1) V oE J dEy '

because the continuity equation is, in / space

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96 The Langevin Equation

q

dp _ dq

Integrating with respect to E between two points A and B along the E-(or /-) coordinate yields

= /3kT[peEI(kT)]BJ]jeEI«T)dE. (1.13.2.2)

The density p is practically constant along lines of constant energy (since a Boltzmann distribution is set up with

P = P0e-Em)

so E = const implies p = const over a range of curves that start at A and that extends to energy curves that cut the x-axis, not at C itself but at some point D very close to C. This restriction is necessary if the potential function has a smooth saddle point as the frequency tends to infinity as E tends to AV. This would mean that the viscosity is no longer small in the sense used in Section 1.5.12. (This restriction is unnecessary if the saddle point is not a smooth function of the space variables). Equation (1.13.2.2) may be written as

(peEIkT)) - U £ / ( * r ) ) q = /3kT- c'

neaiA ^ . (1.13.2.3)

f -eEIkT)dE near A

We avoid integrating from the point A itself because at this point E = I = 0, and so the integral would diverge. We may take it that "near

A" means an energy value of the order of the thermal energy kT, and so corresponds to points in phase space where p is still of the same order as at A itself, and where practically all the particles are trapped. The condition that particles leaving at C practically never reenter the well

means, according to Kramers, that (peEI(kT) j may be taken to be 0, and

that the upper limit of the integral may be taken to be the barrier energy at C. Let us write

pA=(peEKkT)) . (1.13.2.4) r* \ r ''near A

Thus, Eq. (1.13.2.3) becomes:

q = /5kTpA\ j r1eEIkT)dE\ . (1.13.2.5)

Now the main contribution to this integral comes from energy values which differ from the barrier energy AV by a quantity of the order of

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Chapter 1. Historical Background and Introductory Concepts 97

magnitude k T, so that we may take / to have the value Ic corresponding to the energy trajectory through the saddle point C. Hence, the integral in Eq. (1.13.2.5) may be written as

A V , i AV

J LeEIkT)dE~—eAVI(-kT) \ e-&v-E),ikT)dE . (1.13.2.6)

kT * *C kT

Let S s AV - E, then dE = -dZ . We now take the high barrier limit by integrating over E from - ° ° (i.e., deep in the well) to AV so that the above integral governing the current q now becomes

AV 1 1 ° IT J -eE,(kT)dE^~—eAV,kT)je-E,kT)dZ= — eAV,<-kT) (1.13.2.7)

kT1 lC oo 1C

so that the current is

q~/3pAIce-&v/(kT\ (1.13.2.8) Hence, since the number of particles trapped in the well near A is

nA=pA27rkT/0)A, (1.13.2.9) the escape rate is given by

r = J_ = jsnEcX^Le-EcKm (1.13.2.10) nA kT 2x

where Ec = AV . Kramers now roughly approximates the action of the almost periodic motion at the saddle point by

1C=27CECI(DA, (1.13.2.11) which is the action of a harmonic oscillator of energy equal to the barrier energy and natural frequency fA [67], so that Eq. (1.13.2.10) becomes

r = pW-e-AV/(kT), (1.13.2.12)

KJ.

which is Eq. (28) of Kramers [19]. Note the discrepancy between this equation and the low damping limit of the IHD Eq.(1.13.1.19), i.e., Eq. (1.13.1.21), which predicts the TST result as /?—> 0, and so escape in the absence of dissipative coupling.

We remark that Eq. (1.13.2.10) may also be written in the form

T=AEa^e_AV/kT) (1.13.2.13) kT 2K

where AE = /3l(Ec)«kT (1.13.2.14)

is the energy loss per cycle of the motion at the saddle point and

lEc) = j>E pdx (1.13.2.15)

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98 The Langevin Equation

is the action of the almost periodic motion on the saddle point energy contour.

In conclusion of this section, we mention that a first passage time approach to reaction rates is described in detail in Ref. [67] (for magnetic spins). Here, we merely remark that Hanggi et al. [20] rederived the low damping result above using this approach [their Eq. (4.49)], namely Eq. (1.13.2.10)

P tO)Ae-Ecl(kT) T = -^I(Ec)^e-*c'yKl) (1.13.2.16)

kT 2K (Hanggi et al. [20] use ftb instead of coA). We shall also need the escape rate as rendered by the TST theory. That result is rather simple to write down, because, according to Boltzmann statistics, which unlike in the treatment of Kramers in TST, is assumed to hold everywhere, the probability of a jump, that is an event occurring is

P = Ke~EIkT), (1.13.2.17) where E is the energy and AT is a constant. In using this equation for reaction rates the constant is written as fA, where fA is called the attempt frequency, so that according to TST P = F = fA exp[-AV /(kT)]. We now write Eq. (1.13.2.16) in the form given by Kramers. Thus as before in the low damping limit

r=-^/(£c)rT kT

TST • (1.13.2.18)

Critical energy trajectory

Figure 1.13.2.1. Diagram of critical energy curve and separatrix given by Matkowsky et al. [J. Stat. Phys. 35, 443, (1984)]. Shown here are the critical energy curve (E = Ec) and the separatrix (g) in phase space [67]. The critical energy, Ea is the energy required by a particle to just escape the well. When a particle reaches this energy, it may either escape or fall back with equal probability. The separatrix separates the bounded and unbounded motions. In other words, when a particle reaches the separatrix it exits the well. The separation of these curves (greatly exaggerated in the diagram (is infinitesimally small).

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Chapter 1. Historical Background and Introductory Concepts 99

Notice that essentially Hanggi et al. [20] used the mean-first-passage-time method of Matkowsky et al. [J. Stat. Phys. 35, 443 (1984)] (which is explained in detail in the context of magnetic spins in [67]), namely

r - 1 =r (A) + 2T^ s e p . , (1.13.2.19)

where T(A) is the time to go from the well to the critical energy curve [the critical energy is the energy required by a particle in order to escape the well; thus without any extra energy a particle, on reaching the critical energy curve, may either fall back into the region or escape, with equal probability]. Here TA^sep, is the time required to go from the critical energy curve to the separatrix [a curve whose distance from the critical energy curve is infinitesimal, however, all particles reaching the separatrix are on their way out of the region A (see Fig. 1.13.2.1)]. Note that TA _> sep is considered to be negligible.

1.13.3 Range of validity of the IHD and VLD formulae

The IHD escape rate in the limit of vanishing friction tends to the TST result, namely,

r = fkg-AV/(*7-). (1.13.3.1) In

This limiting behaviour is, however, inconsistent with the derivation of the IHD result Eq. (1.13.1.19) because, in the limit of vanishing friction, the variation of x is not the same as the variation of u so that the correct formula to use is Eq. (1.13.2.18), that is,

r = -l(Ec)^e'AV/(kT), (1.13.3.2) kT V c,2n

where /3lEc)«kT. (1.13.3.3)

In order for Eq. (1.13.3.2) to hold, J3 must be small compared with a>A. If /3 = 2ooA, we have aperiodic damping, and we might expect that there would be a plentiful supply of particles near the point C, and so the escape rate would be described by the IHD formula. Kramers [19] confesses (cf. Fig. 1.13.2) that he was unable to extend Eq. (1.13.3.2) (the VLD Result) to values of f3 which were not small compared with 2coA , that is in the crossover region between VLD and IHD formulae.

The approximate formula Eq. (1.13.3.2) for Y in the VLD limit is useful for obtaining a criterion in terms of the barrier height for the ranges of friction in which the VLD and IHD Kramers formulae are

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100 The Langevin Equation

valid. Equation (1.13.3.2) is [with the harmonic oscillator action given by Eq. (1.13.2.11)].

r = r V L D = ^ A V e - < ™ . (1.13.3.4)

If now we define a dimensionless friction parameter cc = 27tf5lcoA (1.13.3.5)

Eq. (1.13.3.4) becomes

T = a^-TTS (1.13.3.6) kT

so that aAV is approximately the energy loss per cycle. Hence, the condition for the validity of the VLD Eq. (1.13.3.4) becomes

aAV/(kT)«l while one would expect the IHD formula Eq. (1.13.1.19) to be valid if

aAV/(kT)»l. The damping region given by

aAV/(kT)~l defines a crossover region, where neither the VLD nor the IHD formula is valid, which is the reason for the calculation of Mel'nikov and Meshkov mentioned above. We shall now give a physical interpretation of the three regions identified above.

We may summarise the results of our calculation. In the mechanical Kramers problem three regimes of damping appear:

(a) Intermediate to high damping: the general picture here [67] being that inside the well the distribution function is almost the Maxwell-Boltzmann distribution prevailing in the depths of the well. However, near the barrier the distribution function deviates from that equilibrium distribution due to the slow draining of particles across the barrier. The barrier region is so small that one may approximate the potential in this region by an inverted parabola.

(b) Very low damping: here the damping is so small that the assumption in (a) namely that the particles approaching the barrier region have the Maxwell-Boltzmann distribution completely breaks down. Thus, the region where deviations from that distribution occur extends far beyond the region where the potential may be approximated by an inverted parabola. Thus we may now, by transforming the Klein-Kramers equation into energy and phase variables [by averaging over the phase and by supposing that the motion of a particle attempting to cross the barrier is almost conserved and is the librational motion in the

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Chapter I. Historical Background and Introductory Concepts 101

well of a particle with energy equal to the barrier energy] obtain an escape rate formula. We remark that the assumption of almost conservative behaviour (meaning that the energy loss per cycle is almost negligible and is equal to the friction times the action of the undamped motion at the barrier energy) ensures that the Liouville term in the Klein-Kramers equation vanishes (unlike in IHD where strong coupling between the diffusion and the Liouville term exists). Thus, only the diffusion term in the energy variable remains (the dependence on the phase having been eliminated by averaging the distribution in energy-phase variables along a closed trajectory of the energy since we assume a librational motion in the well),

(c) An intermediate (crossover) friction region where neither IHD nor VLD formulas apply: in this region neither of the above approaches may be used. In contrast to the VLD case, the Liouville term does not vanish, meaning that one cannot average out the phase dependence of the distribution function, which is ultimately taken account of by constructing from the Klein-Kramers equation a diffusion equation for the distribution function with the energy and action as independent variables. This diffusion equation allows one to express the calculation of the energy distribution function at a given action, as a Fredholm integral equation which can be converted into one (or several) Wiener-Hopf equation(s) [67]. This procedure yields an integral formula the product of which with the IHD escape rate [cf. Eq. (1.13.7)] provides an expression for the escape rate, which is valid for all values of the damping, so allowing the complete solution of Kramers' problem. The interpolation integral derived from the Wiener-Hopf equation effectively allows for the coupling between the Liouville term and the dissipative term in the Klein-Kramers equation written in terms of energy-phase variables, which is ignored in the VLD limit.

The Kramers theory may be verified numerically for large potential barrier heights by calculating the smallest nonvanishing eigenvalue of the Klein-Kramers equation [67]. This procedure is possible because of the exponential nature of the escape rate, so that, in effect, the smallest eigenvalue of the Fokker-Planck equation is very much smaller than all the higher order eigenvalues, which pertain to the fast motion in the well. Thus the Kramers escape rate is approximately given by the smallest nonvanishing eigenvalue if the barrier height is sufficiently large > 5 k T. This method has been extensively used to verify the Kramers theory, in

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102 The Langevin Equation

particular the application of that theory to magnetic relaxation of single domain ferromagnetic particles (see Section 1.16 below). We shall now briefly summarise the extension of the Kramers theory to many dimensions due to Langer [68].

1.13.4 Extension of Kramers' theory to many dimensions in the intermediate-to-high damping limit

We have seen that the original IHD treatment of Kramers pertained to a mechanical system of one degree of freedom specified by the coordinate x with additive Hamiltonian H = p2 /2m + V(x). Thus, the motion is separable and described by a 2D phase space with state variables (x,p). However, this is not always so. For example, the motion of the magnetic moment in a single domain ferromagnetic particle is governed by a Hamiltonian that is non-additive so that the system is non-separable and which is simply the magnetocrystalline anisotropy energy of the particle. The Gilbert equation governing the relaxation process also causes multiplicative noise terms to appear, which complicates the calculations of the drift and diffusion coefficients in the Fokker-Planck equation (see Section 1.17).

The phase-space trajectories in the Kramers problem of the underdamped motion are approximately ellipses. The corresponding trajectories in the magnetic problem are much more complicated because of the non-separable form of the energy. Similar considerations hold in the extension of the Debye theory of dielectric relaxation (see Section 1.15) to include inertia as in this case one would usually (albeit with a separable Hamiltonian) have a six-dimensional phase space corresponding to the orientations and angular momenta of the rotator. These, and other considerations, suggest that the Kramers theory should be extended to a multidimensional phase space.

Such generalisations, having been instigated by H. C. Brinkman [Physica (Utrecht), 22, 149 (1956)], were further developed by R. Landauer and J. A. Swanson [Phys. Rev. 121, 1668 (1961)]. However, the most complete treatment is due to J. S. Langer in 1969 [68], who considered the IHD limit.

As specific examples of the application of the theory, we shall apply it to the Kramers IHD limit and to the calculation of the magnetic relaxation time for a single domain ferromagnetic particle for an arbitrary non-axially symmetric potential of the magnetocrystalline anisotropy in that limit (see Section 1.18).

Before proceeding, we remark that a number of other interesting applications of the theory, which, as the reader will appreciate, is

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Chapter 1. Historical Background and Introductory Concepts 103

generally concerned with the nature of metastable states and the rates at which these states decay, have been mentioned by Langer [68] and we briefly summarise these.

Examples are: 1. A supersaturated vapour which can be maintained in a metastable

state for a very long time but which will eventually undergo condensation into the more stable liquid phase.

2. A ferromagnet, which can persist with its magnetisation pointing in a direction opposite to that of an applied magnetic field.

3. In metallurgy, an almost identical problem occurs in the study of alloys whose components tend to separate on ageing or annealing.

4. The final examples quoted by Langer are the theories of superfluidity and superconductivity, where states of nonzero superflow are metastable and so may undergo spontaneous transitions to states of lower current and greater stability.

According to Langer [68], all the phase transitions above take place by means of the nucleation and growth of some characteristic disturbance within the metastable system. Condensation of the supersaturated vapour is initiated by the formation of a sufficiently large droplet of the liquid. If this droplet is big enough, it will be more likely to grow than to dissipate and will bring about condensation of the entire sample. If the nucleating disturbance appears spontaneously as a thermodynamic fluctuation it is said to be homogeneous. This is an intrinsic thermodynamic property of the system and is the type of disturbance described by Langer [68], which we shall summarise here. The other type of nucleation is inhomogeneous nucleation and occurs when the disturbance leading to the phase transition is caused by a foreign object, an irregularity, for example, in the walls of the container or some agent not part of the system of direct interest.

The above examples have been chosen in order to illustrate the breadth of applicability of the theory. In the present context, we remark that Langer's method, since, in effect, it can be applied to a multi-degree of freedom system, is likely to be of much use in calculating relaxation times for fine particle magnetic systems in which other types of interaction, such as exchange and dipole-dipole coupling, also appear. We also emphasise that Langer's treatment of the homogeneous nucleation problem contains within it the magnetic case of the Kramers' IHD calculation which we shall treat in Section 1.18. The multidimensional Kramers problem was first solved in the VHD limit by Brinkman and Landauer and Swanson, see [67].

We should also mention that Langer's treatment is in effect also a generalisation of a calculation of R. Becker and W. Doring [Ann. Phys.

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104 The Langevin Equation

(Leipzig), 24, 719 (1935)] of the rate of condensation of a supersaturated vapour. A general discussion of this problem is given in Chapter 7 of Frenkel [34] on the kinetics of phase transitions.

1.13.5 hanger's treatment of the IHD limit

In order to achieve an easy comparison with previous work on the subject, we shall adopt the notation of Ref. [67]. Thus, we shall consider the Fokker-Planck equation for a multidimensional process governed by a state vector TJ which is [20,68]

7) 2N2N ?

f /x i i .o=ZZr-^ • + kT- p(r\,t). (1.13.5.1)

Here, when the noise term in the Langevin equation is ignored, the system evolves in accordance with

tf.\=-2X — , (1.13.5.2)

where (My) is called the transport matrix which, for simplicity, we shall assume to be constant.

An example of such a system is provided by Hamilton's equations for the magnetisation of a single domain ferromagnetic particle given in Section 1.18.2. In the non-stochastic limit, where dissipative coupling to the bath is entirely ignored, we have

. _ y dE

Ms d(p (1.13.5.3)

^ T M T ' (1.13.5.4) Ms dp

where p = cos??, E is the energy, yis the gyromagnetic ratio, and Ms is the saturation magnetisation.

In Eq. (1.13.5.1), E(r\) is a Hamiltonian (energy) function having two minima at points A and B separated by a saddle point C surrounded by two wells. One, say that at B, is at a much lower energy than the other. The particles have to pass over the saddle point, which acts as a barrier at C. We again assume that the barrier height EC-EA is very high (at least of the order of 5kT) so that the diffusion over the barrier is slow enough to ensure that a Maxwell-Boltzmann distribution is established and maintained near A at all times. The high barrier also assures that the contribution to the flux over the saddle point will come mainly from a small region around C. The 2N variables r\ = %,%,...,JJ2N) a r e parameters, which could be the coordinates

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Chapter 1. Historical Background and Introductory Concepts 105

and momenta of a point in phase space or coordinates describing the orientation of the magnetisation vector of a single domain ferromagnetic particle as in Eqs. (1.13.5.3) and (1.13.5.4).

Generally, however, the first TV of the 77,'s will be functions of the N coordinates of position [20]

% =?(*,•), i = l,2 iV. (1.13.5.5) The second N of the 77,'s will be the conjugate momenta n-(x) taken at the same points:

tji+N=z(Xi) i = l,2,...,N. (1.13.5.6)

In fact, the 77,'s will often (although they need not) be the coordinates themselves; in which case (obviously):

tj. =x. i = l,2,...,N. (1.13.5.7)

We may define the matrices D and A by the equations

D = - ( M + M r ) (1.13.5.8)

and

A = - ( M - M r ) , (1.13.5.9)

where M is the transport matrix resulting from Eq. (1.13.5.2), viz., M = (Mij). (1.13.5.10)

The matrix D is called the diffusion matrix, which characterises the thermal fluctuations due to the heat bath while the matrix A describes the motion in the absence of the bath, that is the inertial term in the case of mechanical particles, and if D is not identically zero, then the dissipation of energy satisfies [20]

£ = - X | ^ , | ^ - < 0 . (1.13.5.11)

We consider, as before, a single well and suppose that at finite temperatures a Maxwell-Boltzmann distribution is set up and the density at equilibrium is

/V(T|) = - W ( m , (1-13.5.12)

where 00 00

Z= j ••• j e-Em)d?jl--d7]2N (1.13.5.13) —00 —00

is the partition function. The IHD escape rate for this multivariable problem may again be calculated by the flux over barrier method.

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106 The Langevin Equation

We make the following assumptions about yOCfi]): 1. It obeys the stationary Fokker-Planck equation (i.e., p = 0),

which is (on linearisation about the saddle point):

& " ' . k d77j

p(r\) = 0, (1.13.5.14)

where the ejk are the coefficients in the Taylor expansion of the energy about the saddle point truncated at the second term, namely the quadratic (form) approximation

E(ry) = Ec-(l/2)YdiJeij(7?i-7lj!)(Tlj-7]sJ), r\~r\s (1.13.5.15)

and Ec is the value of the energy function at the saddle point (compare Kramers' method above, there the saddle point is a one-dimensional maximum). Equation (1.13.5.15) constitutes the parabolic approximation to the potential in the vicinity of the saddle point. For example, in magnetic relaxation in a uniform field with uniaxial anisotropy, the energy surface in the vicinity of the saddle point will be a hyperbolic paraboloid [65]. [Equation (1.13.5.14) is the multidimensional Fokker-Planck equation "linearised" in the region of the saddle point.]

2. Due to the high barrier just as in the Kramers high damping problem, a Maxwell-Boltzmann distribution is set up in the vicinity of the bottom of the well, i.e., at A, so:

/ K M ) « / V W X H l H I ^ - (1.13.5.16)

3. Practically, no particles have arrived at the far side of the saddle point. So that we have the sink boundary condition

P(in) = 0, !] beyond n s . (1.13.5.17) This is Kramers' condition that only rare particles of the assembly cross the barrier. Just as in the Klein-Kramers problem for one degree of freedom, we make the substitution

(Again, the function £ is known as the crossover function). Thus, we obtain from Eqs. (1.13.5.12) and (1.13.5.14), as before, an equation for the crossover function ^(77), namely

I ,. ' j

-ltejk(Tlk-Tli)-kT-. ^ ( i | ) = 0, (1.13.5.18)

where T| ~ t|A). We postulate that C, may be written in terms of a single variable u

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Chapter 1. Historical Background and Introductory Concepts 107

^ ( M ) = _L_fc-^/(2« ,)<fe (1.13.5.19)

and we assume that u has the form of the linear combination

u = YVi(ili-V?)- (1.13.5.20)

This is Kramers' method of forcing the multidimensional Fokker-Planck equation into an equation in a single variable u (in his original case a linear combination of the two variables, position and velocity, so that u = p-ax). In order to proceed, we have to determine the coefficients Ui of the linear combination u of the rjj. This is accomplished as follows: we define the matrix

M = - M r . (1.13.5.21) Then we shall have the coefficients Ui of the linear combination as a solution of the eigenvalue problem

JLjiUlMiSejk=X+Uk. (1.13.5.22)

The eigenvalue A+ is the deterministic growth rate of a small deviation from the saddle point, and is the positive eigenvalue of the transition (system) matrix of the noiseless Langevin equations, linearised about the saddle point. It characterises the unstable barrier-crossing mode. Thus, in order to calculate A+, all that is required is a knowledge of the energy landscape and Eq. (1.13.5.22) need not, in practice, to be involved.

Equation (1.13.5.22) is obtained essentially by substituting the linear combination u, i.e., Eq. (1.13.5.20), into Eq. (1.13.5.18) for the crossover function and requiring the resulting equation to be a proper ordinary differential equation in the single variable u with solution given by Eq. (1.13.5.19) (the details of this are given in [67]). Equation (1.13.5.22) may also be written in the matrix form

-VMES=A+V. (1.13.5.23) (Hanggi et al. [20] describe this equation by stating that U is a "left

eigenvector" of the matrix -ME S . The usual eigenvalue equation of an arbitrary matrix A is

AX = AX. (1.13.5.24) In the above terminology, X would be a "right eigenvector" of A). In Eq. (1.13.5.23)

E s s ( ^ . ) (1.13.5.25)

is the matrix of the second derivatives of the potential evaluated at the saddle point which is used in the Taylor expansion of the energy near the

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108 The Langevin Equation

saddle point. The determinant of this (Hessian) matrix is the Hessian itself.

The normalisation of £/,• is fixed so that

A ^ X u ^ W O . (1.13.5.26)

which is equivalent to

l u ^ W ^ - 1 - (1.13.5.27)

[This condition ensures that the crossover function Eq. (1.13.5.19) retains the form of an error function and so may describe diffusion over a barrier. Alternatively, one may say that the foregoing conditions require that the entry in the diffusion matrix in the direction of flow (that is, the unstable direction) is nonzero, that is, we have current over the barrier and so particles escape the well.]

The Fokker Planck Eq. (1.13.5.1) is a continuity equation for the representative points just as described earlier so that

_ £ + V-J = 0 . (1.13.5.28) dt

By inspection, we find that the current density takes the form:

/,-(ii,o=-5X- dE-+kT-d ... p(n,t) (1.13.5.29) dtjj dtjj

and we obtain, using Eqs. (1.13.5.12), (1.13.5.19), and (1.13.5.20) for the stationary current density, i.e., dp/dt = 0,

JMV) = J—TMijUjPe,(in)e 2kT • (1.13.5.30) \L7t j

We now take advantage of the condition stated above that the flux over the barrier emanates from a small region around C (the saddle point or col). We integrate the current density over a plane containing the saddle point but not parallel to the flow of particles. The plane u = 0 will suffice here. The total current, that is flux of particles, is

;' = ! J JiUnndSi. (1.13.5.31) ' w=0

Using Eq. (1.13.5.31) with the approximation of Eq. (1.13.5.15) for the energy near the saddle point, the integration for the total flux (current) now yields after a long calculation [67]:

Y . .U:MUU : „ , , , „ X| —1/2 „ ,,,_, ; ~ ''J J ' X,-; U^Ujdct^zkTr'E'j e-EcKkT) .(1.13.5.32)

ZJ/L lit

From Eqs. (1.13.5.26) and (1.13.5.27), we immediately obtain for the current

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Chapter 1. Historical Background and Introductory Concepts 109

/ = -^±-\<tet((2zkTTlEsfU2e-Ec/(kT). (1.13.5.33)

Now, we assume that the energy function near the bottom of the well A may again be written in the quadratic approximation

E = EA+±Xav(l'-rf)(*J-1j) d-13.5.34) 1 'J

and we write EA = (aij) (1.13.5.35)

so that the number of particles in the well is (details in [67])

nA =det[(2Mr)-'EA]]"' / 2Z-1 . (1.13.5.36)

Now the escape rate F, by the usual flux over barrier method, is defined to be

T=jlnA (1.13.5.37) and so from Eqs. (1.13.5.33) and (1.13.5.36) in terms of the positive eigenvalue A+ of the set of noiseless Langevin equations linearised about the saddle point

' - A \teWA0-Ecnm^ (1.13.5.38) 2;ZM detEs

which is Langer's [68] expression in terms of the Hessians of the saddle and well energies for the escape rate for a multidimensional process in the IHD limit. The result again pertains to this limit because of our postulate that the potential in the vicinity of the saddle point may be approximated by the first two terms of its Taylor series so that, once again, that result fails for very small damping because the region of deviation from the Maxwell-Boltzmann distribution in the depths of the well extends far beyond the narrow region at the top of the barrier in which the potential may be replaced by its quadratic approximation. In passing, we remark that rate theory at weak friction is generally known as "unimolecular rate theory" [20] the VLD limit of Kramers treated earlier being an example of this. For a general discussion (see Hanggi et al. [20]).

1.13.6 Kramers' formula as a special case of Langer's formula

As an example of the application of Langer's method, we shall use the method to derive the IHD result of Kramers. To recover the Kramers formula, Eq. (1.13.4), by Langer's method, we take N=\, thus the state variables are the position and momentum, so that

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110 The Langevin Equation

Vi=x, TJ2 = P- (1.13.6.1) The noiseless Langevin equations are

x = —, p = -Pp- — . (1.13.6.2) m dx

Here V denotes the potential energy and (5 = C,lm is the friction coefficient. Now, the energy E is

2

£ = — + V(x). (1.13.6.3) 2m

(1.13.6.4)

Thus, Hamilton's equations are dE__P_ dE _dV dp m dx dx

and • P dE dE . a dV adE dE 7ll=^ = — = - — , TJ2=-J3P- — = -mj3- — . (1.13.6.5)

m dp drj2 dx dr]2 OTJ

Hence, we have the equation of motion in terms of the state variables jJi,rj2) of the general case of Langer's method above, as

0 1 YdE/drjA « k ,-x (1.13.6.6)

-I -mfiJldE/dj],) and so the transport matrix (My) is the negative of the matrix in Eq.

(1.13.6.6), namely

whence

T (0 -1 ^ M = -MT =\

[l -m/3) Here we can take the saddle point as the origin so

(1.13.6.7)

(1.13.6.8)

7 i 5 =0. (1.13.6.9)

(1.13.6.10) The momentum of a particle just escaping is zero also, so

772s=0.

Equation (1.13.5.15) then yields the inverted oscillator approximation for the saddle energy

E = Ec-^eij(rji-TJf)(?]j-T]sj)/2. (1.13.6.11)

'j

However, we have chosen the saddle energy Ec = 0 and

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Chapter 1. Historical Background and Introductory Concepts 111

E = ^--l-mco2cx

2 =~ma^c(rh-Tfx)2+ - ^ 2 " ^ f • (1.13.6.12)

2m 2 2 2mv ' Thus in Eq. (1.13.6.11)

en=m(0^, e22 = -Mm, e12=e21=0, (1.13.6.13) which are the matrix elements of the saddle Hessian matrix.

We now determine A+. We have the linearised noiseless Langevin equation:

0 1

-1 -mfi,

f^c-S dE> ldT]x

dES/d7J2)

0 1

-1 -m/3

^-m(Q2c%

rj2/m J

' 0 1 '

-1 -mp -m(Oc 0

0 Mm Thus

( 0 1/ m mcOc -J3

V

(1.13.6.14)

(1.13.6.15)

or (with A denoting the transition matrix) f] = At| (1.13.6.16)

with secular equation det(A-/lI) = 0. (1.13.6.17)

We thus solve the secular equation, namely

A(A + J3)-O)l=0, (1.13.6.18) to find

A±=±^Jco2+jB2 Z4-/3/2. (1.13.6.19)

We pick the upper sign so that the solution (which is now always positive) corresponds to the unstable barrier crossing mode, hence

A+=ylo)c+p2/4-/3/2. (1.13.6.20)

Now the Hessian matrix of the saddle energy is given by

E * = • 1

2nkT

(mo?c 0

0 -Mm (1.13.6.21)

and the Hessian matrix of the well energy is in like manner given by

EA = * 2nkT

Thus, the Hessians are given by

f 2 n 1 mcoA 0

0 Mm (1.13.6.22)

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112 The Langevin Equation

detEc =-col/(2xkT), dQtEA = co2A/(2nkT) (1.13.6.23) Jc

and so

^detEA /|detE5 \ = a)A/a>c. (1.13.6.24)

The escape rate is given by Eq. (1.13.5.38), which for the problem at hand becomes

p _ ^^A -AVI(kT)

2ncoc 2n L Jl + /32/(44)-j3/(2coc) e-AV«kT\ (1.13.6.25)

Equation (1.13.6.25) is Kramers' IHD Eq. (1.13.4). We will return to Langer's method when we discuss magnetic relaxation in Section 1.18.2 The various applications of the theory of Brownian motion in a potential and of the Kramers theory are summarised in the following section.

1.14 Applications of the Theory of Brownian Movement in a Potential

Amongst the physical phenomena to which the theory has been applied are:

1. The current-voltage characteristics of the Josephson junction. 2. Dielectric and Kerr-effect relaxation in liquids and in

molecular and nematic liquid crystals. 3. The mobility of superionic conductors. 4. Linewidths in nuclear magnetic resonance. 5. Incoherent scattering of slow neutrons. 6. Cycle slips in second-order phase-locked loops. 7. Quantum noise in ring laser gyroscopes. 8. Thermalisation of neutrons in a heavy gas moderator. 9. The photoelectromotive force in semiconductors. 10. Escape of particles over potential barriers. 11. The line shape of single mode semiconductor lasers. 12. Motion of single domain charge-density wave-systems. 13. Dynamic light scattering. 14. Magnetic relaxation of single domain ferromagnetic particles

(superparamagnetism). 15. Magnetic relaxation in ferrofluids. 16. Polymer dynamics. 17. Fluorescence depolarisation.

The applications listed under the headings 3,4, 5, 8,9,11-13, and 16 have been summarised in the review article [5] and in Risken's book [13] in the context of the Fokker-Planck equation for the Brownian motion in a periodic potential. Applications 1, 6 and 7 all rely on the

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Chapter 1. Historical Background and Introductory Concepts 113

same Langevin equations for Brownian motion in a tilted periodic potential, which is described in the context of the Josephson junction in Chapter 5 (see also [13]). Applications to laser spectroscopy are described in [13,63,64]. The reader is also referred to Gardiner's Quantum Optics [31]. The various applications to radio engineering are described by Stratonovich [32]. Fluorescence depolarisation of molecules in solutions is discussed in Ref. [100] as are the applications to polymer dynamics. Dynamic light scattering and nuclear magnetic relaxation in liquids are described by Berne and Pecora [103] and McConnell [104], respectively. The application to escape of particles over potential barriers is the transition state theory of Kramers [13,19,20] which has been described fully in Section 1.13. The magnetic relaxation of single domain ferromagnetic particles and the Kramers transition state method are closely interlinked so we shall summarise the application of the Kramers theory to superparamagnetism in Section 1.18. The application of the theory to dielectric and Kerr-effect relaxation in liquids and nematic liquid crystals is discussed in Chapters 3, 4, 7, 8, and 10. In passing, we finally note that possible new applications of the methods described in the book are the fluctuations of a rubidium atom on the surface of a fullerene cage and orientation moments of spheroids in simple shear flow. These applications are described in Refs. [93,94]. Yet another modern application is to the phenomenon of stochastic resonance which is illustrated in Section 1.21. The simplest example of the rotational Brownian movement is the Debye theory of dielectric relaxation [33,34, 100,102], which we shall summarise before proceeding to our discussion of superparamagnetism.

1.15 Rotational Brownian Motion — Application to Dielectric Relaxation

The Debye theory of dielectric relaxation has as its starting point the Fokker-Planck equation for the rotational Brownian motion in space of a sphere, when the inertia of the sphere is neglected. A detailed derivation of this equation is given by Debye [33]. We shall give here a derivation of his equation based on the vector Euler-Langevin equation of Lewis et al. [35]. This method has the advantage that it can easily be extended to include the dipole-dipole interaction between the polar molecules as well as crystalline anisotropy which is important in the application of the theory to nematic liquid crystals [36,37] (see Chapters 7 and 8).

We study the rotational Brownian movement of a sphere, which is supposed homogeneous, the motion being entirely due to random

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114 The Langevin Equation

couples that have no preferential direction. The sphere contains a rigid electric dipole ft. Then the rate of change of \i(t) is

|i(0 = <o(Oxn(0, (1.15.1) where w (/) is the angular velocity of the body. We remark that Eq. (1.15.1) is a purely kinematic relation with no particular reference either to the Brownian movement or to the shape of the body. We specialise it to the rotational Brownian motion of a sphere by supposing that w obeys the Euler-Langevin equation

/<b(0 + £ w(0 = X(t) + |i(0 xE(0- (1.15.2) In Eq. (1.15.2), / is the moment of inertia of the sphere, £<o is the damping torque due to Brownian movement, and X(t) is the white noise driving torque, again due to Brownian movement so that k(t) has the following properties:

4 ( 0 = 0, (1.15.3)

^(t)Xjt') = 2kT£dijdt-f), (1.15.4)

where the indices i,j = 1,2,3 in Kronecker's delta fy correspond to the Cartesian laboratory coordinate axes X, Y, Z. The term p x E(0 in Eq. (1.15.2) is the torque due to an externally applied electric field. The overbar means a statistical average over an ensemble of dipoles which, bearing in mind our interpretation of the Langevin equation as an integral equation, start at time t with the same angular velocity <o and same orientation ji.

Equation (1.15.2) includes the inertia of the sphere. The noninertial response is the response when / tends to zero or when the friction coefficient £ becomes very large. In this limit, the angular velocity vector may be immediately written down from Eq. (1.15.2) as

<o(t) = Cl[Ut) + \i(t)x'E(t)]. (1.15.5) We combine this with the kinematic relation Eq. (1.15.1) to obtain

C\i(t) = Ut)xp(t) + [ti(t)xE(t)]x\i(t), (1.15.6) which, with the properties of the triple vector product, becomes

cm=ut) x ii(o+//2E(o - (n(o • E(O) (i(o (1.15.7) This is the Langevin equation for the motion of u in the noninertial limit. It should be noted that Eq. (1.15.7) is also valid for a linear molecule rotating in space [8].

Equation (1.15.6) refers to one selected dipole. We may now use that equation to write down the Fokker-Planck equation in spherical polar coordinates (see Fig. 1.15.1) using the intuitive method of Section 1.2 above. As we saw in that section the current density Jd in the absence of thermal agitation is

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Chapter 1. Historical Background and Introductory Concepts 115

: _ ^ K

Fig. 1.15.1. Spherical coordinate system.

Jd=Wii, (1.15.8)

where u is a unit vector along \i and W(&, (p, t) is the density of dipole moment orientations on a sphere of unit radius. The orientation of the unit vector u is described by the polar angle $ and azimuth (p with ux = sin t?cos (p, uy- sin t? sin (p, uz = cos •&. The applied electric field E(f) is the negative gradient of a scalar potential V so that

AW dV dV l dV

E ( 0 = -gradV = • or - e * -d& sin^d<p (1.15.9) = Erer + E#ea + E(peip,

where e# is a unit vector in the direction of z? increasing, ep is a unit vector in the direction of (p increasing and er is a unit vector in the direction of the radius vector r increasing. The vector products in Eq. (1.15.6) are then in spherical coordinates

e, c t9

0

Em

u x E = -a

0 ~ Epe# + Eae?

and

(uxE)xu = -Eq> Ei

0 0

— E&e& + Eq>ep '

so that the drift current density is

J„ = 1 dV

-e,, + -1 dV

Cldtf v sin^a^ W.

(1.15.10)

(1.15.11)

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116 The Langevin Equation

Equation (1.15.11) refers to the current density in the absence of thermal agitation, which has been calculated from Eq. (1.15.6) neglecting the noise term. In order to take account of the thermal agitation we add to Jd

a diffusion term Jdiff=-DgmdW, (1.15.12)

the tendency of which is to make the density of dipole moment orientations more uniform. We then have for the components of the total current density, on writing out the gradient operator in spherical polar coordinates, using the total current J = Jd + Jdiff,

WdV dW + D

C 3z? 3z?

J& -

The continuity equation

where

divj =

Jv=~ W dV D dW

- + -£"sinz?3#> sinz? dtp

W + divJ = 0,

1 sinz? S F C ' - * ) ^ '

(1.15.13)

(1.15.14)

(1.15.15)

then yields the Fokker-Planck (here the Smoluchowski) equation

1 3 ( . „„dV\ 1

dt C sintf3z^ 3t? + -sin2 z? dtp d<p

where the operator A given by

1 d A = 1 sinz?—

s i n z ^ ^ l d$ + -

1

,(1.15.16)

(1.15.17) sin2 z? d<p2

is the angular part of the Laplace operator. Now, in equilibrium dW/dt = 0 so that W must reduce to the Maxwell-Boltzmann distribution

W0 = Ae-V^lkT)

(A is a normalising constant). Substituting the above equation into Eq. (1.15.16), we then find that the rotational diffusion coefficient D is

D = kT/C- (1.15.18) If we define the Debye relaxation time by

TD=l/(2D), (1.15.19) for reasons which will become immediately apparent, Eq. (1.15.16) takes on the form

2T, dW

' dt = AW+' 1_

kT

i a r sinz^dz?

sinzW av . (1.15.20)

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Chapter 1. Historical Background and Introductory Concepts 117

We shall now follow Debye [33] who specialised this equation to a uniform field E applied along the Z axis (so that V=-juEcos&), Eq. (1.15.20) then becomes

dW(0,t) 1 2T, D dt sin7?3z?

sint? "'MAfflM (1.15.21) Bz? kT

which is the equation obtained by Debye [33]. This is simply the Smoluchowski equation written in spherical polar coordinates. We remark that Eq. (1.15.21) is also the correct form of the Smoluchowski equation for prolate or oblate spheroids as may be seen by writing out the Smoluchowski operator in these coordinates (for a good description of spheroidal coordinates see MacRobert [38]). The general solution of Eq. (1.15.21) which is finite at the poles of the sphere is

oo

W(#,t) = Zan(t)Pn(cos#), (1.15.22) n=0

where the Pn are the Legendre polynomials [38] and the an(t) are functions to be determined. However, as far as dielectric relaxation is concerned, we are generally only interested in the linear approximation to the solution. Thus for the after-effect solution in which the steady field is suddenly switched off at time t = 0, we may assume that W has the form with E = E0

W(#,t) = — An

r l + ^-g(t)cos&

kT

//(cosz?) = J 2 * j V w c o s t f s i n t f d ^ = ^ - ^ e ~ ' / r ° . (1.15.24)

which on substitution into Eq. (1.15.21) yields

g(t) = e-"r". (1.15.23) Thus the mean dipole moment is taking account of the azimuthal angle (p

r2* r „w „„„ ,«„;„ ^ M m _ M2E0

3kT Likewise we may deduce that when the field E is alternating so that E = Em e'°"; the value of the mean dipole moment is

/ / 2 F plm

//(costf) = - - a . (1.15.25) 5kT 1 + i(OTD

We see at once that there is a difference in phase between [X (cos i9) and E. This phase difference persists if in place of Em e"°! we take its real and imaginary parts Em coscat or Em s'mcot. Debye made an estimate of the relaxation time TD by assuming that the Stokes formula for the frictional torque on a rotating sphere ^ = 8mja3 applies to the dipole (rj is the viscosity of the liquid, a is the radius of the sphere). For water at room temperature, where rj = 0.01 Po and assuming that a = 2 x 10~8 cm,

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118 The Langevin Equation

he found £ to be 2 x l 0 2 4 which yields a relaxation time TD of 0.25 x 10"10 s. Thus the maximum absorption should occur in the microwave frequency region. This is the principle of operation of the microwave oven.

1.15.1 Breakdown of the Debye theory at high frequencies

As we have pointed out, the Debye theory of dielectric relaxation takes no account of the inertia of the rotating dipoles. Thus, as has been said repeatedly, that theory breaks down at very short periods. (The discussion that follows is largely due to Sack [40]). In order to achieve maximum clarity in what follows we now summarise the main findings of the Debye theory. First of all, by the preceding discussion it is apparent that if a system is subject to an alternating field

Emcos<y? = EmRe(e ,ft"), (1.15.1.1) which is not so large as to cause nonlinear effects, then the steady-state response may be described by a complex polarisability depending on frequency

aco) = a\co) -ia\ co) (1.15.1.2) such that the time-dependent dipole moment is

(jUz(tj)<xEm[a'(a))cosGM + a"'(co)smcot]. (1.15.1.3)

The coefficient cc"co) is a measure of the energy loss per cycle for a constant amplitude Em. For a system obeying the Debye equations the frequency dependence of aco) is given by

aco) = - ^ - (1.15.1.4) 1 + icoz

or

o'(fl)) = - ~ , a"(co) = =-=-. (1.15.1.5) l + G)2T2 1 + fflV

The dielectric loss a"(co) has a maximum for aru= 1. Experimental curves of (/co) vs. log a>usually show an absorption peak broader than that predicted by Eq. (1.15.1.5). This can be explained by a distribution, f(f),of relaxation times rso that we have

* * » = f I™*-. (1.15.1.6) or'(0) J l + im

We shall illustrate later how this distribution of relaxation times may arise naturally. Whatever the mechanism for the relaxation, however, the Debye equations, Eqs. (1.15.1.4) and (1.15.1.5), break down at very high frequencies since the absorption coefficient rc(co) yielded by them is proportional to

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Chapter 1. Historical Background and Introductory Concepts 119

log10(or)

Figure 1.15.1.1. Variation with frequency of the absorption coefficient Ka>). The solid line is >c(aj) for the Debye dielectric, Eqs. (1.15.1.5) and (1.15.1.7). The dashed line corresponds to what is actually observed.

K(0) oc coa\ai) (1.15.1.7) with a"(co) given by Eq. (1.15.1.5). They predict, for example, that in a dielectric a relaxation absorption in the audio or radio frequency range would entail an infinite absorption coefficient (the Debye plateau; see Fig. 1.15.1.1) for visible light with complete opacity in some extreme cases such as water "Black Water") [40] and an infinite rate of energy loss for an electron travelling through the material. They are also unable to account for the resonance or Poley [41,52,56-59] absorption peak occurring in the far-infrared or terahertz band of frequencies. This problem is discussed in Chapter 10.

Furthermore, in the time domain the Debye equations imply that an abrupt change in the field E would produce an instantaneous finite alteration of the rate of change of the dipole moment (\i), which is impossible in view of the finite rotational inertia of the dipoles. In general the product a>c/'(0) must tend to zero in the limit of high frequencies [52] which means that in the frequency range in which a(co) can be expanded in negative powers of CO, i.e.,

a(co) = a0+^ + -^T..., (1.15.1.8) id) (id))

the coefficient ax must vanish. This is compatible with a distribution of relaxation times of the type given in Eq. (1.15.1.6) only if the function a (log T) can take on negative as well as positive values, [40]. In the time domain the stipulation that 0)d'co) must vanish means that the Taylor expansion of the after-effect function must not contain terms of the order of If I. The above considerations show that in relaxation processes inertial effects become important at sufficiently high frequencies. Consequently, there have been many attempts to include them in the Debye theory [8,40]. The earliest attempt was that of Rocard in 1933 [followed by

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120 The Langevin Equation

Dimitriev and Gurevich (1946) and Powles (1948)] [40] who derived the equation now known as the Rocard equation

oXO) l + ia)T-co2Tl/£' which shows the desired return to transparency at high frequencies (the second characteristic time 7/^is of the order 10-13 s). The calculations were all based in one way or the other on the inclusion of an extra term in the Smoluchowski equation thus rendering them only partially correct. Indeed inertial effects can be consistently treated only on the basis of the Klein-Kramers equation which considers distributions in configuration-angular velocity space where both angular positions and angular momenta (or angular velocities) are taken as independent variables, or the inertial Langevin equation. The first investigator to adopt the approach based on the Klein-Kramers equation was E. P. Gross in 1955 [39], who considered the behaviour of rigid dipoles rotating about fixed axes in a viscous medium. The paper of Gross however, does not include a detailed description of how his results were obtained and the reader is referred to the work of Sack [40] who gave a detailed derivation of these results and extended the theory to rotation in space. McConnell [8] has succinctly reviewed all the calculations described above.

We have formulated the Debye theory of dielectric relaxation, utilising the vectorial method based on the kinematic relation of Eq. (1.15.1). This method has the advantage that it may be easily adapted to include the effects of a crystalline anisotropy potential as is required in the theory of dielectric relaxation of nematic liquid crystals [36] (see Chapters 7 and 8 for detailed treatment) and the effect of the electric dipole-dipole coupling between dipolar molecules. If both effects are included the potential energy would be (in the case of uniaxial anisotropy (see Section 1.17 below) for N similar dipoles with common anisotropy axis n

v=-Zft-E<-Z*V<fc-")2+ZE i=l i'=l i=l i<j

J*'

(1.15.1.10) where the first term is arising from the external field , the second term is the uniaxial anisotropy energy, and the third term is the dipole-dipole interaction energy, r,-,- is the vector separation, K is the anisotropy constant. Unless the dipole-dipole interaction energy is zero it is impossible to make the assumption that each dipole of the assembly behaves in the same way as is done in the Debye theory which ignores all interactions. Another advantage of the vector formulation is that it allows

l*«-|»; 3(^-r^)(nrr^)

tf 4

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Chapter 1. Historical Background and Introductory Concepts 121

of an easy comparison with the Langevin equation governing the behaviour of the magnetisation of single domain ferromagnetic particles such as the Landau-Lifshiftz or Gilbert equations [11,42], a topic which we shall now discuss.

1.16 Superparamagnetism — Magnetic After-Effect

In general, a particle of ferromagnetic material below a certain critical size (typically 150 A in radius) constitutes a single domain particle by which we mean [43] that the particle is in a state of uniform magnetisation for any applied field. If we denote the magnetic dipole moment of such a particle by fi and ignore the anisotropy energy and suppose that an assembly of such particles has come to equilibrium at temperature T under the influence of an applied magnetic field H, then we will have for the mean dipole moment in the direction of the field

(n-h) = M c o t h ^ - r 1 ) = / < ^ ) . d-16.1) where h = H / / / ,

£ = /iH/(kT) (1.16.2) and L(£) is the Langevin function [44,61]. The behaviour is exactly analogous to that of an electric dipole in the Debye theory of the static electric susceptibility [44] or the Langevin treatment of paramagnetism, the vital difference, however, is that the moment n is not that of a single atom but rather of a single domain particle of volume v which may be of the order of 104 - 105 Bohr magnetons so that extremely large moments and large susceptibilities are involved and hence the term superparamagnetism. There is no hysteresis, merely saturation behaviour, just as in a Langevin paramagnet as hysteresis is not a thermal equilibrium property. The superparamagnetic behaviour (or thermal instability of the magnetisation) occurs (letting the free energy per unit volume be V•&, <p), where fi and <p are angular coordinates which describe the orientation of the magnetic moment \i), if the differences in v V(&, q>) are very small in comparison with k T. Then the thermal agitation causes continual changes in the orientation of \i and in an ensemble of such particles maintains a distribution of orientations characteristic of thermal equilibrium. Thus the number of particles with orientations within solid angle dQ. = sm&dfidq) is proportional to e~vV"•kT)dQ.. Hence the overall behaviour is just like an assembly of paramagnetic atoms.

The theory that a ferromagnetic material consists magnetically of elementary regions, each magnetised almost to saturation in some direction, was first proposed by Weiss [43] in 1907. He assumed that these regions coincided with the crystals of which the material was composed, a point, which was subsequently refuted by Frenkel and

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122 The Langevin Equation

Dorfman and Heisenberg and Bloch [43], who realised that even a single crystal is comprised of these minute permanent magnets, now known as (magnetic) domains. However, the exact nature of domains remained under debate until 1935 when Landau and Lifshitz [43] discovered them to be in the form of elementary layers (see Figure 1.16.1).

Single domain particles will in general not be isotropic as is assumed in deriving Eq. (1.16.1) above but will have anisotropic contributions to their total energy associated with the external shape of the particle, imposed stress or the crystalline structure itself. If we consider the simplest anisotropy energy, which is the uniaxial one, then the total free energy of the particle, v V, will be (if the applied field H is assumed parallel to the polar axis; this assumption is discussed in detail in Chapters 8 and 9)

Vv = Kvsin23-jjHcos3 (1.16.3) so that the magnetisation curve will no longer be the Langevin function. However, the dominant term governing the approach to saturation will still be [4,43]

1-4T1- (1.16.4) Typical values of the anisotropy constant K are of the order 105 Jm~3. The discussion so far has been concerned with the equilibrium behaviour. We now have to consider magnetic after-effect behaviour, that is, under what conditions, an assembly of single domain particles can achieve thermal equilibrium in a time short compared with the time of an experiment. One way of achieving equilibrium is by physical rotation of

Figure 1.16.1 a) A ferromagnetic material, where the domains of the material are magnetised in the directions of easy magnetisation. The arrows indicate the direction of the magnetisation of each domain, where adjacent domains have opposite directions of the magnetisation, so that the overall magnetisation of the material is negligible. If we apply a uniform external magnetic field, in the direction shown, some of the domains with magnetic orientations perpendicular or opposite to the direction of the applied field, become unstable and quickly rotate to another direction of easy magnetisation in the same direction as the applied field. This is illustrated in Figure b. If we were to increase the applied field to the point of saturation, we would have the situation of Figure c.

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Chapter 1. Historical Background and Introductory Concepts 123

M

Figure 1.16.2 Diagram representing (a) an isotropic particle and (b) a particle with uniaxial shape anisotropy, where & is the angle between the magnetisation vector of the particle and the easy axis of the anisotropy. The stable orientations of the magnetisation M are then z?= 0 and i= K. The arrow indicates the direction of a longitudinally applied external magnetic field. The magnetic moment of the particle is ju = v Ms, where Ms is the constant magnitude of M.

the particles which can occur if they are suspended in a liquid carrier -this is a ferrofluid, here [43] the factor determining the rate of approach to equilibrium is the viscosity of the medium in which the particles are suspended - this mechanism may be treated using the Debye theory which we have illustrated for electric dipoles above. The mechanism is described in more detail in Section 1.19 below which deals with ferrofluids [45]. We remark that the conditions for the validity of the Debye theory are much better satisfied for a ferrofluid than for electric dipoles as the ferrofluid particles, due to their relatively large size, are a much closer approximation to idealised Brownian particles than are polar molecules. In addition the dynamical behaviour of the particles will manifest itself in the mid radio frequency region rather than in the microwave band.

In a solid, physical rotation of the particles cannot take place. However, in 1949, Neel [46,66] pointed out that if a single domain particle were sufficiently small, thermal fluctuations could cause its direction of magnetisation to undergo a type of Brownian rotation so that the stable magnetic behaviour characteristic of a ferromagnet would be destroyed. Here, since the relaxing entity is the magnetic moment inside the particle, the inertia of the particle will of course play no role because no physical rotation of the particle occurs unlike in Debye relaxation of a polar molecule. An example given by Brown [47] of a tape recording is of interest, we expect that if we put this recording on a shelf that it will stay in the same magnetic state and would be surprised if it suddenly jumped from being a recording of Beethoven to that of Brahms. In principle, however, [47] the apparent stability of the recording is only one of many local minima of the free energy; thermal agitation can cause spontaneous jumps from one such state to another. The apparent stability [47] (ferromagnetic behaviour) arises because our tape or magnet cannot

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124 The Langevin Equation

get from one magnetic state to another without passing over an energy barrier which is very large in comparison with kT. Thus, the probability per unit time of a jump over such a barrier is so small that the mean time we would have to wait for it to occur far exceeds our own lifetime so that we perceive stable ferromagnetic behaviour. However, if the barrier is neither very large nor very small in comparison with the noise strength kT (which is the case we shall be concerned with) the specimen neither remains in a single stable state for a long time nor attains thermal equilibrium in a short time after a change in field; it undergoes a change of magnetisation which is not completed instantaneously but lags behind the field. This is called 'magnetic after-effect' or 'magnetic viscosity' or Neel relaxation and occurs only for sufficiently fine ferromagnetic particles. In order to illustrate [43] the Neel mechanism, consider an assembly of aligned uniaxial particles in the presence of a field H, so that the potential energy is given by Eq. (1.16.3). Thus, the particles are fully magnetised along the polar axis which is the axis of symmetry. A sufficiently long time after the field is switched off, the remanence will vanish as

Mr(t) = Mse~"T, (1.16.5) which is the longest lived mode of the relaxation process. Here Ms is the mean magnetisation of a non relaxing particle, t is the time after removal of the field and t is the relaxation time. Neel [46,66] then suggested that the relaxation time is given by (TST theory)

r - l = / o e - * v * H - ) t ( L 1 6 6 )

where/o is roughly the frequency of the gyromagnetic precession [61] so that

Z o s V s l O ^ s , (1.16.7) so that by varying the volume or the temperature of the particles rcan be made to vary from 10~9 s to millions of years (to is often taken as small as 10_u s in practice)

The presence of the exponential factor in Eq. (1.16.6) indicates that in order to approach the zero remanence corresponding to thermal equilibrium, a sufficient number of particles (magnetic moments) must be reversed by thermal activation over the energy barrier Kv. The probability of such a process is proportional to e"

Kvl(kT) (cf. the Kramers transition state method above). For example when H = 0, Eq. (1.16.3) is a symmetric bistable potential with minima at t?=0 and t?=^rand a maximum at •&= n/2.

Neel's calculation of rwas later criticised by Brown [42] on two counts (this is the point where the subject matter of the book comes into

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Chapter 1. Historical Background and Introductory Concepts 125

play) (i) the system is not explicitly treated as a gyromagnetic one and (ii) it relies on a discrete orientation approximation. Brown [42] suggested that both these difficulties could be resolved by constructing the Fokker-Planck equation for the distribution of magnetic moment orientations on the unit sphere from the underlying Langevin equation. He was then able to find, using the Kramers method, an approximate formula for r i n the high barrier limit for the potential of Eq.(1.16.3) which agreed with Neel's formula except for the prefactor f0. This is discussed in detail in Sections 1.17 and 1.18 below.

It is apparent from Eq. (1.16.6) that the Neel relaxation time r depends exponentially on the particle volume, hence, there is a fairly well defined particle radius above which the magnetisation will appear stable. We consider the figures given by Bean and Livingston [43] for a spherical iron particle with uniaxial anisotropy Kv sin2# A particle of 115 A in radius will have a relaxation time of 1CT1 s at 300 K and hence the moment will relax almost instantaneously. A particle of 150 A on the other hand will have a relaxation time of 10 s and hence will be exceedingly stable (i.e., the moment will not reverse in this time; cf. our preceding example above). This situation corresponds to an energy barrier very large in comparison to kT where for any reasonable measurement time [42] we may ignore thermal agitation and may calculate the static magnetisation by simply minimising V with respect to the polar and azimuthal angles •&, (/)) for each value of an applied field H0. This is the well-known Stoner-Wohlfarth calculation [48]; it leads to hysteresis because in certain field ranges there are two or more minima and transitions between them are neglected. Here a typical potential would be [49]

vV&,(/)) = Kvsin2 &-juH0(costfcosyf+smy/sm&cos<p). (1.16.8) The polar axis k is the easy axis of magnetisation and the field H0 is applied in the x-z plane at an angle ^"to the easy axis. Thus, in general, there will be only a narrow range of particle sizes in which the relaxation time will be of the order of experimental times, for which measurable "magnetic viscosity" effects manifesting themselves as an observable change of magnetisation, lagging behind field changes would be expected. Bean and Livingston have given a rough measure of the size of the particle for transition to stable behaviour, taking T = 102 sec, they find that the energy is 25 k T. The temperature at which this occurs for a given particle is called the blocking temperature. They obtain sizes of 40 A for h.c.p. cobalt, 125 A for iron, 140 A for f.c.c. cobalt. We mention that in an assembly consisting solely of single domain particles the remanence at a given temperature should be a measure of the amount of material with particle volume greater than that just stable at this

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126 The Langevin Equation

temperature. Thus [43] by following the increase of remanence with decreasing temperature, we can find out how much material lies in various ranges of volume and so determine the particle size distribution.

It is interesting to recall that Neel [46] was led to his solid state mechanism of relaxation, that is rotation of the magnetic moment inside the particle due to thermal agitation, through the study of paleomagnetism where much work had been done on measuring the intensity and direction of the remanent magnetisation in rocks with the view that such data would yield information on the strength and direction of the earth's magnetic field at the time the rock was formed. As is well illustrated by the elegant exposition of Blackett [50],

"The detailed study of the natural magnetisation of rocks is likely to allow us to trace back to the beginning of geological time both the history of the earth's magnetic field and the motion of continental masses relative to each other and to the geographical pole. The ability of the magnetic rocks to "remember" an earlier magnetic field depends on their ability to exist in thermal equilibrium with the earth's field at a stage early in their formation but to be later "frozen" in a state of magnetisation stable against later changes of the direction and strength of the field".

Thus [43] the establishment of thermal equilibrium of the rock magnetisation may be accomplished by either the Debye or Neel mechanisms. The Debye-like mechanism occurs [43] in sedimentary rocks. There the particles align themselves with the direction of the earth's field by mechanical rotation, while the sediment is still wet and uncompressed. Later the sediment becomes relatively hard rock and this ferrofluid-like behaviour is lost, so that Debye relaxation can no longer take place and the magnetisation is thus stable against later changes in the earth's field, so preserving the direction of the earth's magnetic field from the epoch in which the sediment was laid down.

In igneous rocks, however, the material becomes magnetic by cooling through its Curie point [43] and thus there is no mechanical rotation of the particles. At high temperatures however, the ratio, barrier height / thermal energy, is such that Neel relaxation may occur. Clearly, this will be in the early stages of the formation of the rock (e.g., in mountain building periods). As the rocks cool, the magnetisation will become stable because the particles will have become cooled to a temperature below their blocking temperature and the magnetisation will be thus stable against later changes in the earth's field again preserving the direction of the earth's magnetic field from the epoch in which the rocks have formed. The relaxation times in this case are of the order of geological times. The rocks thus play the role of "magnetic fossils"

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Chapter 1. Historical Background and Introductory Concepts 127

(rocks from widely scattered parts of the world but of the same age showing the same magnetic patterns) giving the evidence of a past reversal of the earth's magnetic field, records of a time when the northern magnetic pole resided where the southern magnetic pole is today. Reversals are believed to have occurred as many as 25 times during the last 5 million years with the last such reversal having occurred about 730,000 years ago. In an apt phase coined in a popular article by Dr. William Reville the earth's magnetic chronicle is written in stone.

Another possible mechanism of magnetic relaxation in the presence of a potential barrier, which is of much current interest is macroscopic quantum tunnelling of the magnetisation so called because of the large number of spins involved, which was originally suggested by Bean and Livingston [43]. By this tunnelling, they mean the possibility of transitions of the magnetic moment at absolute zero from a state of complete alignment to a state of zero overall magnetisation, due to quantum tunnelling [51] of the magnetic moment through the anisotropy potential barrier (see [70] for a recent review of single particle measurements).

Before we proceed to the more sophisticated treatment of Brown based on the Langevin equation, we shall briefly describe the discrete orientation model for the calculation of the Neel relaxation time. We shall suppose that the energy barriers are so large in comparison with kT that the magnetisation is always along one of the directions (fy $) of easy magnetisation, nevertheless, the barriers are not so high as to preclude changes of orientation altogether. Thus, in orientation i, there is a probability Vyper unit time of a jump to orientation;'. The Vy depend on K, H, and kT. If we have a large number of identical noninteracting particles at the same T and H, the number of particles «, in orientation i then changes with time in accordance with the equation

Ai=H(vjinj-vijni^ (1.16.9)

where

£/»,•=«. (1.16.10) i

Let us now suppose that we have only two orientations which is a uniaxial crystal with free energy given by Eq. (1.16.3). Let 1 refer to the positive orientation and 2 to the negative one, then Eq. (1.16.9) reduces to

n\-~ni = vi\ni~vnn\- (1.16.11) Whence [53]

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128 The Langevin Equation

—(n2-nl) = -(v2l+v2l)(n2-nl) + (v12-v2l)n (1.16.12) at

so that nx and n2 approach their final values when v12 and v21 are constant

according to the factor e~(Vn+V2l)t, that is, with time constant

(v12+v21)_1 . (1.16.13)

If vfj is the frequency of oscillation of a particle in a potential well, the

probability per second for the flip of a particle from orientation i to orientation j is given by

Vy = v ^ - ^ - W ^ , (i = \,j = 2 or i = 2,j = 1), (1.16.14)

where V,- is the free energy density in orientation i and Vm is the free energy density at the top of the barrier between the orientations i and j ; v is as usual the particle volume. The frequencies Vy, if they vary with temperature, are assumed to do so slowly in comparison with the exponential factor and are often taken to be constant although Neel [46] has calculated them explicitly (v. Refs. [42,47,66]). We reiterate that

regardless of the precise form of v? , if the ratio v I T changes by a factor

of less than 3 in a certain critical part of its range the time constant, Eq. (1.16.13), changes from 10"1 to 109 s. Thus, to a good approximation [47], we may reaffirm that there is a critical volume vc such that particles with v < vc are superparamagnetic while those with v > vc have hysteresis loops. We remark that the discrete orientation model of overbarrier relaxation was originally proposed in the context of dielectric relaxation by Debye [33] and extensively developed by Frohlich [53].

1.17 Brown's Treatment of Neel Relaxation

The starting point of Brown's treatment of the dynamical behaviour of the magnetisation M for a single domain particle is Gilbert's equation [42], which without thermal agitation is

dM . . („ dM yMx\ H-77

dt \ dt

\\ (1.17.1)

In Eq. (1.17.1), ^is the ratio of magnetic moment to angular momentum (gyromagnetic ratio), 7 is a phenomenological damping constant,

dV H = - — - , (1.17.2)

dM

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Chapter I. Historical Background and Introductory Concepts 129

dv . dv . dv , dv = 1 + j + k - , (1.17.3)

3M dMx JdMy dMz

and V is the Gibbs free energy density (the total Gibbs free energy is v V). In general, H represents the conservative part and rjM the dissipative part of an "effective field". Brown now supposes that in the presence of thermal agitation, the dissipative "effective field" -77M describes only the statistical average of the rapidly fluctuating random forces due to thermal agitation, and that for an individual particle, this term must become

-77M + h(f), (1.17.4) where the random field h(f) has the properties:

h(0 = 0 (1.17.5) and

hi(tl)hj(t2) = (2kTrj/v)SijStl-t2). (1.17.6) Here <% is Kronecker's delta and the indices i, j = 1, 2, 3 correspond to the Cartesian axes X, Y, Z of the laboratory coordinate system. The overbars denote the statistical averages over a large number of moments, which have all started with the same orientation (tf0,<p0) in the configuration space (here we use the spherical polar coordinates, see Fig. 1.15.1). On assuming that the hj(t) obey Isserlis's theorem (Section 1.3), namely,

where the sum is over all distinct pairs, and for 2n + 1 h's

hi(t1)..Mhn+i) = 0. (1.17.8) Brown was then able to derive after a long and tedious calculation using the methods of Wang and Uhlenbeck [11,12,42] as elaborated upon in Section 1.9 the Fokker-Planck equation for the density of magnetisation orientations Wi, (p,t) on the sphere of radius Ms. This procedure may be circumvented however, using an alternative approach given by Brown which appears to be based on the argument of Einstein given above [Section 1.2].

In order to illustrate this, we first write (by cross multiplying vectorially by M and using the triple vector product formula) Gilbert's equation in the absence of thermal agitation as an explicit equation for M, namely (see Chapter 7, Section 7.3.1)

M = g 'M 5 (MxH) + /z ' (MxH)xM, (1.17.9) where

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130 The Langevin Equation

Y OCY g' = '-- , h' = ~ = ag\ a = rjyMs.

(l + a2)Ms (l + a2)Ms s

Equation (1.17.9) has the mathematical form of the earlier Landau-Lifshitz equation, namely

M = 7 ( M x H ) + - ^ ( M x H ) x M , Ms

which may be written from Eq. (1.17.9) by taking the low damping limit, a« 1 (usually, a lies in the range 0.01 to 1). On writing M = uMj, Eq. (1.17.9) becomes

3 3 u = -g 'M s (ux—V) + h'ux(ux—V). (1.17.10)

3u 3u Here instead of M we use the unit vector u, where the Cartesian coordinates are the direction cosines w,- of M so that 3/3 M may be replaced by M^d/du, where 3/3u means the gradient on the surface of the unit sphere [17] so that in the spherical coordinate system (Fig. 1.15.1), the operator 3/3u is

3 3 I d „ ,„ , „ ^ = T-*e#+—TH~V (1.17.11) 3u 3tf sin &d<p v

The instantaneous orientation ( # <p) of the magnetisation of a single domain particle may be represented by a point on the unit sphere (\,$,(p). The probability current of such representative points may be determined as follows. We have

^-f Wdv = -\ J ndS, (1.17.12) 3rv Js

where J is the current density of representative points across the surface S of the sphere, n is the unit vector normal to S, so that by Gauss's divergence theorem we again have the continuity equation for representative points.

—W + divJ = 0. (1.17.13) 3 /

Equation (1.17.13) states that representative points are neither created nor destroyed, merely moving to new positions on the surface of the sphere. Now in the absence of thermal agitation

J = Wu, where u is given by Eq. (1.17.10). Let us again add to this J a diffusion term -k' 3U W (k' is a proportionality constant to be determined later), which represents the effect of thermal agitation; its tendency is to smooth out the distribution, i.e., to make it more uniform. This intuitive

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Chapter 1. Historical Background and Introductory Concepts 131

procedure just as in the dielectric problem of Section 1.15 gives for the components of J (on evaluating u x 9U V, etc. in spherical polar coordinates)

J,« —

Jp

dv sin&d<p;

W + k ,BW

W + k' dW

sin?? d<p

(1.17.14)

(1.17.15) sinz?9^

Equations (1.17.14) and (1.17.15) when substituted into the continuity Eq. (1.17.13), now yield Brown's Fokker-Planck equation namely

dW

dt = k'AW + -

1

+-

sinz?3tf

d

s in •&

K dV

d& sine? 8 ^ ,

dV sin t? d<p ,

W

( 'dV + g — + W

(1.17.16)

sinz?3^

which may be written in the compact vector form as dW , f d „ 8 „,V ,,d r

dt • = k'AW + g'u —Vx—W

du ou + h'-

du w—v du

(1.17.17)

[the angular part of the Laplace operator A is defined by Eq. (1.15.17)]. The terms in g' and h' are the precessional (gyromagnetic) term (it gives rise to ferromagnetic resonance, usually in the GHz range) and the alignment term, respectively. The constant k' is evaluated by requiring that the Maxwell-Boltzmann distribution of orientations should be the stationary (equilibrium) solution of Eq. (1.17.16) so that

W0(#,(p) = Ae-vV(»-mkT\ (1.17.18) where A is a normalising constant. The imposition of the Maxwell-Boltzmann equilibrium distribution of orientations yields

k' = kTh'lv = \l(2rN), (1.17.19) where rN is a characteristic (diffusion) relaxation time (TN is of the order of 10"" - 1CT9 s). One can readily verify that for g' = 0 (i.e., when one may ignore the gyromagnetic terms) Brown's Eq. (1.17.16) has the same mathematical form as the rotational diffusion Eq. (1.15.20).

The early studies of the relaxation process [11,42] were mainly confined to the axially symmetric solutions of Eq. (1.17.16), i.e., when V = V(&) and W = W(i% t) (where the gyromagnetic terms automatically drop out of the Fokker-Planck equation). This is so, e.g., when we wish to calculate the Neel relaxation time (the time required to surmount the potential barrier) T for uniaxial anisotropy in the presence of an applied

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132 The Langevin Equation

field parallel to the easy axis. The Neel relaxation time was then calculated assuming that the process was dominated by a single relaxation mode (the barrier crossing or Neel mode) so that

r = \l\, (1.17.20) where Ai is the smallest non-vanishing eigenvalue of the axially symmetric version of Eq. (1.17.16) when it is converted to a Sturm-Liouville problem [11,42]. In the case of axial symmetry, it is easy to convert Eq. (1.17.16) to a set of differential-recurrence relations [11, 42] by expanding the distribution function W(i% t) as a series of zonal harmonics (Legendre polynomials). These may be arranged as an infinite set of linear equations [11]

X(0 = AX(0, (1.17.21) whence X\ may be determined as the smallest nonvanishing root of the characteristic equation

det(AI-A) = 0, (1.17.22) by taking a sufficiently large number of equations. The restriction to axial symmetry so that only the zonal harmonics are involved, simplifies the problem of solving Eq. (1.17.17) in two important ways.

(a) It radically simplifies the intricate manipulations of the spherical harmonics Yl m(&,<p), which are required in order to obtain the set of differential-recurrence relations as only one index /, the order of the spherical harmonics, is involved.

(b) The restriction to the single index / reduces the number of equations to be solved in Eq. (1.17.22), so eliminating the loss of precision in floating point calculations which so bedevils numerical calculations associated with the non-axially symmetric solutions [47,65].

The two most important problems, which involve non-axially symmetric potentials, are (i) uniaxial anisotropy, where the assumption that the applied field and anisotropy vector are collinear is abandoned [the potential of Eq. (1.16.8) is a special case of this, where the external magnetic field H0 is assumed to be applied along a longitude] and (ii) cubic anisotropy in which V becomes

V(t?,p) = £(sin2 2& + sin4 tfsin2 2<p)/4. (1.17.23) Here, the direction of the magnetisation of a particle is always defined by the two polar angles & and <p measured from one of the easy axes. Equation (1.17.23) is written for cubic anisotropy in zero applied field with [100] as easy axis where K>0. If the easy axis is [111] the same expression can be used with K<0 [47]. For particles with cubic anisotropy, the energy barrier between adjacent easy directions of magnetisation will appear in the exponent, the barrier is Kv IA for K> 0

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Chapter 1. Historical Background and Introductory Concepts 133

and Kv 112 for K < 0. We shall treat in detail these non-axially symmetric problems in Chapter 9.

1.18 Asymptotic Expressions for the Neel Relaxation Time

At the time Brown was writing (1963), the lack of advanced computing facilities [without which k\ cannot be calculated from Eq. (1.17.22)] compelled him to seek simple analytic formulae for X\ in the high energy barrier approximation. This was accomplished by utilising the Kramers transition state theory of Section 1.13 suitably modified for rotation in space and for in general a non separable Hamiltonian (as we shall see in Section 1.18.2), as the theory had originally been formulated by Kramers for translational Brownian motion.

1.18.1 Application of Kramers' method to axially symmetric potentials of the magneto-crystalline anisotropy

The solution for the escape rate for an arbitrary potential of the magneto-crystalline anisotropy will be given in Section 1.18.2 using Langer's method. It is instructive, however, to give at first the solution for the particular case of axially symmetric potentials, as it illustrates the application of Kramers' theory to the magnetic problem, where the escape rate has the interesting particular property that it is valid for all values of the damping parameter, a, unlike the mechanical problem treated in Section 1.13. This is a consequence of the fact that in magnetic relaxation for axial symmetry, the Fokker-Planck equation is always effectively a one-space-variable equation. In Kramers' mechanical problem, on the other hand, the underlying equation, namely, the Klein-Kramers equation, is always an equation in a two-dimensional state space and can only be converted to a one-dimensional equation in the limiting cases (VLD and IHD). We remark that the IHD case is only quasi one-dimensional by virtue of the introduction of the variable u = p-ax'. In magnetic relaxation, the three friction regimes of Kramers' problem, namely, VLD, the crossover region, and IHD will only appear when non-axially symmetric potentials are involved.

For axial symmetry, we have dJ^/dcp-0 referring to Eq.

(1.17.16), and in the stationary case also W = 0, hence, J# is constant. Thus, with Eq. (1.17.16)

d_

(1.18.1.1) and so

fc'sintf dW h'dV„?

d& k'dtf = k'e~/v sin tf—[<?^ w ] = const. = k'J

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134 The Langevin Equation

*|VV|=^ o r ^ = / j £ ^ + C ' . (1.18.1.2) 3??L J sin?? J sin??

where, in this instance, C is a constant of integration; in this section, the abbreviation /?= v / (kT) will be used. Suppose now that W vanishes at the boundary (i.e., particles which arrive at the boundary E = EC are no longer counted) so that W(E = Ec) = 0, that is, all the particles are absorbed. This forces C" = 0 and so

J=e»w/l^ or W = / < r H — • d-18.1.3) / J sin?? J sin??

The number of particles in the well is

= f V sin ??<*?? = ./ j V ^ s i n i ? f ^ — — ^ - d & (1.18.1.4) JO J J oinW' V '

N, 0 sin ??'

so that the mean first passage time Tis [72]

/ 0J J sin??'

This is the time to reach the top of the well provided all particles which reach the top are absorbed, which is the condition that W vanishes at ?? = ??m. In practice, a particle has a fifty-fifty chance of crossing which means that the escape time re is given by

re=2z (1.18.1.6)

that is the escape time is twice the mean first passage time. [This implies, supposing that the integrals are taken at the high barrier limit, that r = r^1 = (2T)~1 ]. These integrals may be approximately evaluated, using the method of steepest descents [72]. In order to accomplish this, we note that for the exact time to go from the well at ??= 0 to the top of the barrier at ?? = ??m we have

mJ]^^±l%-^^M. (U8.LT) 2xN I sin?? J o

Employing Kramers' argument in the manner described in Ref. [72], the integral is now evaluated in the limit of very high potential barriers. Since almost all the particles are situated near the minimum at ??= 0 (because we have imposed a delta function initial distribution), then ??" is a very small angle. The inner integral in Eq. (1.18.1.7) may then be approximated by

J e - ^ s i n ? ? ^ ? ? ^ tfe-WW'^Wdti. (1.18.1.8)

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Chapter 1. Historical Background and Introductory Concepts 135

The integral on the right has been extended to infinity without significant error since the particles are almost all at the origin. Because # ' is very small, the Taylor series in V(#') can be approximated by its first two nonvanishing terms [V'(0) = 0] . Hence, the integral on the right-hand side in Eq. (1.18.1.8) becomes

e-mo) r tfe-evm*'2<2d# = £ . (i.is.1.9)

0J pvm

Thus, the two integrals in Eq. (1.18.1.7) effectively decouple from each other in the higher barrier limit.

We now evaluate the outer integral in Eq. (1.18.1.7). We have near z?m

V(0)~V(0m)-\V"(0m)\(fi-0m)2/2 (1.18.1.10) and hence for the outer integral

V ^ f ^ ' j .-»«LW>M. (1.18„1> 0 S i n Z ? S i n ? ?m 0

The range of integration in Eq. (1.18.1.11) may be extended to -°° since the integral has its main contribution from values $m - e to ??m and almost no contribution from outside these values. Now on noting that

J e-(x-»)l^)dx = (j4^T2, (1.18.1.12)

we have

f « , = . (1.18.1.13) 0J sin?? p\j3V"(0m)\ sintfm

Hence, in the high barrier limit, the mean first passage time r(0) for transitions from the point domain (i? = 0) is

r(0) 1 V ^ eM(*m)-vm

*N PV'm^jSV'X&Jl sinz?m

Likewise, the time to go from the minimum at & = n to &m is

(1.18.1.14)

(1.18.1.15) rN /3V"(K)J\j3V"(#m)\ sinz9m

These are the normalised times to reach the barrier. The corresponding Kramers escape rates from well 1 (#= 0 ) and well 2 (z?= n) are

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136 The Langevin Equation

<K\-hf

*~ tf

Figure 1.18.1 The potential profile of the anisotropy energy Eq. (1.18.1.18), showing a maximum at •&= &m =cos - ' h and minima at z?= 0 and K. Particles in the shallower well are inhibited from crossing into the deeper well by the potential barrier of height o(\-ti) . The particles populating the deeper of the two wells however, must possess greater thermal energy to be able to cross into the shallower well, due to the elevated potential barrier height denoted by <j\ + ti) .

r i2=[2T(0)]_1 and T21 = [2T71)]~X (1.18.1.16) the overall escape time, i.e., taking account of crossings and recrossings, is in the high barrier limit given by

ie=Vn+rixT\ (1.18.1.17) For example, for the axially symmetry potential yielded by Eq. (1.16.3), viz.,

pV = Gsm2$-2hco&i?), (1.18.1.18) we have from Eqs. (1.18.14)-(1.18.18) (details in [67,72])

_ 3 / 2 n ,2s. i a (l-h )

which in the limit h —> 0, reduces to

(l + h)e-a(1+h)2 Hl-h^-V2 (1.18.1.19)

r = -:N4x a3/2e-°. (1.18.1.20)

Equation (1.18.1.20) is Brown's asymptotic formula [42] for the escape rate for the uniaxial potential of the magnetocrystalline anisotropy

v£sin2tf. We shall return to Eqs. (1.18.1.19) and (1.18.1.20) later in Chapters 7 and 8.

We now consider non-axially symmetric relaxation problems so that the various cases of the Kramers calculation will appear.

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Chapter 1. Historical Background and Introductory Concepts 137

1.18.2 IHD formula for magnetic spins

The application of Kramers' escape theory to the problem of magnetic relaxation in non-axially symmetric potentials has been given in detail by Smith and De Rozario [95], Brown [47], Klik and Gunther [96], and Geoghegan et al. [65] (all this work is described in Ref. [67]). Klik and Gunther [96] used Langer's method (described in Section 1.13) and realised that the various Kramers damping regimes also applied to magnetic relaxation of single domain ferromagnetic particles. In this section, we show in detail how Langer's method may be used to solve this problem. Again, we deal with an energy (or Hamiltonian) function, E (or H), with minima at points A and B separated by a barrier (saddle point) at C, see Fig. 1.13.1. We use spherical polar coordinates &, (p), where 1 is the polar angle and (p is the azimuthal angle as usual. The noiseless Gilbert Eq. (1.17.10) can be written as follows [47]

dH ,,( dH) • h ux

and takes the form in these coordinates

p = -hXl-p2)Hp-g'H(p, (1.18.2.2)

<p = g'Hp-h'a-p2TlH<p, (1.18.2.3)

where p = cos $ and the subscripts denote the partial derivatives. We linearise these equations about the saddle point and determine A.+ from the transition matrix as in the Klein-Kramers case of Section 1.13.6. Thus, expanding the Hamiltonian H(-E) as a Taylor series about the saddle point qr, ps = cos if), we obtain

u = -g'ux- / j ' j u x — xu (1.18.2.1)

H = E = HS + 1 Hl(p-psf+2Hsp<p(p-ps)(<p-<ps) + Hs

w(<p-<psf

(1.18.2.4) where the superscript denotes the taking of the values of the relevant functions at the saddle point. We remark, following Klik and Gunther [96], that the Hamiltonian is defined on a phase space which is a closed manifold [the space •&, (p) is the surface of a unit sphere] and thus a local energy minimum is surrounded by two or more saddle points, depending on the symmetry of the problem. The total probability flux out of the metastable minimum equals the sum of the fluxes through all the saddle points. In an asymmetric case, e.g., when an external field is applied, some of these fluxes become exponentially small and may safely be neglected. The total flux out of the metastable minimum is, in this

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138 The Langevin Equation

case, dominated by the energetically most favourable path. Now, if the

coordinates of the saddle point are i(ps, ps), then

dE

dp

dE

= Hspp(p-ps) + Hs

pip(<p-<ps),

^=HUp~psh<(<p-<psy Equations (1.18.2.2) and (1.18.2.3) yield

Thus

,dE ,,dE . JfdE ,dE <p = g- h—, p = -h- g—.

dp d(p dp dq>

'9\_(-h' g'YdEldq?

J) .S' -h')\dEldp/

(1.18.2.5)

(1.18.2.6)

(1.18.2.7)

(1.18.2.8)

So the transport matrix M and the matrix M (see Section 1.13.5) are given by

'h' -g'] m-„ f - h ' -g^ -h'

M = / h' J

M = 8

(1.18.2.9)

Thus the linearised Eq. (1.18.2.8) has the form of the canonical Eqs. (1.13.5.3) and (1.13.5.4), and so Langer's IHD expression, Eq. (1.13.5.38), may be used to calculate the escape rate. The equations of motion linearised at the saddle point may be written as

4> = gH%(p-pS) + H^-9s)]-h'[H^(p-ps) + Hs

w(g>-<ps)_,

(1.18.2.10)

(1.18.2.11) Now, let the saddle point of interest lie on the equator p = 0 and make the transformation <p -^<p- q?, so that the above equations become in the notation of Klik and Gunther [96]

<p = gH%p + H%<p]-h'[H%p + H<»v], (1.18.2.12)

P = -hlH%p + H%<p]-g'[H%p + H%<p], (1.18.2.13)

or, in matrix notation,

^ ( g'Hm-h'Hm <P g'Kl~h'H

P<P

" p<p o l l ifxp •h'Hm-g'Hm

nnpp 5np<pj

<P

J) (1.18.2.14)

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Chapter 1. Historical Background and Introductory Concepts 139

[the superscript (1) denoting evaluation at the saddle point]. Equations (1.18.2.12) and (1.18.2.13) are the noiseless Langevin equations linearised at the saddle point given by Klik and Gunther [Ref. 96, their Eq. (3.2)]. The secular equation of Eq. (1.18.2.14) then leads to (just as in Section 1.13.6)

4=4[«»+^H >-K+HST -V + * l

The Hessian matrix of the system is

\HP<P

HA p<p

H

llpp w \ w)

(1.18.2.15)

(1.18.2.16) pp J

and the Hessian itself is negative at the saddle point, thus, to ensure a growing disturbance at the saddle point, we must again take the positive sign in Eq. (1.18.2.15). The square of the well angular frequency is [the superscript (0) denoting evaluation at the minimum]

< = f Mj I

(0)H(0) _(H(0)) HKU)H pp P9 I

while the squared saddle angular frequency is

a>l = T2

Mi ( l )r/( l ) mm. PP <P9 -«f

(1.18.2.17)

(1.18.2.18)

which, with Langer's Eq. (1.13.5.38), leads to the Klik and Gunther result [96]

r = A^L iKCOr

-PEC (1.18.2.19)

This formula underlines the power of Langer's method and shows clearly how, once the potential landscape is known, all quantities relating to the IHD escape rate may be calculated. We now choose a system of local coordinates, (p,p), in the vicinity of the saddle point (where Hp(p=Q). Then writing a = h'lg', we obtain a more compact expression for A+ namely:

^ s " f W +HwYi[< -<? -«*-2H%H% ,(1.18.2.20)

where we remark that the a' term represents the effect of the preces-sional term g' in the Gilbert equation on the longitudinal relaxation. This mode coupling effect is always present in a nonaxially symmetric

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140 The Langevin Equation

potential as the smallest eigenvalue of the Fokker Planck equation will always intrinsically depend on the damping unlike in axial symmetry.

Equations (1.18.2.19) and (1.18.2.20) were also derived from first principles, without recourse to Langer's work by Brown in 1979 [47] and have been reviewed by Geoghegan et al. [65]. In Brown's calculation [47], the Hamiltonian, Eq. (1.18.2.4), is diagonalised so that [65]

H = HS +-[Ci<p2+c2p2~\, (1.18.2.21)

where c\ and c2 are the coefficients of the second order term of the Taylor series of the expansion of H at the saddle point. Thus, ignoring the second well of the bistable potential, Brown's result [47] reads (cf. Eq. (5.60) of Geoghegan et al. [65], where a detailed derivation is given)

r=-*l An

-Ci-c2+Jc2-Clf-4^ iW-e-m-*), (L18.2.22) C 1 C 2

where c[l) and c2l) are the coefficients of the second order term in the

Taylor series expansion of the energy in the well. We have illustrated the application of Langer's method by considering the solution of the original Kramers particle problem and the solution of the problem of magnetic relaxation of single-domain ferromagnetic particles. Langer's method provides a powerful means of calculating the escape rate in the IHD case for a multi-dimensional potential. All that is required being to evaluate the determinants of the energy function at the bottom of the well and at the saddle point. The positive eigenvalue, A+ (which effectively gives the correction to the TST result) characterising the deterministic growth rate of a small deviation from the saddle point (known [20] in the chemical physics literature as the Grote-Hynes frequency) is simply obtained from the secular equation of the transition (system) matrix of the noiseless equations of motion linearised about the saddle point. The eigenvalue formally corresponds to the left eigenvector U, of the matrix MES formed from the transport matrix of the coefficients of the saddle energy and is formally obtained from the matrix ME using

-UMES = A+V. In practice, this procedure will be quite difficult to carry out because of the problem of determining the stationary points and the second derivatives of the potential at the stationary points corresponding to the well and saddle point energies. The reader is referred to Geoghegan et al. [65] where the calculation is explicitly carried out for a uniform field at an angle to the easy axis of magnetisation of a single-domain ferromagnetic particle.

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Chapter 1. Historical Background and Introductory Concepts 141

Equation (1.18.2.22) derived for a non separable Hamiltonian, which is the free energy, applies like the separable Hamiltonian result Eq. (1.13.1.19) when the energy loss per cycle of the almost periodic motion at the saddle point energy Ec, namely AE» k T. If AE « k T, one may prove using the method of first passage times (full details in [67]) that for the escape from a single well

r - > = r ^ ~ - ^ ^ v . (1.18.2.23) w 0)AAE

Hence,

AE = aj> (l-p2)Hpd<p-[l-p2Jl H9dp , (1.18.2.24)

which on inversion is effectively the same as the corresponding Kramers result for mechanical particles Eq. (1.13.2.13).

We remark that Eqs. (1.18.2.22) and (1.18.2.23) may be used to verify experimentally the Kramers theory for magnetic particles. This has been accomplished using the sophisticated single particle measurement techniques developed by Wernsdorfer [70]. In the frictional crossovers region, AE~kT, where neither formula applies, a bridging formula similar to Eq. (1.13.7) must be used in order to obtain accurate results for r . We further remark that a second bridging formula problem arises in the magnetic version of the Kramers problem, namely, how to join axially asymmetric and non-axially symmetric asymptotic expressions for the greatest relaxation time in the limit of small departures from axial symmetry. This problem has been described in detail in Ref. [67] by considering the asymptotes generated by Eq. (1.16.8) for yf= 0 and y/±0.

We shall return in Sections 1.20 and 1.21 to the Kramers theory in connection with two universal effects, which occur in bistable potentials. These are (i) the effect of a uniform bias force on the relaxation time and (ii) the stochastic resonance phenomenon. In the meantime, we shall briefly indicate how the formulae developed in Section 1.18 may be applied to ferrofluids.

1.19 Ferrofluids

Ferrofluids are [45] stable colloidal suspensions of single domain magnetic particles in a liquid carrier. These liquids are composed of small (-150 A) particles of ferromagnetic material, coated with a molecular layer of a surfactant and suspended in an ordinary liquid. The coatings prevent the particles from sticking to each other, and thermal

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142 The Langevin Equation

agitation keeps the particles suspended because of the Brownian motion ensuing from it.

In the simplest model of a magnetic fluid, it is considered to be a gas of non-interacting ferroparticles suspended in a liquid carrier. The magnetic properties of such a system are analogous to those of a paramagnetic gas: the magnetisation curve is described by the Langevin function, the static susceptibility %, by Curie's law % ~T _1 ; and its dispersion, by the Debye formula

X(a>) = 1 % \ + ianc '

where Tc is a characteristic relaxation time. In ferrofluids, both Debye relaxation with the Debye relaxation time rD (due to mechanical rotation of the fluid particles) and Neel relaxation with a characteristic time T (due to rotation of the magnetic moment inside the particle) may occur. Theoretical studies have assumed [45] for the most part that the Debye and Neel mechanisms may be treated separately, that is, one has two extreme types of behaviour: (a) the Debye relaxation mechanism, where the Neel relaxation mechanism is blocked or frozen in the particle, and (b) where the mechanical rotation of the particles is frozen and only the Neel mechanism is operative. The overall characteristic relaxation time Tc is then supposed to obey the equation

JTD__ ( 1 1 9 1 )

T + TD

A popular formula used in analysis, for example, is Brown's formula for r, which is the inverse of the escape rate F given by Eq. (1.18.1.20) combined with the Debye time TD. The relaxation process has been described succinctly in Ref. [45] as follows:

"There is a finite coupling between the orientation of the magnetic moment n of a ferroparticle and the position of the particle itself (orientation of its crystal axes). Without this coupling the moment u would be similar to a compass needle, where rotation of the instrument frame does not influence the behaviour. Because of the above coupling, the reorientation of the vector |i may take place in two different ways: (1) rotation of u within the particle with respect to its crystal axes, and (2) rotation of \i together with the particle with respect to the liquid matrix. Both processes - they proceed simultaneously - are of the rotary diffusion type. The efficiency of the internal (Neel) diffusion of the magnetic moment strongly depends on particle size, but that of the external one (the Brownian) depends strongly on the viscosity of the liquid carrier".

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Chapter 1. Historical Background and Introductory Concepts 143

Along with these, two other factors are of great importance: Polydispersedness of ferrofluids. The actual distribution of the

particle size in colloids results in a moderate (1-2 orders of magnitude) extension of the Debye relaxation spectrum and enormous (up to 13-15 orders!) extension of the Neel relaxation spectrum. Averaging with a particle size distribution function may change the low-frequency Debye susceptibility xi®) unrecognizably.

Blocking of the rotational degrees of freedom of the particles on solidification of a carrier liquid. As the liquid matrix freezes, the suspended particles lose their mechanical mobility. The Debye relaxation mechanism thus becomes ineffective.

We reiterate that one of the outstanding advantages of the ferrofluid system as opposed to polar molecules as a test of the Debye theory of relaxation is that the ferrofluid particles closely approximate in terms of size to actual Brownian particles unlike electric dipoles. Thus the conditions for the validity of the Debye theory are more closely satisfied by the ferrofluid particles.

In view of the above considerations a major development in the theory of ferrofluids would be of a model of particle reorientation which avoids the two extremes of a frozen Debye or a frozen Neel mechanism so taking account of both mechanisms simultaneously. Such a development has been inspired [60] by the recent discoveries of Fannin et al. [54,55] which suggest that both relaxation and ferromagnetic resonance behaviour appear in magnetic fluids. The "egg" model of magnetic fluids [45,60] which is a form of the itinerant oscillator model (and is treated in Chapter 10) represents an attempt to consider the composite behaviour.

1.20 Depletion Effect in a Biased Bistable Potential

In a biased bistable potential such as that shown in Fig. (1.18.1), the population in the shallower of the two potential wells may be substantially decreased by the application of the uniform bias force. This has a profound effect on the relaxation time because at a certain critical value of the bias force (which is much less than that required to destroy the bistable character of the potential) a switchover of the overall relaxation time from Arrhenius to non-Arrhenius behaviour will take place. Thus, the behaviour of the relaxation time ceases to be dominated by the inverse of the Kramers escape rate. The effect was originally discovered in the context of magnetic relaxation in 1995 [74] (see Chapter 8, Section 8.3.2) for the biased uniaxial anisotropy potential

yf?y = -cr(cos2*?-l-2/jcosz?) = -(7cos2tf-£costf (1.20.1)

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144 The Langevin Equation

by numerical solution of the Fokker-Planck equation. The solution of that equation for the integral relaxation time that is the area under the curve of the decay of the magnetisation indicated that at fields well below the critical field at which the bistable character of the potential is destroyed, the integral relaxation time ceases to be dominated by the smallest non vanishing eigenvalue X\ of the Fokker-Planck equation (or equivalently the smallest non vanishing eigenvalue of the hierarchy of recurrence relations generated by the Langevin equation averaged over its realisations). Thus, the inverse of the Kramers escape (which for sufficiently high barriers is proportional to /If1) no longer dominates the integral relaxation time for bias fields in excess of a certain critical field. Hence, the contribution of relaxation modes to the integral relaxation time, other than the barrier crossing one becomes significant. The effect having been discovered by numerical solution for the Fokker-Planck equation was later elegantly explained by Garanin [73] (see [72] for a review). It appears to be a universal feature of the relaxation in biased bistable potentials [72,74,75]. In the analysis which follows, however, we shall demonstrate using the magnetic anisotropy potential of Eq. (1.20.1) that the effect is essentially independent of the precise nature of the prefactor in the Kramers escape rate and may be predicted by the transition state theory coupled with the definition of the integral relaxation time and the partition function.

In Section 1.18.1, we have shown how Kramers escape rates may be derived using the method of first time passage times. The other time which is used to characterise the decay of a distribution of magnetisation or electric polarisation and which we have mentioned, is the integral relaxation time z (see Chapter 2, Section 2.9) This time is the area under the curve of the decay of magnetisation following a sudden change in the amplitude of an applied external stimulus such as a uniform magnetic field, etc. The integral relaxation time takes into account the contribution of all the modes of the decay of the magnetisation to the relaxation process and so is a global characteristic of the process. It is virtually identical with the escape time if the configuration of the system is such that the contributions of all the other modes, save the longest lived one, are negligible. In other cases, however, the integral relaxation time rmay differ exponentially from the escape time, and so may not be used to estimate the escape time ze. The two characteristic times, ze and rhave been extensively discussed in Refs. [67,72,74-76] and will be described in more detail later in this book. For our present purpose, it will be sufficient to consider the integral relaxation time for linear response which is identical with the correlation time (of linear response theory) as we now describe.

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Chapter 1. Historical Background and Introductory Concepts 145

We begin our approximate calculation of the integral relaxation time by recalling that escape over the barrier in an asymmetric bistable potential is a very slow process and the time dependence of the probability distribution is exponentially slow. This quasi-stationary behaviour (to all extents and purposes, the behaviour of the system appears stationary) allows us to make use of the partition function to calculate the dynamic quantities pertaining to the over-barrier relaxation process and ultimately to characterise the depletion effect. If we have a random variable, the quantities of interest are its expectation value and its variance and as far as the dynamics are concerned, the autocorrelation function (ACF). The autocorrelation function (Section 1.6) measures the correlation between the value of a random variable at time t\ = 0, and its value at time t2 = t. It may be shown from linear response theory [102] (see Chapter 2, Section 2.8) that the decay of the magnetisation, following a small change AH in the applied field is

b(t) = pAHMsUcosi(0)cos&(t))0-(cos#(0))20 , (1.20.2)

where the zeros on the angular braces represent the averages in the absence of the small perturbation AH. The quantity inside the square braces is the ACF of the longitudinal component of the magnetisation, c (t). Clearly, if we set t = 0, we have

c(0) = (cos2 0(0)) -(cos 0(0))*. (1.20.3)

This is the variance of the magnetisation. Moreover, the process is assumed stationary, which means that the statistics describing the process do not change over the course of time.

The important property associated with the ACF is the area under the curve of c (t), which is

1 °° T = \ct)dt, (1.20.4)

c(0) J where r is the correlation time, which is the integral relaxation time for linear response. In order to evaluate T, we would have to determine all the eigenvalues of the system which may be accomplished by solving the recurrence relations generated by the Fokker-Planck or averaged Langevin equations. However, if the barrier is high as in the present problem, we may suppose [75] that c (t) is the sum of the fast relaxation processes in the deep well plus the slow overbarrier process from the shallower well into the deep well. Thus (see Chapter 2, Section 2.13)

c(t)~Awe~"Tw + ABe~* (1.20.5)

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146 The Langevin Equation

and we note that c(0) = Aw+AB, (1.20.6)

where Aw is the population of particles in the deep well and AB arises from the population of particles crossing over the barrier. The relaxation time zw is associated with the fast relaxation processes in the well. The inverse of the well time z^ is approximately equal to 2<JZ^l + h) [73]. This equation follows from the fact that all the intrawell modes have approximately the same relaxation time, which is approximately the diffusion time divided by the effective barrier height. Ab which is the smallest non-vanishing eigenvalue of the Fokker-Planck equation, is essentially proportional to e~

ffl~h)2 [cf. Eq. (1.18.1.19)] by the Arrhenius approach and represents the escape rate over the barrier from the shallower of the two wells.

The simple analytical expression Eq. (1.20.5) may be used to determine the integral relaxation time, whence the depletion effect may be explained using approximate expressions for the partition function. We proceed as follows, we have

T - - U (Awe-^+ABe-*)dt = T"A"+A°/A*. (1.20.7) c(0)J

0 v ' Aw+Af i

Because escape over a barrier is a slow process, i.e., few particles escape due to the height of the barrier, Aw» AB , thus we can write

z = zw+(AB/Aw)A[l. (1.20.8)

Since /If1 is exponentially large in the high barrier limit, one would expect that z would be dominated by the second term, unless ABI Aw has negative exponential type behaviour so that z would be dominated by the fast relaxation time zw • We shall show that this is indeed the case if the reduced field h exceeds a critical value [73,75] and we shall further show that the behaviour is independent of the precise nature of the prefactor.

To illustrate this, we will derive an expression for Aw + AB based on our analysis of the ensemble averages using the partition function. We have from Eqs. (1.20.3) and (1.20.6),

Aw+AB=(z2)0-(z)20 (1.20.9)

2\ with z = cos*9. We can [73] evaluate (z)0 and (z )0 in terms of the

partition function Z given by

Z = ]e-pv*)&miH0=\eo*l+tldz. (1.20.10)

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Chapter I. Historical Background and Introductory Concepts 147

We have , . 1 dZ Z' , 1 2\ 1 d2Z Z"

Substituting these expressions in Eq. (1.20.9), we have

AW+AB=Z"/Z-(Z'/Z)2. (1.20.12)

The right-hand side of this equation is formally an exact expression for the variance.

Thus, the vital quantity in calculating averages (and hence the integral relaxation time), is the partition function Z from Eq. (1.20.10). We shall now indicate using the asymptotic results how approximate expressions for Z may be calculated for the biased uniaxial potential Eq. (1.20.1) which of course hold for potential barriers heights well in excess of kT. We have from Eq. (1.20.10) evaluated at the i'h minimum, i = 1,2,

Z(0i)~e-fivW/[l3V"0i)]. (1.20.13)

We now take the second derivative of the potential and make the appropriate substitutions for the numerator and denominator, thus we can express Z in terms of the dimensionless anisotropy and external field parameters a and £ respectively. On noting that (for convenience, we follow Ref. [73])

^ " ( t f ) = 2<7cos2tf-£cosz?, the evaluation of Eq. (1.20.13) at $= 0 will yield an expression for the portion of the partition function corresponding to the deeper well, which we call Z+, viz.,

Z+=ea+f/(2a + £). If we evaluate Eq. (1.20.13) at •&= K, where the particles are located in the shallower well, we will find the portion of the partition function corresponding to the shallower well which we call Z_:

Z_=e f f-*7(2o--£). The overall partition function, however, is effectively the sum of these two individual contributions, i.e., Z = Z+ + Z_ . The asymptotic method which we have employed is useful since it is difficult to evaluate the integrals exactly when anisotropy is included, because they are not expressible as elementary functions. (The exact solution for Z can be expressed in terms of the error functions of imaginary argument [72]).

The population AW + AB can now be evaluated using our high barrier approximation for Z,

7" ,j" ( 7' +7' \ AW+AB=±±±^- ±±±±^ . (1.20.14)

W B Z++Z_ [Z+ + Z_)

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148 The Langevin Equation

Z+ = / \ = AxeaH and Z_ = e = A2e

a^, (1.20.15)

The essence of our calculation is also to evaluate Aw + AB from our separate expressions for Z+ and Z_. It is important that we are mindful of the fact that very few particles cross over the barrier once the relaxation process has been initiated. Therefore, the variance may also be approximately estimated as being the sum of the variances in each well if it were isolated. The terms in this expression would not contain any exponential terms, as they will always cancel in a formula such as y7 y or y"ly, if y is given by a single exponential term, as is so, for each isolated well. Therefore, the difference between the variance of the overall process calculated by using the exact Z = Z+ + Z_ and the variance calculated by using Z+ and Z_ separately gives AB. Thus, by determining which terms correspond to Aw and, which correspond to AB, we can calculate the integral relaxation time t, as expressed by Eq. (1.20.7). For example, let us consider Z+ and Z_ as

'-— = V"* and z =—— 2cr + £ l " 2<7-£

where A\ and A2 represent the algebraic prefactors of this particular

system. It is obvious that Z~[dZ/d^ contains exponential factors while the isolated well moments Z+ldZ±/d^ do not. We need not be concerned with the precise form of Ai and A2, as they will simply be algebraic prefactors for any given system.

We now give our final result by considering the second term on the right of Eq. (1.20.14). The relevant representation is

Z'+ + Z'_= Z'+/Z+ + Z'_/Z_ (1.20.16) Z+ + Z_ 1 + Z_/Z+ 1 + Z+/Z_

If we ignore the irrelevant algebraic prefactors, then it is obvious that

Z_ /Z+ s e~2* and Z+ IZ_ = eH . (1.20.17)

By using these approximations, we can rewrite Eq. (1.20.16) as

K + z-^K+e-uiLm (1.20.18) z+ + z_ z+ z_

z"+z" Clearly then also, if we apply the same method to —± , we obtain Z+ + Z_

z"++z:„z; , ^ z : - + e-

Zf — . (1.20.19) z++z_ z+ z_

By substituting these two equations into Eq. (1.20.14), we have our final expression for Aw + AB, which is

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Chapter 1. Historical Background and Introductory Concepts 149

z 'z;f z: _„ _z; z: v z+y

+ f ^ e - 2 ^ _ 2 ± ± ± ^ e - ^ . (1.20.20) z_ z+ z

We remark that in writing this equation, we have declared an e~4^ term small and irrelevant. Equation (1.20.20) now yields by inspection, the terms, which correspond to Aw and the terms, which correspond to AB. Clearly, the first two terms in the right hand side correspond to the variance in the deeper well and must then correspond to Aw. The third and fourth terms of Eq. (1.20.20) must correspond to AB because, from our initial reasoning, they can only have their origin in the partition function Z of the entire system and so, must be due to the combined effect of the two wells.

We can thus infer that ABI Aw ~ e~2^ = e~4ah and X \x is of the order of exp [<x(l -2h)] [see Eq. (1.18.1.19)], so that the overall contribution of the second term is exp [cr(l -6h)], which verifies the analytical result first given by Garanin [73] in 1996 and confirmed by earlier (1994) [74] numerical calculations of the integral relaxation time. Thus, the relaxation time can change its sign from positive exponential behaviour to negative exponential behaviour if h= 1/6 = 0.17. Thus, the overall relaxation time switches from growing exponential behaviour (/i < 0.17), to being dominated by the (algebraic) first term, which pertains to relaxation in the deep well. Such behaviour appears to be a universal feature of bistable potentials subjected to a bias force, see Refs. [72,75]. Another more precise method of evaluation of the parameters Aw, AB, and % will be given in Chapter 2, Section 2.13.

The depletion effect we have just described is an example of a particular effect, which arises as a result of the bistable nature of the potential which is considered. Yet another effect, which may arise in systems which possess multistable states, is the phenomenon of stochastic resonance [69].

1.21 Stochastic Resonance

The mechanism of stochastic resonance which is intimately bound up with the Kramers escape rate, as it is rooted in a physical synchronisation between the inter-well (Kramers escape) time scale and the periodic time of the weak ac modulation which acts as an external clock, is relatively easy to explain [69]. We consider a Brownian particle moving in a symmetric double well potential V(x). The thermal forces due to the bath cause transitions between the neighbouring potential

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150 The Langevin Equation

wells with escape rate from one well to the other given in the heavily damped case (where the energy loss per cycle at the saddle point energy i s » i t r ) b y E q . (1.13.1.20), viz.,

T = SWc_e-ww)m ( L 2 L 1 ) 27T/3

If we now apply a weak periodic forcing of frequency Q, the double well potential will be tilted up and down periodically raising and lowering the potential barriers AV [69]. The periodic forcing is too weak to let particles roll periodically from one potential well into the other one, however, noise induced hopping between potential wells may become synchronised with the weak periodic forcing [69]. This statistical synchronisation takes place when the averaged waiting (escape) time (cf. Section 1.18.1 above)

Te(*D = l / r (1.21.2) between two noise induced transitions is comparable with half the period TQ of the periodic forcing. This yields [69] the time scale matching condition for stochastic resonance namely

2re(kT)=ra (1.21.3) Thus, stochastic resonance in a symmetric double well potential manifests itself in a synchronisation of activated hopping events with reaction rate described by Eq. (1.21.1) in the heavily damped case with the weak periodic forcing [69]. For a given period of the periodic forcing namely

Fig. 1.21.1 Double well potential function as used in stochastic resonance. The minima are located at A and B. In the absence of a periodic forcing function, the barrier heights AVi and AV2 are equal to AV so that the potential is symmetric. The periodic forcing function causes the double well potential to tilt back and forth thereby raising and lowering the potential barriers of the right and left wells, respectively in an antisymmetric cyclic fashion.

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Chapter 1. Historical Background and Introductory Concepts 151

r f l = 2 * / Q , (1.21.5) the time scale matching condition can be fulfilled by altering the noise level kTmax.

An attractive account of the discovery of stochastic resonance has been given by Gammaitoni et al. in Ref. [69]. According to Gammaitoni et al. [69], the stochastic resonance phenomenon was first noted by C. Nicolis and G. Nicolis [97] and Benzi et al. [98] (details in Ref. [69]) in a discussion of the problem of the periodically recurrent ice ages. In the model of Benzi et al. [98] formulated in 1981, the global climate is represented by a double-well potential where one minimum represents a well temperature corresponding to a largely ice-covered earth. The small modulation of the earth's orbital eccentricity (the orbits of the minor planets are nearly circular) is represented by a weak periodic forcing. Short term climatic fluctuations, such as the annual fluctuations in solar radiation are modelled by Gaussian white noise. If the noise is tuned according to Eq. (1.21.4), synchronised hopping between the cold and warm temperatures (i.e., the other potential well) could significantly advance the response of the earth's climate to the weak perturbations caused by the earth's orbital eccentricity, according to Benzi et al. [98]. A very complete account of stochastic resonance phenomena in physical and biological systems is given in Ref. 69. For example, stochastic resonance has been observed [69] in bistable ring lasers, analogue electronic simulators, in neurophysiology, where the firing of periodically stimulated neurons appears to exhibit stochastic resonance, in single domain ferromagnetic particles [77], etc.

In the context of single domain ferromagnetic particles with

bistable uniaxial anisotropy potential A\»sin # , the basic concept of stochastic resonance has been rather well described by Raikher and Stepanov [77]. In such a potential in the presence of noise, a weak alternating spatially uniform field of frequency Q favouring the Kramers transitions between the equilibrium positions at t? = 0, r is applied. Under these conditions the signal-to-noise ratio (of the magnetisation) determined from the spectral density <pM (a>) (see Section 3.2 below) of the magnetisation (i.e., the frequency response to the applied field) evaluated at the frequency Q of the weak applied ac field, first increases with increasing noise strength kT, then passes through a pronounced maximum and decreases again. This is the stochastic resonance effect whereby the periodic response in both amplitude and phase may be manipulated by altering the noise strength. For example, on decreasing the driving frequency Q, the peak amplitude of the periodic component

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152 The Langevin Equation

of the response of a system moves to smaller noise strengths [69]. Later (Chapter 8, Section 8.3.3), we shall illustrate the phenomenon in detail by describing the calculations for such single domain particles. The fact that the phenomenon occurs for these particles also indicates that one would expect stochastic resonance to be exhibited by nematic liquid crystals since Neel or longitudinal relaxation of single domain ferromagnetic particles in axially symmetric potentials is directly analogous to relaxation of nematics [71].

1.22 Anomalous Diffusion

The theory of the Brownian motion, which we have described, is distinguished by a characteristic feature namely the concept of a collision rate which is the inverse of the time interval between successive collision events of the Brownian particle with its surroundings; we recall the words of Einstein [2].

"We introduce a time interval rin our discussion, which is to be very small compared with the observed interval of time, but, nevertheless of such a magnitude that the movements executed by a particle in two consecutive intervals of time t are to be considered as mutually independent phenomena".

This concept which is based on a random walk with a well defined characteristic waiting time (thus called a discrete time random walk) and which applies when collisions are frequent but weak leads to the Smoluchowski equation for the evolution of the concentration of Brownian particles in configuration space. If inertial effects are included see Note (8) of Ref. [2], due to Fiirth, we obtain the Klein-Kramers equation for the evolution of the distribution function in phase space which describes normal diffusion.

The random walk considered by Einstein [2] is a walk in which the elementary steps are taken at uniform intervals in time and so is called a discrete time random walk. The concept of collisions which are frequent but weak can be clarified by remarking that in the discrete time random walk, the problem [5] is always to find the probability that the system will be in a state m at some time t given that it was in a state n at some earlier time. Referring to the Smoluchowski integral Eq. (1.9.4), viz.,

P2(x3,t3\xl,tl) = J P2(x2,t2\xl,tl)P2(x3,t3\x2,t2)dx2, (1.22.1)

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Chapter 1. Historical Background and Introductory Concepts 153

X\ and x3 can only have discrete values n and m, and the time t can only have discrete values ST with 5=1,2,3. . . . The discrete form of the Smoluchowski equation is thus [12]

P(m,sT\n,T) = YJkP[k,(s-l)T\n]Q(m,k),

where Q(m,n) = P(m,T\n). It is usual to drop the -rand just write the

equation as

P(m,s\n) = YlkP(k,s-\\n)Q(m,k). (1.22.2)

Now ]T Q(m,k)-\, thus

Q(k,k) + ^Q(m,k) = l, (1.22.3) m

where the prime means that the value m-k must be omitted from the summation. If we use this in Eq. (1.22.2) and drop the initial value n, we can write Eq. (1.22.2) in the form Pm,s)-P(m,s-\) = -P(m,s-\)Y^Qk,m) + YJ'Pk,s-l)Q(m,k).

k k

(1.22.4) According to Wang and Uhlenbeck [12], one may interpret this by saying that the rate of change of P(m,s) with time (where time is given by s) arises from the "gains" of P due to transitions from k to m minus the "losses" of P due to transitions from m to all possible k. This provides a complete analogue of the Boltzmann equation for the case where the molecules of the gas can collide only against fixed centres or against other molecules that have a given velocity distribution. Equation (1.22.4) must be solved for P given an initial distribution for P. Also, a "mechanism" or "physical cause" (stosszahlansatz) for the random process must be given, that is, Q must be specified. The initial condition for Eq. (1.22.4) is

P(m,0\n) = Pm,0) = Sm,n, (1.22.5)

where 5 is Kronecker's delta. This is just the mathematical statement of the fact that the particle was certainly in state n at the start of the process.

We consider the random walk problem in one dimension. We imagine a particle that moves along the x-axis in such a way that in each step it can move either A to the right or A to the left, the duration of each step being T. We wish to evaluate

P(mA,sT\nA) = P(m,s\n), (1.22.6) which is the probability that the particle is at mA at time ST if at the beginning it was at nA. The fact that the particle is free is now introduced by writing the transition probability Q as

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154 The Langevin Equation

Qm,k) = \8m^+\SmMl. (1.22.7)

If we substitute Eq. (1.22.7) into Eq. (1.22.2), we find that P(m,s) satisfies the difference equation (dropping the initial state n)

P(m,s)=\P(m + \,s-X) + \Pm-\,s-\). (1.22.8)

This must be solved subject to the initial condition (1.22.5). The solution is [12]

P(m,s\n) = . (1.22.9) 2s[(\n-m\+s)/2]\[(\n-m\-s)/2]\

Equation (1.22.9) is readily verified by direct substitution into Eq. (1.22.4).

Now referring to Eq. (1.22.8), if A is the step length and f the duration of the step,

P(mA,sz\nA) = ±P[(m+1)A,(s-\)zInA]+\P[(m-1)A,(s-l)rInA].

Let us now subtract P[mA,(s - \)z I nA] from both sides of this equation,

so it becomes P (mA, sz\nA)-P [mA, (s - \)z I nA]

1 P[(m + l)A,(s-l)t\nA]-2P[mA,(s-l)T\nA]\ (1.22.10)

~1[ +P[(m-l)A,(s-l)r\nA] J

Equation (1.22.10) may be written in the equivalent form 2 f [ m A , J r l n A ] - f [ m A , ( J - l ) r l n A ] = ^

r 2x

-2P[mA,(s-l)T\nA] + P[(m-l)A,(s-l)r\nA]-^-. (1.22.11) A

Let us now consider a large number of small steps of short duration. More precisely, we suppose that A and z approach zero in such a way that

A2/(2z) = D, nA—»x0, mA—»x, sz = t. The Eq. (1.22.11) (by the definition of the derivative) goes over formally into the partial differential equation

^ = D*£, (1.22.12) dt dx2

where P is now writtenP(x,11 x0,t0). This equation is the basis of Einstein's theory of Brownian movement. It shows how in a certain limit the solution of the random walk problem may be reduced to solving a

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Chapter 1. Historical Background and Introductory Concepts 155

diffusion equation like Eq. (1.22.12). The conditions imposed on P, the probability density, are

oo

j P(x,t\x0)dx = l, (P>0), (1.22.13) —oo

and limP(x,t\x0) = S(x-x0). (1.22.14) l ->0

For convenience, we have taken fo = 0. Condition (1.22.13) is the usual ones that a probability density function must satisfy. Condition (1.22.14) expresses the certainty that at t = 0 the particle was at XQ. These conditions imply that

p(x,t,\Xo) = —^e-(x-x°)2K4Dt). (1.22.15)

2yhvDt Thus, the position x (t) of the Brownian particle is a Gaussian random variable with

mean value (x(t)) = xQ and variance a2 = ([x(t) - x0]2) = 2D\t\.

If this approach is applied to the orientational motion of dipoles as described in Section 1.15, we have seen that the theory describes normal relaxation with mean dipole moment given by Eq. (1.15.24) and complex polarisability given by Eq. (1.15.1.3). The T in these two equations means of course the Debye relaxation time and not the duration of a step. Thus, the complex permittivity £ = e\(0)-ie"co) has behaviour described by the Debye equation

e(e» = e„ + f*~S~ , (1.22.16) 1 + l(OXD

where £*, is the relative permittivity at very high frequencies, es= f'(O) is the static permittivity, and TD is the Debye relaxation time which for the rotating sphere model is given by Eq. (1.15.18). Moreover, a (Cole-Cole) plot [52] of €"oi) versus e'(Qi) is a semicircle with radius (££-£„) /2 and centre on the e' axis at e' = (es + £») / 2 with maximum when (OTD = 1. Thus, TD may be determined by measuring the angular frequency when e" is a maximum. In practice, however, many disordered substances such as [80] glass forming liquids, amorphous semi conductors and polymers show very significant departures from the Debye behaviour. In addition, the mean square displacements or angular displacements associated with such transport phenomena are always proportional to a fractional power of the time resulting in anomalous relaxation behaviour. Such behaviour has led to the description of these anomalous phenomena in the language of continuous time random walks

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156 The Langevin Equation

(CTRW) (originally [79,91] introduced by Montroll and Weiss in 1965) which we will very briefly summarise.

First, we reiterate that that the random walk on which the theory of the Brownian motion is one in which the successive jumps in position of the particle are later at uniform intervals r in time. In the CTRW on the other hand, the times between the successive steps are themselves random variables [79]. Hence, the prediction of position at the following step at any given time requires not only a knowledge of the location (as in the discrete time random walk) of the random walk at that time but also the time at which the last step has occurred. This dependence on the state of the system and its past history means that the CTRW is in general not a Markov process [79]. Thus referring to the quotation of Karl Pearson given in Section 1.2, the probability that after n stretches a random walker is at distance between r and r + dr from his initial point r = 0 is a function not only of r but also [79] of the intervals Tn = tn- tn-\ between successive steps of the walk. The concept of the CTRW is essential in the explanation of the various types of anomalous relaxation behaviour of eco) which we now describe. (A more detailed discussion of the CTRW is given in Chapter 11; see also [100,101]).

1.22.1 Empirical formulae for £(G))

From almost the earliest days of dielectric relaxation measurements [52] marked departures from the form of eco) predicted by the Debye equation have been observed. The best-known empirical formulae, which have been used to describe such experimental data, are the Cole-Cole equation [52]:

e(eo) = s00 + £s~£"la, (0<«<1) , (1.22.1.1)

which again produces a circular arc however the centre lies below the horizontal axis, the Davidson-Cole equation [52]:

g(ftj) = goo+ £s~£~ (0</?<l ) , (1.22.1.2) (l + imDr

which produces a skewed arc, and the Havriliak-Negami formula, which is a combination of the Cole-Cole and Davidson-Cole equations [52]:

e(a) = e„ + £ 5 " g T g B. (1.22.1.3)

Each of these formulae exhibit anomalous relaxation (i.e., departure from the Debye pattern) behaviour. Just as in the Debye case, our task is to provide a theoretical justification for these empirical formulae. Before

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Chapter 1. Historical Background and Introductory Concepts 157

illustrating how this may be accomplished using an approach based on the CTRW, it is important to demonstrate that the above equations naturally give rise to a distribution of relaxation mechanisms. This idea originally advanced by von Schweidler and later by Wagner [52] is in the notation of Frohlich [52,53].

e(co)-eoa= f Jy ' dr. (1.22.1.4) ' 1 + ion

This is a superposition integral and embodies the idea [52] that the dielectric behaves as if it were a collection of individual Debye time mechanisms with relaxation time T and distribution function / ( r ) . One may show [52] that for the Debye equation (1.22.16)

fD(T) = (£s-^)S(r-TD), (1.22.1.5) thus only one relaxation mechanisms is involved as is obvious by definition, while [52] for the Cole-Cole function, Eq. (1.22.1.1)

fcc(r) = ( g j - O s i n a a r CC 7TT[TlTDta+ TDlTta -2COS71CC]

and for the Davidson-Cole Eq. (1.22.1.2) [52] (£o —e )smftft / x / >

fDc(T)= S, 7 J , (T<TD), fDC(T) = 0, T>TD .(1.22.1.7)

Thus, it is apparent that the anomalous relaxation behaviour may be characterised by a superposition of an infinite number of Debye-like relaxation mechanisms, with distribution functions given by Eq. (1.22.1.6) and (1.22.1.7). For the particular cases that we have considered, these empirical equations of course take no account of high frequency effects such as those due to the inertia of the molecules. These will be considered in the context [78] of anomalous relaxation later in the book. We shall now illustrate how the empirical anomalous relaxation equations we have described may be theoretically explained [82] by invoking a CTRW description in which each step of the random walk occurs at a random time which is chosen from a random distribution of waiting times (replacing the fixed uniform waiting time T of the Einstein theory) so broad that it does not possess a characteristic time scale. In other words, the mean waiting time is divergent. The origin of the parameter 1 - a, taking the Cole-Cole Eq. (1.22.1.1) as an example, must be sought in this "fractal time" waiting time distribution.

1.22.2 Theoretical justification for anomalous relaxation behaviour

The fact that the temporal occurrence of the motion events performed by the random walker is so broadly distributed that no characteristic waiting

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158 The Langevin Equation

time exists has been exploited by a number of investigators [82] in order to generalise the various diffusion equations of Brownian dynamics to explain anomalous relaxation phenomena. The resulting diffusion equations are called fractional diffusion equations because in general they will involve fractional derivatives of the probability density with respect to the time. For example, in fractional diffusion, the simple Einstein Eq. (1.4.11) for the Brownian motion of a free particle becomes (see Chapter 11, Section 11.2)

9 c kT d2

T-f(x,t)= -f- 0D1

t-<T—-Yf(x,t), (1.22.2.1)

df v '"" , kT) u ' $ dx2

where cris the anomalous exponent, the fractional derivative QD]~ff is

given by (the Riemann-Liouville definition) [82,86,87]

0D,1-£ 7=f0D /-f f (1.22.2.2)

at in terms of the convolution (recall Cauchy's integral formula [16])

0D;*= — \ ^^'Idt', (1.22.2.3)

Yz) denoting the gamma function. Equation (1.22.2.1) with 0 < c r < l describes slow diffusion or subdiffusion and with 1 < cr< 2 describes enhanced diffusion (<x= 2 defines the ballistic limit); normal diffusion occurs when a= 1 (see Chapter 11).

The derivation of fractional diffusion equations such as Eq. (1.22.2.1) hinges on the observation (cf. ref. [87], p. 118) that fractional diffusion is equivalent to a CTRW with waiting time density w(t) given by a generalised Mittag-Leffler function (see below Section 1.22.3 and also [86,87]). The fact that w (t) is given by a generalised Mittag-Leffler function amounts to assuming an asymptotic (long-time) power law form for the waiting time probability distribution function, namely considering, slow diffusion,

w(t)~AaTarl-a, (0<o-<l) , (1.22.2.4)

(A a is a constant). The characteristic (mean) waiting time oo

(Tw) = l tw(t)dt (1.22.2.5) o

then always tends to °o except in the limit a—> 1 (the classical Brownian motion), where w(t) = S(t - T) , so that Tw = r .

A famous example [101] of a distribution function with a long time tail like Eq. (1.22.2.4) is the Cauchy distribution

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Chapter I. Historical Background and Introductory Concepts 159

W(t) = - ^ — (1.22.2.6) 7V a + t

with infinite second moment. This distribution is just one example of a whole class of distributions which if applied to a sum of random variables do not converge to the Gaussian distribution as the number of random variables tends to infinity. Thus, the central limit theorem, on which the theory of Brownian motion is based, is not obeyed because the long time tails preclude convergence to the Gaussian distribution. Nevertheless, limiting (now called Levy [101], Chapter 4) distributions may exist (see also Chapter 11).

The divergence of the waiting time associated with the long time tailed nature of the waiting time probability distribution function Eq. (1.22.2.4) is according to Metzler and Klafter [82] a manifestation of the self-similar nature of the waiting time process. This has prompted [82] many investigators to use in the present context, the term fractal time processes to describe anomalous relaxation. Returning to the fractional diffusion equation Eq. (1.22.2.1), that equation will now follow from Eq. (1.22.2.4) and CTRW theory because (Ref. [87], p. 118) the integral equation for the probability density/(x, t) for a continuous time random walker to be in a position x at time t starting from x = 0 at t = 0 with waiting time density given by Eq. (1.22.2.4) is equivalent in the diffusion limit to the fractional diffusion equation Eq. (1.22.2.1) as we illustrate in Chapter 11.

We remark that postulating w ( 0 as a generalised Mittag-Leffler function with long time behaviour given by Eq. (1.22.2.4) so that fractional diffusion may be described as a CTRW is (just as the postulate of the existence of a discrete time t, the duration of an elementary jump in the Einstein theory of the Brownian movement) equivalent to a stosszahlansatz for the Boltzmann equation which must of necessity underpin the entire theory. In other words, the transition probability or "mechanism" of the fractional diffusion process is that of the CTRW. In Chapter 11, we shall see in detail how the introduction of a waiting time density of the form of Eq. (1.22.2.4) allows one to generalise the Klein-Kramers equation of normal diffusion to fractional diffusion. We shall also demonstrate how the anomalous diffusion problem may be treated within the framework of the generalised Langevin equation or memory function formalism due to Mori, reviewed in Ref. [44]. The memory function approach is attractive in the context of fractional diffusion as it quite clearly underlines the non-Markovian nature of fractional diffusion. Furthermore, the CTRW provides a microscopic model, which justifies the introduction of particular memory functions into the Mori theory

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160 The Langevin Equation

which describe anomalous diffusion. In the meantime, we will confine ourselves to the fractional diffusion equation in configuration space (here a generalised Smoluchowski equation) for an assembly of rigid non-interacting dipoles and to demonstrating how that equation can predict [88] the Cole-Cole behaviour, Eq. (1.22.1.1) of the relaxation function.

1.22.3 Anomalous dielectric relaxation of an assembly of fixed axis rotators

In the fixed axis rotation model of dielectric relaxation of polar molecules, a typical member of the assembly is a rigid dipole of moment \i rotating about a fixed axis through its centre. The dipole has moment of inertia / and is specified by the angular coordinate <j) (the azimuth) so that it constitutes a system of one (rotational) degree of freedom. The fractional diffusion equation for the time evolution of the probability density function W(0,t) in configuration space is then the same as that previously written, Eq. (1.22.2.1), for a particle of one translational degree of freedom. However, rotational quantities replace translational ones and a potential energy term V<j),t) is added so that

±(w±v)+kTyw

Here £ is the viscous drag coefficient of a dipole, T is the intertrapping time scale, which is identified with the Debye relaxation time

T = £/(kT), (1.22.3.2) and V(<f>,t) = -juE(t)cos<f> is the potential arising from an external applied electric field E(t). The operator QD)~a is defined as before by the Riemann-Liouville fractional integral definition, Eq. (1.22.2.3), viz.,

Equation (1.22.3.3) means that Eq. (1.22.3.1) now contains a slowly decaying memory function with a power law kernel so that the process is no longer Markovian and so depends on the history of the system. We shall consider two classes of solution of Eq. (1.22.3.1), the first is the after-effect solution following the removal of the constant field.

Here a uniform field E0 having been applied to the assembly of dipoles at a time t = -oo so that equilibrium conditions prevail by the time t = 0, is switched off at t = 0. In addition, it is supposed that the field is weak (juE0 « kT). Thus, Eq. (1.22.3.1) becomes for t > 0

7) T 1 _ < T

(1.22.3.1)

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Chapter I. Historical Background and Introductory Concepts 161

2W(<f>,t). (1.22.3.4) d-W<p,t) = T-\D)-" ^

Equation (1.22.3.4) must be solved subject to the initial condition 1

W(</>,0) = 2K

1 + — - c o s 0 kT

(1.22.3.5)

where lit is the normalising constant. Just as in normal diffusion, the form of the initial condition Eq. (1.22.3.5) suggests that the time dependent solution should be of the form

W((f>,t) = ^-271

i + s(0 • c—^-cos^ kT

(1.22.3.6)

Substitution of Eq. (1.22.3.6) into Eq. (1.22.3.4) then yields the following fractional differential equation for the function g (t)

d ^ dt

•J>. \-o. o", '8(0. (1.22.3.7)

The solution of this fractional relaxation equation is [82]

g(t) = Ea[-(t/Tf], (1.22.3.8)

where Ea(z) is the Mittag-Leffler function defined by [86,87] oo n

E*iz)=Z — v — • (1-22.3.9) n=0 r ( l + an)

The Mittag-Leffler function interpolates between the initial stretched exponential form [82]

Ea[-tlT)a] ~ e-i'^f'r^ (1.22.3.10) and the long-time inverse power-law behaviour [82,86,87]

EJ-itltf] . (1.22.3.11)

The Debye result for g(t) corresponds to a= 1, viz.,

Ex(-tlT) = e-t,T (1.22.3.12) that is a special case of the Mittag-Leffler function [82].

We may now calculate the mean dipole moment. The mean moment due to orientation alone is given at any time t > 0 by, (e is a unit vector in the direction of field E0)

(|i-e)(f)= j jUcos0W(0,t)d</> (1.22.3.13) o

so that with Eq. (1.22.3.6) we have

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162 The Langevin Equation

(»-e)(t) = ^E(7[-(t<Tr] (1.22.3.14)

in contrast to the Debye result embodied in Eq. (1.22.3.12) A practically much more important result than that treated above is

the behaviour of the system in a periodic field E(t) = E0e'w , so that the fractional diffusion equation Eq. (1.22.3.1) for the distribution function becomes

£w = •".*<- S("E«s^e"" d0 kT

+ -d02 -W

Following Debye, let us try as a solution [88]

W(/>,t) = ~ IK kT Y

. (1.22.3.15)

(1.22.3.16)

where Bco) is a constant to be determined. Substitution of Eq. (1.22.3.16) into Eq. (1.22.3.15) yields

i0B(C0)eim = T'a 0Dl-a[l-B(G))]eia' (1.22.3.17)

Equation (1.22.3.17) may be further simplified if we recall the integration theorem of Laplace transformation as generalised to fractional calculus, viz., [82]

„i-<r / ( * ) , (1<<7<2), (1.22.3.18)

where

~f(s) = Lf(t) = \e-s,ft)dt.

(This theorem is of fundamental importance in fractional dynamics as we shall see later when the use of it coupled with continued fraction methods allows recurrence relations associated with normal diffusion to be generalised to fractional dynamics in an obvious fashion.) Regarding the ac response, if we assume that the above result may be analytically continued into the domain of the imaginaries or equivalently noting that if D denotes the operator d/dt and Gco) denotes an arbitrary function of co, then [88]

N - 1 JCOt/ D-leio"G(oo) = (i0))-le,a"G(0)). The generalisation of the above operator equation is

D-qeimGQ)) = (iooYqeiwG(D), (1.22.3.19)

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Chapter 1. Historical Background and Introductory Concepts 163

where q denotes an fractional index. Equation (1.22.3.17), assuming Eq. (1.22.3.19), then simplifies to

£(©) = - l

l + (icor)c

Thus, we have in linear response

2K 1 + ^ - c o s ^

kT \ + (iO)t)c

and, as before, the mean moment is

(|i-e)(0 = M2EQ

2kT l + (im)c

The complex susceptibility is then

X®) = M

lN0 1

(1.22.3.20)

(1.22.3.21)

(1.22.3.22)

(1.22.3.23) 2kT l + (im)a

where N0 is the concentration of dipoles. Equation (1.22.3.23) may also be derived from the after effect solution Eq. (1.22.3.14) with the help of linear response theory (see Chapter 2, Section 2.8). According to that theory, the complex dynamic susceptibility is given by

Z(6)) = zX0)-ia>jb(t)e-imdt, (1.22.3.24)

Z(G>) = M2N0

2kT

where b (t) is the after-effect function. (In the context of the use of this theorem we note the non-stationary nature of the stosszahlansatz or mechanism underlying the fractional dynamics). Equation (1.22.3.24) with the after-effect function b (t) given by Eq.(1.22.3.14), yields

OO 2 AT 1

l-iQ)\EJ-(t/Tf]e-i0Xdt =OJ^ , (1.22.3.25) J 2kT \ + (imf

where we have noted that the one-sided Fourier transform of the Mittag-

Leffler function Ea[-(tIz)a] is [82,86,87] 1

io)+T'a(iO)f-a ' Equation (1.22.3.25) is retrieved, demonstrating that, as far as the present problem is concerned, linear response theory is obeyed in the fractional dynamics despite the non-stationary character of the mechanism underlying the fractal time relaxation process.

The above approach may be carried over in all respects to rotation in space. Here, the space coordinate is the polar angle & (the

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164 The Langevin Equation

colatitude) and the fractional diffusion equation assumes the form (cf. Eq. 1.15.21) (referring to the a.c. solution)

- W = r - 0 D , \-a 1

2sintf3r? s i n t f l i ^ s i n i f e t o W + J L ^ ,(1.22.3.26)

kT " 3tf where the time T = £ IlkT is the Debye relaxation time for rotation in space. Recalling the work of Debye, it is now apparent that Eq. (1.22.3.26) may be solved just as Eqs. (1.22.3.4) and (1.22.3.15) which pertains to rotation about a fixed axis, to yield the corresponding result for rotation in space, viz., the after effect solution

M2E0

3kT '-t/tf

ju2E0

3kT l + iicorf

M2N0 1

(1.22.3.27)

(1.22.3.28)

(1.22.3.29)

(|i-e)(f) =

the ac stationary response

(H-e) =

and the complex susceptibility

Y(CO) =

3kT \ + (imf One can see that just as in normal diffusion Eqs. (1.22.3.28) and (1.22.3.27) differ from the corresponding two-dimensional analogues [Eqs. (1.22.3.23) and (1.22.3.14)] only by a factor 2 / 3 and the appropriate definition of the Debye relaxation time. These results indicate clearly how the Debye theory of dielectric relaxation of polar molecules based on the concept of normal diffusion may be generalised to anomalous diffusion.

This brings to an end our long discussion of the various applications of the theory of the Brownian movement, which we hope will serve as both an introduction and as a motivation for the study of various detailed aspects of the theory and its applications which are presented in the remainder of the book.

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4. J. W. S. Rayleigh, Theory of Sound, 2nd Edition, Vol. 1, Macmillan, London 1894, reprinted Dover, New York, 1945.

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Chapter 1. Historical Background and Introductory Concepts 165

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25. M. Born, Natural Philosophy of Cause and Chance, Oxford University Press, London, 1948.

26. R. Barnes and S. Silverman, Brownian Motion as a Natural Limit to All Measuring Processes, Rev. Mod. Phys. 6, 162 (1934).

27. M. Kac, Random Walk and the Theory of Brownian Motion, Am. Math. Mont. 54, 369 (1947), also reprinted in Wax loc. cit.

28. B. Gnedenko, The Theory of Probability, Mir, Moscow, 1988. 29. M. R. Spiegel, Real Variables, Schaum's Outline Series, McGraw Hill, New York,

1969. 30. J. L. Doob, Stochastic Processes, Wiley, New York, 1953.

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166 The Langevin Equation

31. C. W. Gardiner, Quantum Optics, Springer, Berlin, 1991. 32. R. L. Stratonovich, Topics in the Theory of Random Noise, Gordon and Breach,

New York, Vol. 1, 1963, Vol. II, 1967. 33. P. Debye, Polar Molecules, Chemical Catalog Co. New York, 1929; reprinted by

Dover, New York, 1954. 34. J. Frenkel, The Kinetic Theory of Liquids, Oxford University Press, London, 1946.

Reprinted by Dover, New York, 1955. 35. J. T. Lewis, J. R. McConnell, and B. K. P. Scaife, Proc. R. Ir. Acad. A 76, 43

(1976). 36. A. J. Martin, G. Meier, and A. Saupe, Symp. Faraday Soc. 119, 5 (1971). 37. W. H. de Jeu, Physical Properties of Liquid Crystalline Materials, Gordon and

Breach, New York, 1980. 38. T. M. MacRobert, Spherical Harmonics, 3rd Edition, Pergamon, Oxford, 1967. 39. E. P. Gross, J. Chem.Phys. 23, 1415(1955). 40. R. A. Sack, Relaxation Processes and Inertial Effects, I. Free Rotation about a

Fixed Axis, II. Free Rotation in Space, Proc. Phys. Soc. Lond. B 70, 402, 414 (1957).

41. J. P. Poley, Appl. Sci. Res. B 4, 336 (1955). 42. W. F. Brown, Jr., Phys. Rev. 130, 677 (1963). 43. C. P. Bean and J. D. Livingston, Superparamagnetism, J. Appl. Phys. 30, 120S

(1959). 44. M. W. Evans, G. J. Evans, W. T. Coffey, and P. Grigolini, Molecular Dynamics,

Wiley, New York, 1982. 45. M. I. Shliomis and V. I. Stepanov, Adv. Chem. Phys. 87, 1 (1994). 46. L. Neel, Influence des Fluctuations Thermiques sur I'Aimantation de Grains

Ferromagnetiques tres Fins, C. R. Acad. Sci. Paris 228, 664 (1949); Theorie du Trainage Magnetique des Ferromagnetiques en Grains Fins avec Applications aux Terres Cuites, Ann. Geophys. 5, 99 (1949).

47. W. F. Brown, Jr., Thermal Fluctuations of Fine Ferromagnetic Particles, IEEE, Trans. Mag. 15, 1196, (1979).

48. E. C. Stoner and E. P. Wohlfarth, A Mechanism of Magnetic Hysteresis in Heterogeneous Alloys, Phil. Trans. R. Soc. Lond. A 240, 599 (1948).

49. H. Pfeiffer, Phys. Stat. Sol., 118, 295 (1990); 122, 377 (1990). 50. P. M. S. Blackett, Lectures on Rock Magnetism, Weizmann Science Press,

Jerusalem, 1956. 51. E. M. Chudnovsky, J. Appl. Phys., 73, 6697 (1993). 52. B. K. P. Scaife, Principles of Dielectrics, Oxford University Press, London, 1989;

2nd Edition 1998. 53. H. Frohlich, Theory of Dielectrics, 2nd Edition, Oxford University Press, London,

1958. 54. P. C. Fannin, B. K. P. Scaife, and S. W. Charles, /. Mag. Magn. Mat. 122,

159(1993). 55. P. C. Fannin, J. Mag. Magn. Mater. 136, 49, (1993). 56. W. T. Coffey, P. M. Corcoran, and M. W. Evans, Proc. R. Soc. Lond. A 410, 61

(1987). 57. W. T. Coffey, P. M. Corcoran, and J. K. Vij, Proc. R. Soc. Lond. A 412, 339

(1987). 58. W.T. Coffey, P.M. Corcoran, and J. K. Vij, Proc. R. Soc. Lond. A 425,169 (1989).

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Chapter 1. Historical Background and Introductory Concepts 167

59. G. A. P. Wyllie, Dielectric Relaxation and Molecular Correlation, in Dielectric and Related Molecular Processes, Vol. 1. Specialist Periodical Reports, The Chemical Society, London, 1972.

60. W. T. Coffey and P. C. Fannin, J. Phys. Condens. Matter, 14, 3677 (2002). 61. G. H. Wannier, Elements of Solid State Theory, Cambridge University Press,

Cambridge, 1960. 62. F. W. Sears, Thermodynamics, Addison Wesley, Reading, Mass., 1953. 63. H. Haken, Laser Theory, Springer, Berlin, 1984. 64. S. Stenholm, Foundations of Laser Spectroscopy, Wiley, New York, 1984. 65. L. J. Geoghegan, W.T. Coffey, and B. Mulligan, Differential Recurrence Relations

for Non-Axially Symmetric Fokker-Planck Equations, Adv. Chem. Phys. 100, 475 (1997).

66. A. Aharoni, Introduction to the Theory of Ferromagnetism, Oxford Press, London, 1996

67. W. T. Coffey, D. A. Garanin, and D. J. McCarthy, Crossover Formulas in the Kramers Theory of Thermally Activated Escape Rates - Application To Spin Systems, Adv. Chem. Phys. 117, 483 (2001).

68. J. S. Langer, Statistical Theory of the Decay of Metastable States, Ann. Phys. (N.Y.) 54, 258 (1969).

69. L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Stochastic Resonance, Rev. Mod. Phys. 70, 223 (1998).

70. W. Wernsdorfer, Classical and Quantum Magnetisation Reversal Studied in Nanometre-sized Particles and Clusters, Adv. Chem. Phys, 118, 99 (2001)

71. W. T. Coffey and Yu. P. Kalmykov, Rotational Diffusion and Dielectric Relaxation in Nematic Liquid Crystals, in: Advances in Liquid Crystals. Ed. J. K. Vij, Adv. Chem. Phys. 113, 487 (2000).

72. W. T. Coffey, Finite Integral Representation of Characteristic Times of Orientational Relaxation Processes: Application to the Uniform Bias Force Effect in Relaxation in Bistable Potentials, Adv. Chem. Phys. 103, 259 (1998).

73. D. A. Garanin, Phys. Rev. E. 54, 3250 (1996). 74. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and J.T. Waldron, Phys. Rev. B,

51, 15947 (1995). 75. W. T. Coffey, D. S. F. Crothers, and Yu. P. Kalmykov, Phys. Rev. E. 55, 4812

(1997). 76. A. N. Malakhov and A. L. Pankratov, Evolution Times of Probability Distributions

and Averages - Exact Solution of the Kramers' Problem, Adv. Chem. Phys. 121, 357 (2002).

77. Yu. L. Raikher and V. I. Stepanov, Stochastic Resonance in Single Domain Particles, J. Phys. Condens. Matter 6, 4137 (1994).

78. E. Barkai and R. S. Silbey, J. Phys. Chem. B 104, 3866 (2000). 79. G. H. Weiss, Aspects and Applications of the Random Walk, North Holland

Amsterdam, 1994. 80. P. Lunkenheimer, U. Schneider, R. Brandt, and A. Loidl, Glassy Dynamics,

Contemporary Phys. 41, 15 (2000). 81. E. C. Titchmarsh, An Introduction to the Theory of Fourier Integrals, Oxford

University Press, London, 1937. 82. R. Metzler and J. Klafter, Anomalous Stochastic Processes in the Fractional

Dynamics Framework: Fokker-Planck Equation, Dispersive Transport and Non-

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168 The Langevin Equation

Exponential Relaxation, Adv. Chem. Phys. 116, 223 (2001); Phys. Rep. 339, 1 (2000).

83. M. Schroeder, Fractals, Chaos, and Power Laws, W. H. Freeman, New York, 1991. 84. B. B. Mandelbrot, The Fractal Geometry of Nature, W. H. Freeman, New York,

1983. 85. J. Feder, Fractals, Plenum, New York, 1988. 86. K. B. Oldham and J. Spanier, The Fractional Calculus, Academic Press, New York,

1974. 87. R. Hilfer (Ed.), Applications of Fractional Calculus in Physics, World Scientific

Singapore, 2000. 88. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov, J. Chem. Phys. 116, 6422 (2001) 89. F. Bloch, Fundamentals of Statistical Mechanics, Manuscripts and notes of F.

Bloch prepared by J. D. Walecka, Imperial College Press, World Scientific, Singapore 2000.

90. P. Wilmott, S. Howison, and J. Dewynne, The Mathematics of Financial Derivatives, Cambridge University Press, London, 1995, Reprinted 1999.

91. D. ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered Systems, Cambridge University Press, London, 2000.

92. L. A. Pars, A Treatise on Analytical Dynamics, Oxbow Press, Connecticut, 1979. 93. P. Dugourd, R. Antoine, D. Rayane, E. Benichou, and M. Broyer, Phys. Rev. A 62,

011201(2000). 94. K. Asokan, T. R. Ramamohan, and V. Kumaran, Phys. Fluids 14, 75 (2002). 95. D. A. Smith and F. A. de Rozario, /. Magn. Magn. Mater. 3, 219 (1976). 96. I. Klik and L. Gunther, J. Stat. Phys. 60, 473 (1990). 97. C. Nicolis, Sol. Phys. 74, 473 (1981); C. Nicolis and G. Nicolis, Tellus 33, 225

(1981); C. Nicolis, Tellus 34, 1 (1982). 98. R. Benzi, A. Sutera, and A. Vulpiani, J. Phys. A 14, L453 (1981); R. Benzi, G.

Parisi, A. Sutera, and A. Vulpiani, Tellus 34, 10 (1982); R. Benzi, A. Sutera, G. Parisi, and A. Vulpiani, SIAM (Soc. Ind. Appl. Math.) J. Appl. Math. 43, 565 (1983).

99. K. S. Miller, Engineering Mathematics, Holt, Rinehart, and Winston, New York, 1956; reprinted by Dover Publications, New York, 1963.

100. R. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications, Oxford University Press, Oxford, 2002.

101. W. Paul and J. Baschnagel, Stochastic Processes from Physics to Finance, Springer Verlag, Berlin, 1999.

102. R. Kubo, M. Toda, and N. Nashitsume, Statistical Physics II. Nonequilibrium Statistical Mechanics, Springer Verlag, Berlin, 1991.

103. B. J. Berne and R. Pecora, Dynamic Light Scattering with Applications to Chemistry, Biology and Physics, Wiley, New York, 1976.

104. J. R. McConnell, Nuclear Magnetic Relaxation in Liquids, Cambridge University Press, Cambridge, 1987.

Note: The full titles of articles, which have the character of a review, have been given in this Chapter.

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Chapter 2

Langevin Equations and Methods of Solution

2.1 Criticisms of the Langevin Equation

In Chapter 1, we have introduced the Langevin equation for a free particle (treated in detail in Chapter 3) supposing that the white noise driving force satisfies the condition

F(tl)Ft2) = 2DStl-t2). (2.1.1) This approach is open to criticism on several grounds, one obvious one [1] being that it is impossible to plot a realisation with a ^-correlated noise, i.e., one, where F(tx) and F(t2) are completely independent for arbitrarily small \tx-t2\. The criticisms of the Langevin equation have been succinctly described by Doob [2] who was the first investigator to show how that equation should properly be interpreted as an integral equation and not as a differential equation. This approach has led to far-reaching mathematical and conceptual simplifications, the latest of which constitutes the central theme of this book - that the hierarchy of differential-recurrence relations may be directly derived from the Langevin equation so bypassing the Fokker-Planck equation entirely. Doob's observations on the Langevin equation are best described in his own words [2]:

"Since 1905 the Brownian movement has been treated statistically on the basis of the fundamental work of Einstein and Smoluchowski. Let X (t) be the coordinate of a particle at time t, Einstein and Smoluchowski treated X (t) as a random variable. They found the distribution of [X(t)-X(0)] to be Gaussian with mean zero and variance a\t\, where or is a positive constant which can be calculated from the physical characteristics of the moving particles and the given liquid. More exactly such a family of random variables, X(t)\ is now described as the family of random variables determining a temporally homogeneous differential stochastic process: the

169

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170 The Langevin Equation

distribution of [X (s + t) - X (t)] is Gaussian with mean zero, variance cc\t\ and if tx<....<tm Xt^-Xt), ..., X(tn)-X(tn-\)< are mutually independent random variables. Wiener who was the first to discuss this stochastic process rigorously proved in 1923 [3] that the sample functions X(t) of this process are continuous with probability 1. This is of course a desirable result which makes the process somewhat more desirable as the mathematical idealisation of the Brownian movement. It was not expected that the above distribution of [X (s + t) - X s)] would prove correct for small t, even if the derivation did not break down for small t, the mathematical fact that [X (s + t) - X (s)] has standard deviation ~\t\m so that [X (s + t) -X (s )] is of the order of magnitude of \t\m

implying that dX(s) I ds cannot be finite would suggest that the Einstein-Smoluchowski result should be modified."

"A different stochastic process describing the X(t) was in fact derived in 1930 by Ornstein and Uhlenbeck (see Chapter 3). This new distribution of [X(s + t)-X(s)] is Gaussian with mean zero and variance

(a//3)e-W-l + fi\t\), which is approximately 111 for large t but a/3t212 for t small. (Here J3 is a second physically determined constant). The displacement function X(t) as discussed by Ornstein and Uhlenbeck has derivative u (t) and all the probability distributions needed can be derived from those of u (t). However, the variance of [u (s + t) - u(t)] is

Var[u(s + t)-u(s)]2 = 2a20(l-e-m)~2a2

oj3\t\. (2.1.2)

Thus [u (s + t) - u (s)] is of the order of magnitude of 1111/2

and du/dt cannot exist. Physically this means that the particles in question do not have a finite acceleration. Ornstein and Uhlenbeck base their investigation on the Langevin equation

^ 1 = -J3u(t) + A(t), (2.1.3) dt

which is simply Newton's law of motion applied to a particle after dividing through by the mass. The first term on the right is due to the frictional resistance imposed by the medium surrounding the particle on the particle. The second term represents the random forces (molecular impacts); probability hypotheses are imposed on the A (t) including relations between A (t) and u (t) in order to determine the u (t) distribution. Unfortunately, the u (t) distribution has, as we

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Chapter 2. Langevin Equations and Methods of Solution 171

have seen, the property that the velocity function has no time derivative, thus the solution can hardly satisfy Eq. (2.1.3)."

The question that now poses itself is what exactly do we mean by Eq. (2.1.3)? We now show (following Doob) how Eq. (2.1.3) may be interpreted.

2.2 Doob's Interpretation of the Langevin Equation

To interpret the Langevin equation, we write it as dut) = -/3ut)dt + dBt) (2.2.1)

and try to give these differentials a meaning. We shall suppose that if, tx < .... < tn,

Bh)-Bis) , B(tn)-B(tn_x) are mutually independent random variables. We shall also suppose temporal homogeneity, i.e., stationarity, so that the distribution of B (s + t) - B (s) is independent of the initial value s. In addition we shall suppose that

B(s + t)-B(s) = 0 (2.2.2) and that

[B(s + t)-B(s)]2=c2t, t>0. (2.2.3) Thus, B(t) satisfies the requirements for a Wiener process. Let us integrate both sides of Eq. (2.2.1) having first multiplied both sides by a continuous function/(O of the time t, we then interpret Eq. (2.2.1) as meaning that, for/a continuous function of time and for all t,

b b b

J f(t)du(t) = -p\ f(t)u(t)dt + J f(t)dB(t). 2.2A) t=a a a

All these integrals are well defined for f(t) a continuous function of time. See, for example, Wiener and Paley [3]. If/(f) = 1 Eq. (2.2.4) gives

u(b) - u(a) = -j3J u(t)dt + B(b) - Ba). (2.2.5) a

In the Ornstein-Uhlenbeck process, u(t) = X(t) exists [X(t) is the displacement], so Eq. (2.2.5) may be written (writing a = 0, b = t)

Bt) - 5(0) = u(t) - w(0) + j3[X(t)-X (0)]. (2.2.6) Let us now suppose that in Eq. (2.2.5), b-a is long compared with the time between impacts so that B(b)-B (a) may be written as the series

B(6)-#(«) = I [B(tk+1)-B(tk)] (2.2.7) k=0

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172 The Langevin Equation

(with t0=a< t\ < ?2 •••• < tn__\ <tn = b).

We see from this discussion that the second term on the right of Eq. (2.2.1) may be represented as a Wiener process. In future, when we write down a Langevin equation,

u(t) = -Put) + Bi), (2.2.8) we shall always interpret it as the integral equation (2.2.5). Setting fit) = / ' in Eq. (2.2.4), we have

t i t

\ e/3sdu(s) = -/3J efisu(s)ds + l e^dB(s) s=0 0 0

and since integration by parts is permissible, we obtain t

u(t)ep> - K ( 0 ) = J efi'dB(s)

,s=0

or t

u(t) = u(!0)e~fi' + J e~^''s)dB(s). (2.2.9) ,s=0

This is Doob's treatment of the linear Langevin equation. His interpretation of the Langevin equation as an integral equation is central to all that follows.

2.3 Nonlinear Langevin Equation with a Multiplicative Noise Term: I to and Stratonovich Rules

The Langevin approach has been used by many authors in order to treat nonlinear systems. This is of importance to us since the equations of rotational Brownian motion are intrinsically nonlinear. The concept of a nonlinear Langevin equation is also subject to a number of criticisms. These have been discussed extensively by van Kampen ([4], Chapters 8 and 14). We summarise these in so far as they are relevant to the present work. In our calculations, we shall encounter stochastic differential equations of the form

£(t) = h[ftt),t] + g[frt),Mt) (2.3.1) with

1 (0 = 0, Mt)A(t') = 2DS(t-t'). As above, the overbar denotes a statistical average over an ensemble of particles starting at time t with the same ^(t) = x (see Chapter 1, Section 1.10). It contains a multiplicative noise term. The noise may be represented, according to van Kampen [4], by a random sequence of delta functions. Thus each delta function jump in A(t) causes a jump in £(?)•

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Chapter 2. Langevin Equations and Methods of Solution 173

Hence, the value of £ at the time the delta function arrives is indeterminate and consequently so is g at this time also. A problem arises, as the equation does not indicate which value of £, one should substitute in g, whether the value of cf before the jump, the value after or a mean of both.

We explain as follows. Equation (2.3.1) is a nonlinear Langevin equation with multiplicative noise, which must be interpreted as the integral equation

t+At t+At

£(t + At)-x = j h[(t'),t']dt'+ j g[Z(t'),t']dB(t'), (2.3.2) / t

where in our notation dB(t') = X(t')df. In Eq. (2.3.2), B(t) is a Wiener process so that the increment

w(At) = B(t + At)-B(t) is stationary and Gaussian. On eliminating the stochastic variable <f (?) in the right hand side of Eq. (2.3.2) by the iteration procedure described in Section 1.10, one obtains [1]

t+At t+At

&t + At)-x= j h[Z(t'),t']dt'+ f g[%t\t']dB(t')

= h(x,t + &lAt)At + g(x,t + &2@^At)—g(x,t + QzAt) f w(t')dw(t')+o(At), dx J

0

(2.3.3) which is simply Eq. (1.10.12) in another notation; here 0 < 0, < 1. In Eqs. (2.3.2) and (2.3.3), g(t+ At) is the solution of Eq. (2.3.1) which at time t has the sharp value £,t) = x. We remark that x itself is a random variable with probability density function W(x, t) defined such that Wdx is the probability of finding x in the interval x,x + dx. It will be necessary, when applying the Langevin equation method, to take a second average over the probability density function of x at time t. The dual average is often denoted by a single pair of angular braces for economy of notation. The stochastic integrals in Eq. (2.3.3) are of the form

At

\ <S>[w(t'),t']dwt'). o

Ito [5] interpreted such stochastic integrals by prescribing that in <t> the value of w before the jump should be taken so that (cf. Section 1.8)

(I)J <S>[w(t'),t']dw(t') = \im £ * M ^ ] [ w ( W - w ( * i ) ] (2-3-4) o <^° i=o

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174 The Langevin Equation

in contrast to Stratonovich [6] who takes the mean of the values before and after the jump and writes

At N-\

(S) f <P[w(t'),t']dw(t') = lim Y 3> ' - * M >

w(r,.) + w(r.+1)

2 'T; [w(r,.+1)-w(T,.)].

(2.3.5) Here S = ma\(Ti+l-ri); 0 = T0 <T 1 <...<TN = A?. On applying the above definitions to Eq. (2.3.3), we have [1]

(I) J W(t')dw(t') = £ w(Tt) [W(TM ) - W&i)] 0 ,=0 (2.3.6)

(=0 1=0

and

(S)J w(f)rfw(0 = E [ w ( t M ) + w(tt)][w(ti+l)-w(ti)] 0 2 '=0

1 w-i . -.

2 i=o

= \ t (^ + 2^i - 2^ - 2ti) = I te+i -*i) = At- (2-3-7) 2 i=o ;=o

Thus, it follows from Eqs. (2.3.3)-(2.3.7) that in the Ito definition gg,t)Mt) = 0 (2-3-8)

and that in the Stratonovich one

g£,t)Mt) = Dg(x,t)dg(x,t)/dx. (2.3.9) The Stratonovich Eq. (2.3.9) leads to the concept of noise induced or spurious drift as

gtf,t)Mt)*o. It is evident that the averaged Langevin equation (2.3.1) interpreted as a Stratonovich stochastic equation will differ from that obtained by the Ito approach. Therefore, in physical applications, in order to guarantee the equivalence of the result for the observed variable x, one must use the equivalent form of the Stratonovich and Ito stochastic equations. The rules of mutual transformation of Stratonovich and Ito equations are described in detail by Gardiner [7]. Here, we simply present these rules which formally follow from Eqs. (2.3.8) and (2.3.9), namely: the Stratonovich equation

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Chapter 2. Langevin Equations and Methods of Solution 175

£(t) = h[Z(t),t] + g[£(t),t]Mt) (2.3.10) corresponds to the Ito equation

(t) = h[£(tU] + Dg[£(t)j]-^g[£(t),t] + g[Z(t),t]Mt). (2.3.11) °g

Likewise, the Ito equation

i(t) = h'[g-(t),t] + glfrtUM) (2.3.12) corresponds to the Stratonovich equation

i(t) = h^(tU]-DgX£(t)j]^g'[g-(t),t] + g'[£(t),t]Mt). (2.3.13) °g

Thus, on averaging Eqs. (2.3.10) and (2.3.11) according to the appropriate rules, we have the unique result

x = h(x,t) + Dg(x,t)—g(x,t) • ox

In like manner, the averaged Eqs. (2.3.12) and (2.3.13) yield the unique equation

x-h\x,t). In the Stratonovich definition, we may use the ordinary rules of calculus, whereas in the Ito definition this is not so. One has to use new rules for differentiation and integration (the Ito calculus). In the problems which we shall treat, the white noise is the limiting case of a physical noise with finite noise power. In this case, it appears that the Stratonovich definition is the correct one to use in various physical applications such as dielectric or magnetic relaxation [1,4].

By way of illustration, let us evaluate the moments v" (t) of the

velocity vt) for n > 2 in the context of the translational diffusion of a free Brownian particle of the mass m (i.e., the Ornstein-Uhlenbeck process, see Chapter 1, Section 1.7 and Chapter 3, Section 3.1). Here, the velocity v(t) obeys the Langevin equation in the Stratonovich definition:

—v(t) + /3vt) = Xt)lm, (2.3.14) dt

where as before X (t) is a white noise driving force with D = mk Tj3. The corresponding Langevin equation for the random variables v" (t) may be obtained by multiplying across Eq. (2.3.14) by nvn~\t) and by using the rules of conventional calculus. Thus, one has

—vn(t) + nj3v"(t) = nvn-l(t)A(t)/m. (2.3.15) dt

We proceed by noting that Eq. (2.3.15) may be regarded as a stochastic differential equation for ^(t) = vn(t), viz.,

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176 The Langevin Equation

—£(t) + nff(t) = n?n-l)ln(t)A(t)/m. (2.3.16) dt

Equation (2.3.16) contains multiplicative noise term g(£) = «<f(n_1)/" lm . This term, in accordance with the Stratonovich rule Eq. (2.3.9), yields a nonzero contribution on statistical averaging, viz.,

m dx

= okTw(w-l)x(n-2)/n _ fikTnjn-l)^

m m Thus

-vn+nj3vn = kTn(-n~l)pvn-2 ( 2 3 1 7 ) dt m

For example, for n = 2, we have from Eq. (2.3.13)

— 7 + 2^7 = 2 — ^ . (2.3.18) dt m

The solution of Eq^ (2.3.18) is the known result (see Section 1.7) v2(t) = \y2(0)-(kT/m)]e-2/j'+kT/m. (2.3.19)

In like manner, all higher order moments can be evaluated using the differential-recurrence Eq. (2.3.17).

In order to apply to this problem the alternative (Ito) approach, which is based on yet another rule of averaging of multiplicative noise terms, one must first transform the Stratonovich Eq. (2.3.15) into an Ito equation. The corresponding Ito equation is

dt vnt)_nzWLvn-2t)

n .n-1/ +—v n ' l ( t )A(t ) , (2.3.20) m m

where the multiplicative noise term must be averaged in accordance with the Ito calculus, viz.,

v"-\t)A(t) = 0. Thus, each of the two stochastic differential Eqs. (2.3.15) and (2.3.20)

predict the same results for the averages vn(t) (n > 1) provided the rules of the corresponding calculus are used. Moreover, the results for all the

statistical moments are identical. Hence, the averages vn(t) are unique. The above example underlines the importance of the consistent

use of a rule for the averaging of the multiplicative noise terms in stochastic differential equations.

We have illustrated the problem of multiplicative noise by referring to motion in one dimension, however, the dilemma is present

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Chapter 2. Langevin Equations and Methods of Solution 111

mainly for motion in several dimensions, since a one-dimensional equation with multiplicative noise can always be transformed into an equation with additive noise [1].

2.4 Derivation of Differential-Recurrence Relations from the One-Dimensional Langevin Equation

As an example of how the average of the noise-induced drift, Eq. (2.3.9) may be used to derive the hierarchy of differential-recurrence relations for statistical averages directly from the Langevin equation, we consider the Brownian motion of a particle in a tilted periodic potential. This problem arises in a number of physical applications, e.g., current-voltage characteristics of the Josephson junction, mobility of superionic conductors, a laser with injected signal, phase-locking techniques in radio engineering, dielectric relaxation of molecular crystals, etc. [1]. As a particular application, we consider here the dynamic model of a Josephson tunnelling junction in the zero capacitance (noninertial) limit. This is treated in detail in Chapter 5. The relevant Langevin equation is [1]

^-^[I^-Ismm + W)], (2.4.1) at n

where the white noise driving current L (t) satisfies [note the difference in the spectral density in comparison with Eq. (1.4.25) since we are dealing with a current source]

1(0 = 0, L(tl)L(t2) = (2kT/R)S(tl-t2),

R is the resistance of the junction, / the amplitude of the supercurrent, Idc

is the bias current, <f> is the phase, e is the charge of the electron and h = hl27C, where h is Planck's constant.

In order to proceed we change the variable in the Langevin equation (2.4.1) by means of the transformation

rn=e-in*, (n = 0,±l,±2...), so that

±r"(t) = \r"-\t)-r^t)]-l^r"(t)[ldc + L(^. (2.4.2) dt n L J n

The multiplicative noise term r"(t) L(t) in Eq. (2.4.2) contributes a noise-induced drift term to the average. Equation (2.4.2) is of the form of Eq. (2.3.1), namely:

£(0 = h[(t),t] + g[£(t),t]L(t). (2.4.3) Hence, noting Eq. (2.3.9), we have

x = h(x,t) + Dg(x,t)—g(x,t), (2.4.4)

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178 The Langevin Equation

where £(t+ f), r > 0 is a solution of Eq. (2.4.3) which at time t has the sharp value %t) = x. The sharp value x in Eq. (2.4.4) is itself a random variable with probability density function W(x, t) defined such that W(x, t) dx is the probability of finding x in the interval (x, x + dx). Thus on averaging Eq. (2.4.2) over W (x, t), we obtain

—(x) = (h(x,t) + Dlg(x,t)—g(x,t)\ , (2.4.5)

where the angular braces mean the relevant quantity averaged over W(x, t).

We may now evaluate the average of the multiplicative noise term in Eq. (2.4.2). We have

8(rn) = -i2enRrn

g(rn)—grn)-. or

2neR

whence

d_r„ = eInRin-

dt h ' „n+l in2eR

Idc+kTR 2en

rn. (2.4.6)

After the second averaging over W, we obtain the hierarchy of differential-recurrence relations for the averages

iW-'fiW))- inleR

h Idc+kTR

2en

IT ( r " ) . (2.4.7)

We remark again that rn(t) in Eq. (2.4.2) and r" in Eqs. (2.4.6) and (2.4.7) have different meanings, namely, r"(t) in Eq. (2.4.2) is a stochastic variable while r" in Eqs. (2.4.6) and (2.4.7) is the sharp (definite) value r" (t) = r" at time t. (Instead of using different symbols for the two quantities we have used Risken's [1] notation and have distinguished the sharp values at time t from the stochastic variables by deleting the time argument).

Equation (2.4.7) may also be obtained from the relevant Fokker-Planck equation [1]. This equation is in the noninertial limit [1]

dW _ d

dt ~ dip + kT

d2W

d(p2 (2.4.8) W—U

where W(/),i) is the transition probability of the phase, ^is the damping coefficient defined as

( * \2

c= 2e R

and U is the tilted cosine potential given by

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Chapter 2. Langevin Equations and Methods of Solution 179

U = -hldc4> + lco^)l(2e). (2.4.9)

The function W is periodic in <j) so that it can be expanded in a Fourier series as [1]

W(<j>,t) = £ anf)ein*. (2.4.10)

Substituting Eq. (2.4.10) and Eq. (2.4.9) into Eq. (2.4.8) and using the orthogonality properties of the circular functions, one finds that the Fourier coefficients an(t) satisfy

«„(0 = k - , ( 0 - a „ + , ( 0 ] . (2.4.11) n

It can be easily shown that the an(t) of Eq. (2.4.10) are related to (rn) by

an(t) = rn)l(2n). (2.4.12) Thus Eq. (2.4.11) coincides precisely with Eq. (2.4.7). This is the first example of how a hierarchy of differential-recurrence relations for the statistical averages may be obtained directly from the nonlinear Langevin equation by using the definition of the noise induced drift and the interpretation of the Langevin equation not as a differential equation but as an integral equation. Thus the Fokker-Planck equation may be bypassed entirely. Further examples, will be given in Chapters 4-10.

We shall now extend the method to several dimensions.

2.5 Nonlinear Langevin Equation in Several Dimensions

We shall first state the result that enables us to construct the hierarchy of differential-recurrence relations for the multi-dimensional Langevin equation. We commence by interpreting the Langevin equation for a set of N random variables = ^1,...,^JV, namely,

4(0 = ^[^(0,?] + ^[^(0^]r ;(0 (2.5.1) with

f~(0 = 0, (2.5.2)

ri(t)rj(t') = 2D8ijS(t-t'), (2.5.3) as the integral equation [1]

r+r

£•(* + T) = xt + J [h, [W)A+Sij W)A?j(t')\dt', (2.5.4)

where £,- (t + T) (T > 0) is a solution of Eq. (2.5.1) which at time t has the sharp value £,(?) =x,^ for k= 1,2,..., N. We then apply the Stratonovich rule to show that

d .. kT —a„(t) + — dt " C

n +i-,rihl dc

2ekT

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180 The Langevin Equation

[Ut + t)-X:\ dgMx,t) l i m i ^ L_1\ =/z.(x,0 + Dg,.(x,0 * " " (2.5.5) «•->« T y dxk

(we use Einstein's summation convention, i.e., the summation is performed over indices appearing twice in the equations without writing down the summation sign). The r,-(f) are again Gaussian random variables with zero mean and with correlation functions proportional to the 8 function. We normalise these Langevin forces T, (t) in such a way that the correlation functions for different indices i are zero and that the factor in front of the £ function is 2D. Thus [bearing in mind the integral equation (2.5.4)], our procedure will show that the average ofEq. (2.5.1) may be expressed as an equation of motion for the set of sharp starting values x. Thus, we may generate directly the hierarchy of differential-recurrence relations which governs the time dependence of the transition probability. The first term in the right-hand side of Eq. (2.5.5) is the deterministic drift, while the second one is the noise-induced drift. It arises from the multiplicative noise term in Eq. (2.5.1).

Following Risken [1], we now proceed as in Section 2.3 first inserting the expansions

hi[$(t'),t'] = hi(x,t') + (Zk(t')-xk)r?-hi(x,t') + - , (2.5.6) dxk

gij[^t'),t'] = gij(x,t') + ^k(t')-xk)^gij(x,t') + .... (2.5.7) dxk

into the integral equation Eq. (2.5.4). We then obtain t+T t+T -\

^(t + T)-Xi=f hi(x,Odt'+j [£k(f)-xk]—htdxWdf ' ' (2.5.8)

t+T t+T -\

+ J gij(x,t')Tjt')dt'+ j rj(t')[£k(t')-xk]—gij(x,Odt'+... t t dXk

We iterate for [§£/)-x*\ under the integral signs in Eq. (2.5.8). By taking the average and recalling Eqs. (2.5.2) and (2.5.4), we then obtain

^ p. t+T t' -\

x, =/%(x,0 + l im— J J SjlSt'-t')gld(x,n^gy(x,ndt''dt'.

(2.5.9) On using the property of the ^-function, namely,

b

j S(b-x)y(x)dx = y(b)/2, (2.5.10)

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Chapter 2. Langevin Equations and Methods of Solution 181

we have

\ gkjx,t")2St'-t")dt"=gkjx,t') (2.5.11)

(2.5.12)

[similar to Eq. (1.10.14) again]. Thus, we finally obtain -\

xi=hi(x,t) + Dgkj(x,t)--gij(x,t) dxk

which is Eq. (2.5.5). In like manner, we can determine the averaged equation for an

arbitrary differentiable function/(% ). Indeed, noting that the rule for a change of variable in Stratonovich differential equations is the same as in ordinary analysis [7], the equation of motion for an arbitrary differentiable function/(\ ) may be obtained by cross-multiplying the ith Eq. (2.5.1) by <?/(§(f))/^-» respectively, and then summing them. Thus, we obtain a stochastic equation for/( \ ):

^/(§(0) = [/>»-(§(0.0 + ^(§(0.0r ;(0]^-/(§(0). (2.5.13)

As we already know, from a mathematical point of view, the stochastic differential equation (2.5.13) with the ^-correlated Langevin forces T/f) must be regarded as the stochastic integral equation

/(^+T))=/(x)+J[^(^0.0+^(R(0.0r/0]^§^*'. t °W

(2.5.14) Thus, supposing that the integrands in Eq. (2.5.14) can be expanded in Taylor series, we obtain

f(m+T))=f(x)+'J h^xit'^f^dt' t

t+T t

dx:

+ J[&(0-*J dxu OX:

dt'

+ 7 M W . O ^ ^ r , ( , > ' (2.5.15)

+ J [&(0-*J-

dxt

dX: Tj(t

,)dt' + ...

On substituting %kt')-xk fromEq. (2.5.8) into Eq. (2.5.15), we iterate

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182 The Langevin Equation

f(W+T))=fax)+Jhi(x,t')*fjx])dt' . oxt

t'

jhk([x,t")dt'dff t+T -,

t dxk *,<w,^/(W)

ix,

t+T -,

dxk OX:

t

\gknx,t")Tnt')dt'dt'

+7 **(M.O^^r/*>' (2.5.16)

t+T -,

, dxk ^ W . ^ / ( W )

dX; rj(t')]hk(x,t")dt'dt'

t+T -,

t 9xk OX; Tjt')\gkn(x,t")Tn(t")dt"dt'+..

Then, averaging Eq. (2.5.16), taking account of the white noise properties, Eqs. (2.5.2) and (2.5.3), and retaining only terms of the order of T, we have

, d /(§(*+ *)) = /(*)+ f \x\t')--fx\)dt'

t+T -\

+DSjnj gkn(x,t')—

dX:

gijax,t')j-f(x)

(2.5.17)

dt' + o(T).

After obvious transformations in Eq. (2.5.17), we obtain

T OXt

+Dgkj(x,t + l®$)~-OXu ' ^ lik) dX;

gMx,t + l®™)—f(x)

(2.5.18)

+ o(l),

where © ^ are constants (0< © ^ < 1; cf. the mean value theorem). On

taking the limit r-> 0 in Eq. (2.5.18), we have

,. f(m+r))-f(x))_df(x) hm T->0

•hi(x,t)-^-f(x) + Dgkj(x,t)-f-OX: O Xj,

dt

^ • ( x ] , 0—/ ( x ) OX:

(2.5.19)

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Chapter 2. Langevin Equations and Methods of Solution 183

For a single variable Langevin equation, Eq. (2.5.19) becomes

. (2.5.20) df (x) , , . d ., . ^ . N d JK J=hx,t)—f(x) + Dg(x,t)— g(x,t)—f(x)

ox dt dx dx Equations (2.5.12), (2.5.19) and (2.5.20) are the most important results of the Stratonovich calculus. We shall them use throughout the book.

2.6 Average of the Multiplicative Noise Term in the Langevin Equation for a Rotator

As a simple example of the application of the noise induced drift, Eq. (2.5.12), let us consider the planar rigid rotator including inertial effects which is treated in detail in Chapter 10. The Langevin equation for a rigid electric dipole of moment of inertia / and dipole moment ju, rotating about an axis normal to itself and under the influence of an external electric field E(r) is [8]

16t) + 0t) + juE(t) sin 9(t) = X(t), (2.6.1)

where 6t) is the angle the rotator makes with the direction of E(t), (6t) and Alt) are the frictional and white noise torques due to the surroundings and Mt) has the properties:

1(0 = 0, A(tl)A(t2) = 2kTCS(tl-t2) (2.6.2)

[the overbars denote the statistical averages over a large number of rotators which have all started from the same (sharp) point in the configuration-angular velocity space (#,#)]. The A's obey Isserlis's theorem (Chapter 1, Section 1.3):

*( ' iM( ' 2 „ + i ) = V A , + i = 0 , (2.6.3)

^ ) 4 M ( ^ ) = A = Z I 1 ^ J ' (2-6-4) ki>kj

where the sum is over all distinct products of such pairs, each of which is formed by selecting n pairs of subscripts from In subscripts.

Let us now write in Eq. (2.6.1) r = e-w (2.6.5)

so that the Langevin equation becomes

Ift) + CKt) + I02(t)r(t)--juE(t)[l-r2(f)] = -iX(t)r(t). (2.6.6)

Equation (2.6.6) unlike Eq. (2.6.1) contains a multiplicative noise term on its right-hand side. Let us now introduce the random state variables

6 = r, & = '> (2-6.7) so that Eq.(2.6.6) becomes

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184 The Langevin Equation

£(0 = 6(0, (2.6.8)

&«) = - ^ « ) + f! + f [ l - £ w ] - ^ p . (2.6.9) Here Mt) =Mf). It is evident that Eqs. (2.6.8) and (2.6.9) are a set of nonlinear stochastic differential equations for the random variables £i and %2,

£ ( 0 = hv [£(*),*] + By [fc(0, ']r , .(0 (2.6.10) with i, ; '=1 and 2, and

Mt) = 2(t), h2(t) = -^2(t)+&^+^[i-•tfw], (2.6.11)

8n(t) = gl2(t) = 0, g21(t) = 0, g22(t) = -i^(t)/I. In accordance with the integral equation (2.5.4), we must identify r(t),r(t) with ^(O.&CO and r,r with the sets of sharp values x = xux2. Thus we have from Eqs. (2.6.8)-(2.6.11) and (2.5.12)

xl=hl + kT£

i 2 = /ij + kT£ Sn

3;

3x,

611 - , +512 - , +521 -^ + <522 -N

dxl ax2 ox2 j

+ 8 ^ + ^ + 8 ^ * dx. 3x, 3x,

(2.6.12)

(2.6.13) Vj WJV.J U A 2 " ^ 2 J

According to Eq. (2.6.11), the noise induced drift term vanishes, thus £ and X are independent random variables and the average value of £ satisfies

Ir + £r + I02r-ilE(l-r2)/2 = O. (2.6.14) If we now take a second average over the distribution W of (r,r) at time t, we have (the angular braces denoting this second average)

/ ( r ) + ^(r) + ( / ^ 2 r - / / £ ( l - ( r 2 ) / 2 = 0. (2.6.15)

This is the first member of the hierarchy of differential-recurrence relations for ( r ) generated by the relevant Fokker-Planck equation [8,9]. The complete hierarchy is derived in Chapter 10.

Yet another example is the evaluation of the multiplicative noise term for the three-dimensional rotator including inertial effects.

2.6.7 Multiplicative noise term for a three-dimensional rotator

The Euler-Langevin equations describing the Brownian rotation of a thin rod representing a polar molecule, which is subjected to an external electric field E(0 making an angle z?(0 with the dipole axis are in the notation of McConnell [10] (see also Chapter 10, Section 10.4)

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Chapter 2. Langevin Equations and Methods of Solution 185

Id)x(t) = Ia);(t)cotiKt)-£(Ox(t)-iEsm&(t) + Ax(t), (2.6.1.1)

Icoy (t) = -lay t)Q)x (t) cot 0(r) - Ccoy (0 + Xy (t). (2.6.1.2)

Here we take rotating orthogonal coordinate axes through the centre of the rod, the rod lying along axis z so that the components of the angular velocity are (cox,a>y,a>z) = (•&, #>sin&, ^>cosd) and the components of the angular momentum are (Id, I<p sin t?, 0) (/ is the moment of inertia of the rod and (p is the azimuthal angle). We suppose, with / and m denoting different axes,

XxV) = XyV) = 0, MijIjf) = 2kTCSlmS(t-f). (2.6.1.3) We now introduce the state variables

6 = cos •&, £, = -tfsin tf, Zi=G)y, (2.6.1.4)

so that Eqs. (2.6.1.1) and (2.6.1.2) become

6(0 = 6(0, (2.6.1.5)

--^(062(0+^(l-^(0)-yVl-^(0^(0,

(2.6.1.6)

(2.6.1.7)

6(0- 76<o w

t(t. cut), 6(06(0 , (o

where /^(O = /^(O and /^(f) = Xy(t). The averaged Eq. (2.6.1.6), where

xp x2, JC3 are the sharp initial values of £,, <f2, £,> is

X> = -^ * 2 1 2

l-xt -x^tUliO + ^l-xt)

+kT£ Sn^ + + S ^ + zJ-^+sJ822

dx. dx. dx. dx-, dx-,

+to^a+«..^L+ta^+«»-,!B

(2.6.1.8)

3x9 3x, 3x, 3x 3 ;

J 1-Xj /

where we note that all gy's are zero except

g22=-r\i-x?f2, g33 = rl. (2.6.1.9) Thus the noise induced drift term also vanishes for this system showing that X(t) and p(t) are again independent random variables. The above

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186 The Langevin Equation

discussions pertained to the inertial response whence the noninertial response may be found as described in Ref. [11] by either letting / tend to zero or 6 become very large in the averaged equations. We now consider the noninertial limit, i.e., we let the moment of inertia 1 = 0 prior to the averaging procedure and we demonstrate that now the average of the multiplicative noise term is not zero.

2.6.2 Multiplicative noise terms with I taken as zero prior to averaging

In order to study the contribution of the multiplicative noise terms to the averaged Langevin equation for the rotational Brownian motion of a linear molecule with the dipole moment u , where I is taken as zero prior to the averaging process, it will be convenient to use the kinematic equation (see Section 1.15)

ji(t) = o>(t)x\i(t), (2.6.2.1) where (a(t) is the angular velocity of the rotating body which is supposed homogeneous and which is subject to random couples which have no preferential direction, and to an external electric field E (t). The angular velocity oo obeys the Langevin equation

76(0 + C<o(t) = Ut) + pit) x E(0 . (2.6.2.2) In the noninertial limit, if we set 1 = 0 in Eq. (2.6.2.2), we have with Eq. (2.6.2.1)

C\i(t) = Ut) x \i(t) + [fi(0 x E(0] x |i(r). (2.6.2.3) The vector Langevin Eq. (2.6.2.3) has the same mathematical form as that of the rotating sphere model [see Section 1.15, Eq. (1.15.6); so we need not confine ourselves to the linear rotator].

By introducing the stochastic variables

6(0 = MO, &(0 = M0, £3(0 = "z(0, where ux(t), uY(t), and uz(t) are the Cartesian components of the vector u (u is a unit vector along ji) in the laboratory coordinate system OXYZ, we have from Eq. (2.6.2.3) (E = E k, i.e., E is directed along the Z axis),

6(o=rl4(o6(o-4(o&(o]-r^£6(o6(o, (2.6.2.4) 4 (0 = r ' [4(06 (0-4 (06 (0] - ClME£2 (06(0 , (2.6.2.5)

6(0=r1K(062(0-4(06(0]+6_1^[62(0+#22(o]. (2.6.2.6)

Equations (2.6.2.4)-(2.6.2.6) contain^ multiplicative noise terms. It is convenient to evaluate the averages 6 , i= 1,2, 3 by using Eq. (2.5.12). The gyAj terms give rise to noise induced drift components. On noting that the components of the tensor gy are given by

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Chapter 2. Langevin Equations and Methods of Solution 187

Sn=° 812=%3/C 8n=-^/C

82i=-&'£ ^22=0 823=&'£ (2-6.2.7)

831=^2^ 832 =-^>C 833=° we find

£ + (2*77 0 £ = - £ £ / / £ / £ \ (2.6.2.8)

£2+(2kT/£)&=-&3ME/C, (2-6.2.9)

4 + (2*770& = ( 1 - ^ ) ^ / ^ (2.6.2.10)

since the noise induced drift associated with the £ variable is, from Eqs.

' n 3&i J. „ 3&2 ^ „ 3&3 j . „ a&i j . „ d8i

8n-rr+8n^r+8i3^rr+821^?-+822 2

(2.5.12) and (2.6.2.7),

+ S a ^ + S 3 , ^ + &ii + ft3^] = - 2 ^ f i . (2.6.2.1!)

d£2 d£3 d£3 d£3 J £

Thus, we have proved that if / is set equal to zero prior to the averaging process, the average of the noise induced terms is not zero.

In order to formally specialise Eq. (2.6.2.3) to the two dimensional rotator in the XY plane, we suppose that in Cartesian coordinates (x,y, z) s (1,2,3)

o = cok, fi = //(£i + £2j), E = Ei, X, = Ak .

Equation (2.6.2.3) then reduces to the simultaneous differential equations

£l(t) = -ClMt)£2(t) + ClM(l-g(t))E, (2.6.2.12)

& (0 = C'Mt)^ (0 - C'ME^ (*)& (0 (2.6.2.13)

or d . . . uEr „„ . . ,-, A(Osin0(O .. , . . .. —cos 0(0 + —[cos 20(0 -1] = — - L — — — , (2.6.2.14) dt 2(, (,

d . n, . U.E . -n, . ,1(0COS0(0 , ^ 0 1 ^ —sin 0(0 +—-sin 20(0 = —— — , (2.6.2.15) ^ 2£ £

i.e., with r = e ld,

rt)-Efi(\-r2(0)/(2O = -iA(t)r(t)/C • (2.6.2.16)

On evaluating the noise-induced drift term in Eq. (2.6.2.16) as

-iAt)r(t)l C = kT£-il (f rd rr = -kT I £)r, (2.6.2.17) we have

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188 The Langevin Equation

r + f r = ^(l-r>). (2.6.2.18)

For E = 0, the solution of Eq. (2.6.2.18) is r = roeHtr,0'=e-io0-i^O't (2.6.2.19)

where 60 is the sharp initial value. All the results above have been obtained by using the average of

the noise induced drift Eq. (2.5.12). In certain cases an explicit solution of the Langevin equation may be written down, hence the above results may be obtained directly. We illustrate this by considering the two-dimensional rotator model in the absence of an external field. We utilise the results of the Ornstein-Uhlenbeck process introduced in Chapter 1, Section 1.7 and described in detail in Chapter 3, Section 3.1 et seq.

2.6.3 Explicit average of the noise induced terms for a planar rotator

The Langevin equation for the planar rigid rotator in the absence of an external field is (see Section 2.6)

W(t) + C0(t) = Mt) (2.6.3.1) or

/ r(t) + £r(t) + W2(t)r(t) = -iA(t)r(t). (2.6.3.2) We suppose, following Uhlenbeck and Ornstein [12], that we have an assembly of rotators each of which starts from an initial orientation 60

with initial angular velocity 60 . (These are sharp initial conditions so that we are again concerned with the transition probability). We suppose [12] that the mean of A (t) at a given value of t over an assembly of similar, but independent rotators which have started with the same angle 60 and

angular velocity 60 is zero. We denote this by 1(0 = 0 . (2.6.3.3)

We also suppose once again that

A(tl)At2) = 2kT£S(tl-t2) (2.6.3.4)

and that the A's obey Isserlis' theorem. We wish to prove that A(t)r(t) = 0. We have

d(t) = 0oe-/"+-\ e - ^ ' - ' U f o ) * ! . (2.6.3.5) ' o

and

0(t) = eo+0j-i(l-e-/*) + jp\ [l-e-^y(tl)dh, (2.6.3.6)

where /? = £"//. Introducing notations

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Chapter 2. Langevin Equations and Methods of Solution 189

rt) may be explicitly written as r(t) = e-ig(t)e-m. (2.6.3.7)

Thus, on averaging over a large number of rotators, which all start from the same initial point, and remembering that g (t) is deterministic for the

purposes of this calculation since 00,60) is a sharp point we have

(2.6.3.8) r(t)Mt) = e _„ - * ( ' ) „-if(0 A(t). Equation (2.6.3.8) is

5 0 Z-t> 00

-'g(t) 1

•3 lit

->-<s 000

XA(?1)/t(f2)^(f3)/l(?)6fr1^2^3 +... (2.6.3.9)

We evaluate the averages in Eq. (2.6.3.9) term by term. The average of the first term is equal to zero according to Eq. (2.6.3.3). We have for the second term

j[l-e~^M|)~U(ji);i(0dir, =2DJ\l-e-fi['"l)~\s(ti -t)dtx = 0. (2.6.3.10)

The average of the third term vanishes because A(t )X(t2 )X(t) - 0 by Isserlis's theorem. Likewise, the average of the fourth term is

AD2?'''

3-£ oo \\\ ['-"*"": l-e rPi'-h) l-e -P('-h)

x\_S(t3 -12 )S(tl-1)+S(t3 - tx) S(t2 -1)+S(t3 -1) S(t2 -tx jjdt^dt^.

Now

J [«-• -0('-hY S(t3-t)dt3 = 0 ,

so that by symmetry the average of the fourth term vanishes and so on. Hence, we conclude that

Mt)r(t) = 0, (2.6.3.11) as has been demonstrated implicitly above by regarding Eq. (2.6.3.2) when written in state variable form as a Stratonovich equation.

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190 The Langevin Equation

There remains the special case where / is taken as zero prior to averaging. The Langevin equation (2.6.3.1) becomes

£0(0 = X(t) or Cr(t) = -iA(t)r(t). (2.6.3.12) On taking the average for a sharp initial conditions 0(0) = #0 , we have

-(i/Oj Hh)dh

r(t)Mt) = e'l0° X(t)e ° f • / -3 t t t ]

=e-l6°\x(t)--\ mMhWi—rjff j Kh)X(t2)Xm)X(tyhdt2dh+..\ [ 5 o 3!£ 000 J

( kT \ = -ie-,e°kT\l-—t + ... =-ikTe-,0°HkTI°' =-ikTr, (2.6.3.13)

thus, also showing that if / is taken as zero prior to the averaging r(t)A(t)^0, so that Xt) and r(t) are no longer independent random variables as we concluded from our implicit calculations above.

2.7 Methods of Solution of Differential-Recurrence Relations Arising from the Nonlinear Langevin Equation

In general, the equations that are generated by averaging the nonlinear Langevin equation or the accompanying Fokker-Planck equation take the form of three- (or higher-order) term differential-recurrence relations between the set of statistical averages (moments) describing the dynamical behaviour [cf. Eq. (2.4.7) above]. Thus, the behaviour of any selected average is coupled to that of all the others so forming an infinite hierarchy of the averages. The time behaviour of the first-order average, for example, involves that of the second-order average, which in turn involves the third-order average, and so on. If the recurrence relation between the averages is a three-term one, the Laplace transform of the solution for a step stimulus may be expressed in terms of infinite continued fractions [1] (see Chapters 4-7). In an actual numerical solution of the problem, successive convergents of the continued fraction are calculated until convergence is reached when a solution is deemed to have been obtained. If the underlying recurrence relation is not a three-term one, such multi-term recurrence relations may often be reduced to a three term matrix recurrence relation, which may be solved in terms of a matrix continued fraction [1], the scalar continued fraction being a particular case of this. Numerical solutions may always be obtained (albeit not in as convenient a representation as the continued fraction) by writing the set of differential-recurrence relations as a first-order matrix differential equation,

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Chapter 2. Langevin Equations and Methods of Solution 191

X(t) = AX(t). (2.7.1) The size of the system matrix A is then successively increased until convergence is attained. We shall first describe the matrix method of solution of Eq. (2.7.1).

2.7.1 Matrix diagonalisation method

We consider the system of equations (2.7.1), where

' * i ( 0 N

x(0 = xM)

and A -Ml is an n x n matrix with constant real elements a </•

We recall some concepts of linear algebra [13]. A vector g^O is called an eigenvector of matrix A, if

Ag = Ag. Number X is called the eigenvalue of A corresponding to the eigenvector g and is a root of the characteristic equation

d e t ( A - a i ) = 0, (2.7.1.1) where I is the identity matrix.

We assume that all the eigenvalues Xu Xj, ..., X„ of A are distinct. Thus, the eigenvectors gi, g2, •••, gn are linearly independent and there exists a n n x n matrix T that reduces A to diagonal form, i.e., such that

'4 0 : (P 0 X1 \ 0

A = T "AT =

0 0 I

(2.7.1.2)

•nj

The columns of T are the components of the eigenvectors gi, g2,..., g„. We introduce the following concepts. Let B(?) be an nxn

matrix whose elements by (t) are functions of t defined on a set Q.. The matrix B (?) is said to be continuous on Q., if all of its elements bjj(t) are continuous on Q.. The matrix B (?) is said to be differentiable on Q., if all the elements by (t) of the matrix are differentiable on Q. The derivative d B ( 0 / dt of the matrix B(0 is a matrix whose elements are the derivatives dby (t)l dt of the corresponding elements of B (t). Using the rules of matrix algebra, we observe by direct calculation that the following formula holds:

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192 The Langevin Equation

A [ B W X ( 0 ] = ^ X ( 0 + B ( 0 ^ . (2.7.1.3) dt dt dt

Specifically, if B is a constant matrix, then

|[BX<,)] = B f . We shall now prove the theorem that if the eigenvalues Au X,...,

Xn of a matrix A are distinct, then the general solution of the system (2.7.1) has the form [13]

X(t) = c ^ ' g , + Cze*g2 +... + cne*»'gn, (2.7.1.4) where gi, g2, ..., g„ are the column eigenvectors of A, and cu c2, ..., cn are constants to be determined from initial conditions.

We introduce a new unknown column vector Y(/) by X(0 = TY(0 , (2.7.1.5)

where T is a matrix that reduces A to diagonal form. Substituting X(0 from Eq. (2.7.1.5) into (2.7.1) yields as shown

TY = ATY. - l

Multiplying both sides of this by T and defining the matrix A by the - l

equation T AT = A, we have Y(r) = AY(0

or, taking into account Eq. (2.7.1.2),

* 2 = ^ 2 ' (2.7.1.6)

jn=Kyn-We have thus obtained a system of n independent equations which is easily integrated to yield

?i = qe*' . 0ht • yn v y2 = c2e

Here c\, c-i,..., cn are arbitrary constants. Introducing unit n-dimensional column vectors

(0^

e, =

0

0

vOy

e, =

1

vOy

..., en =

(0

0

0

Kh

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Chapter 2. Langevin Equations and Methods of Solution 193

we can represent the solution Y(7) as

Y(0 = cxe** zx + c2e h*t + ... + c„eAn'e„ (2.7.1.7) By Eq. (2.7.1.5), we have X(f) = TY(f)- Since the columns of T are eigenvectors of A, then Te^ = gk, where gk is the kth eigenvector of A. Therefore, substituting Eq. (2.7.1.7) into Eq. (2.7.1.5), we will obtain Eq. (2.7.1.4) for X(0-

Thus, if the matrix A of the system (2.7.1) has distinct eigenvalues, to obtain the general solution of the system it is necessary:

(1) to find the eigenvalues X\, A%,..., A„ of the matrix as roots of the algebraic (characteristic) equation det (A - A, I) = 0;

(2) to find all the eigenvectors gi, g2,..., g„; (3) to write the general solution of (2.7.1) using Eq. (2.7.1.4).

To illustrate this procedure, we take the simple example given in [13] d(x~\

or 3 T 2 2, y)

J x = 3x + y |>> = 2x + 2y dtyj

(1) We set up the characteristic equation 3-A 1

det(A-Al) = 2 2-X

The roots of the equation are X\ = 4 and Xj = 1. (2) We now find the eigenvectors

= A2-5/1 + 4 = 0.

gi = §2

r8n

\822J

For k\ = 4, the system becomes

-Sn+8i2 =

\2Sn-28i2

Hence, gn = gn and we may take g

For Xi= 1, we similarly find g2 =

(3) Using Eq. (2.7.1.4), we obtain the general solution of the system as (xt)

y(t). = c,e

4t n vi ;

+ c7e

or x(t)- • C i c- ~t~ C ^ c •

• 2 ;

yt)-ceAt -2c2e'.

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194 The Langevin Equation

2.7.2 Initial conditions

The constants cn in Eq. (2.7.1.4) are to be determined from the initial conditions of the problem under consideration. We illustrate how this is accomplished by referring to the particular problem (treated in detail in Chapter 4) of dielectric relaxation of a single axis rotator with two equivalent sites. Here a rotator is constrained to rotate about an axis normal to itself under the influence of the potential

V(0) = £ / s in 2 0- / /£cos0 . (2.7.2.1) Here, Usin20 and -juE cos 8 are the potentials due to the crystalline field and an applied small uniform field E, respectively, 0 is the angle describing the orientation of the dipole u about its axes of rotation. As shown in Chapter 4, having switched off the field E at an initial time t = 0, the Langevin equation leads to the set of differential-recurrence relations for the after-effect functions

f2p+l(t) = (coS(2p + l)e)(t), (p = 0,+l,...), (2.7.2.2)

namely

*b/2P+i(0 + (2P + D2 /2 p + 1(0 = °VP + D [ / 2 p - i ( 0 - / 2 p + 3 (0] , (2.7.2.3) where a= U/(2kT) is the barrier height parameter (the factor 2 in the denominator arises so as to avoid 2a in the arguments of the Bessel functions below).

The set of Eqs. (2.7.2.3) may be solved numerically in the manner used for Eq. (2.7.1) by forming the vector

X(0 = hit)

fiP+i(t)j

and writing them in the form

where

A =

l - O "

-3a

0

X(r) = AX(0,

a 0 9 3o-

-5a 25

0

0

5o-

0 0

0

V "

(2.7.2.4)

(2.7.2.5)

(2.7.2.6)

•• J

In Eq. (2.7.2.6), p is taken large enough (equal to P, say) to ensure convergence of the set of Eq. (2.7.2.5). For o-less than 10, a 25x25

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Chapter 2. Langevin Equations and Methods of Solution 195

matrix is sufficient [14]. Assuming that the eigenvalues Ap of Eq. (2.7.2.6) are all distinct, the solution may now be represented in the form

X(0 = cxe-*% + c2e~*g2 +... + c„e-**'gn , (2.7.2.7) where the constants cp are to be determined from the initial conditions. Here, we have used the fact that all the eigenvalues Ap of the matrix A are negative. In this notation, the eigenvalues correspond to the eigenvalues Ap of the corresponding Fokker-Planck operator [1]. In general, one can prove that the real parts of eigenvalues Ap of the Fokker-Planck operator are always positive [1]. Below, we shall explicitly use a minus sign for the eigenvalues Ap of A.

The initial value vector X(0) is determined as follows. At time t = 0, the steady field E (t) is switched off. Thus

2n

\ cos[(2p + l)^>CTCOs2e[l+^cos^]^

/2p+1(0) = (ccK[(2p + l)fl)(0)=-2- 2n

f „ercos20| \+€<x&e\do

(2.7.2.8) where <f = juE/(kT) and we have supposed that the external field is weak so that £ « 1. We also have [15]

„<TCOS20 = Z Li*)* ilm0 (2.7.2.9)

where Im(d) is the modified Bessel function of the first kind of order m [15]. Whence, using the orthogonality property of the circular functions and noting that I-m(cr) = Im(cr), we have

~2n

/ 2 p + i ( 0 ) = - iZ'm(^''2 m* 0 m=0

ei2(P+l)0 +e-i2(P+l)0 de

2n oo

+ J Z Im(o-)ei2m9[ei2pe +e-i2pe]dd 0 m=0

2n

IZij&xt^do 0 m=0

-1

(2.7.2.10)

/•W<r) + W<r) , Lp((r) + Ip«T) ^lp^(a) + Ipa)

2 /0(cr) I0(a) I0(a) so yielding the initial value vector X0.

The set of Eqs. (2.7.2.5) may now be solved to any desired degree of accuracy to yield the decay of the longitudinal component of the dipole moment (after-effect function) in the form

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196 The Langevin Equation

jufl(t) = Mcos0)(t)=^-fd a2*+i«~vW- (2.7.2.11) *•*• * = o

The numerical values of the first three eigenvalues and amplitudes a2k+x

are given in Table 4.4.2.1, Chapter 4.

2.7.3 Matrix continued fraction solution of recurrence equations

The solution of the Langevin equation (or the Fokker-Planck equation) can be reduced to the solution of an infinite hierarchy of equations for the moments (the expectation values of the dynamical quantities of interest) describing the dynamics of the system under consideration. Some examples were given above. Further examples will be presented in Chapters 4-11. In general, hierarchies of moment equations take the form of three- (or higher-order-) term differential-recurrence relations. Thus, the behaviour of any selected average is coupled to that of all the others so forming an infinite system of moment equations. The time behaviour of the first-order average, for example, involves that of the second-order average, which in turn involves the third-order average, and so on. If the recurrence relation between the averages is a scalar three-term one, the solution may be expressed in terms of ordinary infinite continued fractions [1]. In the majority of problems, however, the Langevin equation may not be reduced to a scalar three-term recurrence relation. Hence, the method of solution of the recurrence relation in terms of ordinary continued fractions no longer applies. Examples of this are problems involving diffusion in phase space and diffusion in configuration space, where the potential gives rise to a four- (or higher-) order term recurrence relation. These difficulties may however, be circumvented since we have a method of converting a multi-term scalar recurrence relation to a three term matrix one [1]. In other words, a multi-term recurrence relation may be reduced to a matrix three-term recurrence relation, which may be solved in terms of matrix continued fractions [1,16] (see Chapters 8-11). In the present section, we shall confine our discussion to matrix continued fractions as the scalar continued fraction is simply a special case of the matrix one.

Such a three-term matrix recurrence relation, in the notation of Risken [1], may be written down in general form as

T£Cp(t) = Q-pCp_1(t) + QpCp(t) + Q+pCp+l(t), (p>l ) , (2.7.3.1)

where zE is a characteristic relaxation time and Cp(t) are column vectors

which in general have P components [with C0(t) = 0] and the Q~, Q*,

and Qp are time independent noncommutative matrices. A general

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Chapter 2. Langevin Equations and Methods of Solution 197

method of solution of a tridiagonal vector recurrence relation in terms of matrix continued fractions is described by Risken [1]. We now present another method for the solution of Eq. (2.7.3.1) [16], which has the merit of being considerably simpler than the previously available algorithm.

The solution of the three-term recurrence relation above may be accomplished as follows. On taking the Laplace transform of Eq. (2.7.3.1), we have

Q;Cp_, (s) + (Q , - STel) Cp (s) + Q;Cp+1 (S) = -TeCp (0), (2.7.3.2)

where I is the unit matrix and oo

Cp(s) = \ Cp(t)e-S'dt. o

Let us seek the solution of Eq. (2.7.3.2) in the form

Cp(s) = Sp(s)Cp_l(s) + Rp(s), (2.7.3.3)

where the matrix continued fraction Sp(s) is defined as

Sp(s) = [sreI -Qp- Q;Sp+l (s)Jl Qp. (2.7.3.4)

The recurrence Eq. (2.7.3.4) allows us to represent Sp(s) as the infinite matrix continued fraction

M * ) = — i Qp T£sI-Qp-Q

+p j Q;+1

Vi-QP + i ~^U—Tn Q '+ 2

T£SL - Klp+2 ~ • • • (2.7.3.5)

(the fraction lines designate the matrix inversions). The infinite matrix continued fraction Sp(s) represents the complementary solution of Eq. (2.7.3.2). The particular solution ~Rp(s) may be found as follows. On substituting Eq. (2.7.3.3) into Eq. (2.7.3.2), we obtain

Q;Cp_,(S) + (Qp -sr£l)[Sp(*)£,_,(s) + Rp(s)]

+Q; Sp+1 (S) [Sp (*)Cp_, (s) + Rp (S)] + Rp+l (s) = -TeCp (0)

or, noting Eq. (2.7.3.4),

[ST£1-QP -Q+pSp+l(s)]Rp(s)-Q+

pRp+l(s) = T£Cp(0). (2.7.3.6)

Thus, we have

R ^ f w . I - Q , -Q+PSP+1(S)]1[T£CP(0) + Q+

PRP+1(S)]. (2.7.3.7)

The two-term recurrence relation for Rp(s), Eq. (2.7.3.7), can be solved by iteration to yield

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198 The Langevin Equation

Rp(s) = Te[srel-Qp-Q+

pSp+l(s)] C„(0)

+X f l QWi[s^-Qk+P~QlPsk+l+P(s)]x cp+n(0) n=\ \k=\

The substitution of Eq. (2.7.3.8) into Eq. (2.7.3.3) yields [16]

Cp(s) = Sp(s)Cp_l(s) + Te[sTeI-Qp-Q;Sp+l(S)Jl

(2.7.3.8)

C„(0)

oo f n

+i no; n=l V * = l

p+k-l [sTel-Qp+k-Q+

p+kSp+k+l(s)J p+n

(0)

(2.7.3.9)

Equation (2.7.3.9) constitutes the solution of Eq. (2.7.3.1) rendered as a sum of products of matrix continued fractions in the s domain.

Thus, the exact solution for the Laplace transform of Ci(f) of Eq. (2.7.3.1) with C0(O = 0 is given by [16]

x

Cl(s) = Te[sreL-Ql-($S2(s)]'

Ci(°)+Z \t\Qt[^-Qk+i-QlA+2(s)] |c„+1(0) n=\ \k=\

The solution (2.7.3.10) may also be written as

(2.7.3.10)

Cl(s) = Te\l(s)\Cl(0) + YJ k=\

CB+1(0) , (2.7.3.11)

where the matrix continued fraction \s) is defined as

Ap(s) = [sT£I-Qp-Q+

pAp+l(s)Qp+l]~l. (2.7.3.12)

In many physical applications, the initial conditions Cp(0) can be expressed in terms of equilibrium (stationary, in the general case) averages, which are in fact the solutions of the time-independent vector recurrence relation (2.7.3.1) [1]:

Q ^ , +QpC°p +Q;cJ + 1 = 0, (2.7.3.13)

where the column vector C°p is formed from the equilibrium (stationary) averages in the same manner as the vector Cp(t). Equation (2.7.3.13) has a tridiagonal form, so that it is possible to express the equilibrium averages C^ in terms of the matrix continued fraction Sp(0) from Eq. (2.7.3.5). In order to accomplish this, let us consider Eq. (2.7.3.13) for p = 1. We have

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Chapter 2. Langevin Equations and Methods of Solution 199

Qrc°+Q,c?+Qrc°=o or, equivalently,

where the column vector CQ is given by

(2.7.3.14)

(2.7.3.15)

2 c o (2.7.3.16)

We may always choose a particular element of CQ , for example, CQ = 1, because of the normalisation condition. Equation (2.7.3.15) then constitutes a set of inhomogeneous linear equations; hence, the other components CQ,C^,...,CQ~1 and thus the vector CQ can be completely determined. We can then find the other vectors C°p by iteration

C° =Sp(0)Sp_1(0)...SI(0)Cg (2.7.3.17)

and thus calculate all the Cp(0). As an example of the formulation of a problem in terms of matrix

continued fractions, we consider the problem of rotational Brownian motion in a double well cosine potential treated in Section (2.7.2). However, this time, the applied electric field E is not small, neither is it switched off and we wish to determine the relaxation behaviour following a small change in E [16]. The differential-recurrence relations are

^(t) + P2fP(t) = crp[fp_2(t)-fp+2(t)] + ^[fp_l(t)-fp+l(t)]

(2.7.3.18) with

fp(t) = (cos p0)- (cos pd)0, (2.7.3.19)

where the zero denotes that the average is to be evaluated in the absence of the perturbation, and/_p(0=^(0- Equation (2.7.3.18) is a five-term scalar recurrence relation. However, it may be readily cast into the form of the three-term matrix recurrence Eq. (2.7.3.1) by using the method described by Risken [1] (see Chapter 8). Here, it will be convenient to define the column vector

CD(t) =

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200 The Langevin Equation

Hence, Eq. (2.7.3.18) becomes

T Z , C P ( O = Q ; C P _ 1 ( O + Q / , C P ( O + Q ; C P + 1 ( O , (P>2), (2.7.3.20)

where the matrices Q~, Qp, Q+p are given by

Q"P = M 2 p - 1 ) #(2p- l ) /2 N

0 lap ) ' r - ( 2 p - l ) 2

£p

Q: = -£P

0 s^

-lap

% = -<f(2p- l ) /2

This model is treated in more detail in Chapter 8, Section 8.2. For the particular case of a scalar three-term differential-

recurrence relation, Eq. (2.7.3.1) may be written as

T£Cp(t) = q-pCp_l(t) + qpCp(t) + q+pCp+l(t), (p>\), (2.7.3.21)

where the Cp(t) are the appropriate relaxation functions [with C0(t)-0],

and the q~, q*,and qp are time-independent coefficients. On applying

the Laplace transformation to Eq. (2.7.3.21), we obtain the three-term recurrence equation in the s domain:

q-pCp_l(s) + (qp-STe)Cp(s) + q+pCp+l(s) = -TeCp(0). (2.7.3.22)

The exact solution of the scalar recurrence Eq. (2.7.3.22) can be found by the same method as that of the matrix three-term recurrence Eq. (2.7.3.2). Formally, the solution for the Laplace transform C\(s) can be obtained from Eq. (2.7.3.10) by replacing the matrix quantities

C ^ X C / O X Q ^ Q ^ a n d Q , ,

by the corresponding scalar ones

C,(5), Cp(0), qp,q+

p, and qp,

respectively. This exact solution is given by

Q(s) = v - ? i -qts2(s)

C i W + I X ^ o ) ! ! ^ 5 ^ n=\ t=l ft+1

(2.7.3.23)

where the infinite scalar continued fraction Sn(s) is defined as

<Q _ In Sn(s) = V - 9 n - 9 i X n ( * )

v-ft.— qnQn+l

v- 1n +1 %+\Qn+2

*es-qn+2--(2.7.3.24)

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Chapter 2. Langevin Equations and Methods of Solution 201

As an example, we consider the recurrence Eq. (2.7.3.18) with the parameter £=0 (in order to have a scalar three-term recurrence equation). Thus, we have the recurrence equation for the odd-index statistical moments f2p+i(t) = (cos(2/? + \)0)t), (p > 0), only:

*b/2p+i 0) + (2p + D2 f2p+i (» = °Vp + 1 ) [ / 2 H (t) - f2p+3(t)] ,(2.7.3.25)

which, on noting that fMO - fxt), may equivalently be written as

T D / 2 / , + I ( 0 = <tip+\hP-\V) + <lip+\hp+\(0 + ^ + i / 2 / , + 3 ( 0 , (2.7.3.26) where

l2P+i=rfop-(2P + V2> l2P+i=-<r(2P + V> qlp+l=<y2p + \). (2.7.3.27)

Thus, according to Eq. (2.7.3.23), we have the exact solution

%/i(0) Ms)

where

TDs- •9i -4iSi(s)

i , y/2n-n^)rrfa-r2t+iW n=\ / l ( 0 ) *=1 <?2A+1

>2it+l ( 5 ) = - Qlk+l

, (2.7.3.28)

(2.7.3.29)

rD5- ' ?2*+l ?2i+1^21+3

TDS- ?2*+3' ^2t+3^2<:+5

TDs~l2k+5 " • • •

Before proceeding, we remark that the mathematical principles underlying continued fraction methods are described in the books of Wall [17] and Jones and Thron [18]. Moreover, a very useful theorem on the convergence of matrix continued fractions, which is relevant to the present work, has been proved by Denk and Riederle [19]. A variety of applications of matrix continued fractions in physics are given in the Risken'sbook [1].

We now introduce a concept which is of central importance in the calculations of the time behaviour of statistical averages from the Langevin or Fokker-Planck equation, that is, the linear response [61] of a system to an applied stimulus.

2.8 Linear Response Theory

The after-effect and alternating field solutions may be related by means of a method which has been described by Scaife [20,21]. Let us suppose that at a time t = 0, say, a unit electric field is applied to a dielectric body; then an electric dipole moment a (t) is induced in that body. The quantity a (0 is called the response function of the body. Let us now suppose that the induced electric field at time t is E(i) and 0 < t' < t; then the increase

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202 The Langevin Equation

in the field intensity in an infinitesimal time St' is (d E / dt')8t' and this in turn contributes

W)St'at-f) (2.8.1) to the induced moment at time t. Thus at time t, the instantaneous dipole moment of the body, m (t), is

f <mf) dt'

m(f) = J —^-r-a(t-t')dt'. (2.8.2) o

Now on integrating the above equation by parts,

m(t) = E(t')a(t-t%=0-\ W)^p[a(t-t')]dt'. (2.8.3) o

We now suppose that E ( 0 vanishes for t' < 0 and so E(0) = 0. Also we suppose that when a field is switched on, there is no instantaneous response so that a (0) = 0. Thus the first term on the right-hand side of the above vanishes and we may write with t - f = x

' da(x) m " — " " " ' ( 0 = f E(t-x)^^-dx. (2.8.4)

o dx

Our discussion so far refers to the case where the field is being switched on. Let us now consider the opposite case where a constant field E0 had been operative for a long time so that the induced moment is a (ao) and let us suppose that E0 is switched off at time t = 0. For t > 0, the induced moment m(/) is E0 [a (<»)-a (t)]. This leads us to define the after-effect function b (t) by the relation

\a°°)-at), (?>0), b(t)= ' ' , *' (2.8.5)

0, (/<0), whence

Let us now put m(0 = EM) • (2.8.6)

fEmcostftf, (t>0), , „„„s E(t) = \ m (2.8.7)

| 0 , (r<0),

whence

m = Emcos<y(f-x) dx

i dx

^ 7 da(x) , „ . 7 • da(x) = Emcosd)t\ coscox dx + Emsmcot\ smcox dx.

(2.8.8)

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Chapter 2. Langevin Equations and Methods of Solution 203

Here we have noted that if t is very large, da (x) I dx is negligibly small at t < x < oo, and thus the integrals

oo oo

J [da(x)/dx]coscoxdx and J [da(x)/dx]smcoxdx (2.8.9) i t

are also negligible. Thus we may write m(0 = Emc/(co)coscot + Emc/Xco)smcot, (2.8.10)

where

c/(co)=[ ——coscoxdx, a"(co)=\ ——sincoxdx (2.8.11) 0 dx J dx

both being real functions of CO. We now define the complex polarisability aco) = c/co)-ia\co) as

a0)) = l ^ l e - ^ d t = J ^le-'°»dt (2.8.12) Jo dt I dt

and we see from Eqs. (2.8.11) that a-co) = a'(co) + icc"(co). (2.8.13)

On integrating by parts, Eq. (2.8.12) becomes oo

a0)) = b(0)-iO)j b(t)e~ia"dt. (2.8.14) o

where b(0) = a(°°)~ a(0) = or'(O). Thus setting co= 0, we see that or'(O) is the static polarisability due to the field E so we express Eq. (2.8.14) as

^ = 1 - icol Rt)e-imdt, (2.8.15)

where R(t) = b(t)/b(0).

This equation effectively connects the alternating and after-effect solutions provided the response is linear.

We may now make use of the Kramers-Kronig dispersion relations [22]

# o ir-or

«"(fl,)=lpJ ^ 0 ^ , (2.8.17) n J

0 co -fi where the P indicates that the Cauchy principal value (see Jeffreys and Jeffreys [23], Art. 12.02) of the integral is to be taken. Let us put co= 0 in Eq. (2.8.18) and since ft)and ju may be interchanged, we must have

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204 The Langevin Equation

^0) = ag=l] SC^E. (2.8.18) * o °>

This is a most important relation, which connects the equilibrium polarisability as with the dissipative part of the frequency-dependent polarisability; in other words, it provides a link between the equilibrium and the nonequilibrium properties of the body. We now go on to describe one of the most fundamental theorems of statistical mechanics known as the fluctuation-dissipation theorem [24,25,61], of which Eq. (2.8.18) is one form.

The method we now describe follows closely that of Scaife [20,21]. It is known [22] that the static polarisability of a dielectric body may be written as

f, = — (M2) , (2.8.19) a. 3kT

where (M2) = ( M - M ) Q (2.8.20)

is the ensemble average of the square of the fluctuating dipole moment M of the body in the absence of an external field. By the ergodic hypothesis we must have

, Til (M2\ =\\m—t \ M(t)-M(t)dt. (2.8.21) \ /o r->°oT J

1 -T'll

Let us now write the Fourier transform pair:

M(ffl) = j Mt)e-iwdt , (2.8.22) —oo

M(0 = — \ MQ)elmdco. (2.8.23) —oo

Inserting Eq. (2.8.23) into Eq. (2.8.21), we have as a direct consequence of Parseval's theorem and the ergodic hypothesis (see also Chapter 3, Section 3.2 for details)

(M2) = — f M(co)dO), (2.8.24) \ /o in J

—oo

M(o) = lim -^(M(o)-M*(-co)) = lim —,(M(CO))2 (2.8.25)

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Chapter 2. Langevin Equations and Methods of Solution 205

is the spectral density of the fluctuations in M(0 [M(<y) is an even function of to]. We have from Eqs. (2.8.18), (2.8.19), and (2.8.24)

a = - f ^-^J-dco = f M(o))do) (2.8.26)

from which it follows that 6kTa"((o) = oM(co). (2.8.27)

We have now related the dissipative part of the frequency-dependent complex polarisability to the spectral density of the spontaneous fluctuations in the dipole moment of the body at equilibrium. This is the fluctuation-dissipation theorem [24,25,61]. In identifying the integrals in Eq. (2.8.26), we have asserted that macroscopic fluctuations decay according to macroscopic laws. The dipole autocorrelation function is the time average of M ( 0 with M(t +1') that is

, r/2 CM t) = lim — f M(r') • M(f' + t)dt.

Moreover, the Wiener-Khinchine theorem (Chapter 1, Section 1.6.6) states that the autocorrelation function and the spectral density are each others Fourier cosine transforms. Thus

1 °° CM(t) = — M(co)coscotdco, (2.8.28)

which with Eq. (2.8.27) gives

CM(0 = — ? ^^-coscotdco (2.8.29)

so that on inversion K J CD

or"(6>) = <y f CM(t)cos cotdt. (2.8.30) 6AT ^ ^

Now from Eq. (2.8.14) which yields oo

a"(<v) = ®\ b(t)coscotdt o

and Eq. (2.8.30), we have

b(t) = ^-CM(t). (2.8.31) 3kT

Hence

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206 The Langevin Equation

a(co) = as ICO

3£T J CM(t)e ndt (2.8.32)

or

a(o)) = 3kT

(M • M)Q - iO)j (M(f) • M(t' + 0)0 e~imdt (2.8.33)

where the subscript zero denotes that the average is to be evaluated in the absence of the driving field (at the equilibrium or stationary state). This is the Kubo relation [61], and is the generalisation of the Frohlich [22] relation (2.8.19) to cover the dynamical behaviour of the dielectric. Either the Kubo relation or the fluctuation-dissipation theorem [21,61] may be used to calculate a((0). Similar considerations apply to the linear magnetic and other linear responses.

In many problems, which we shall consider, (M(0))0 will not vanish. This would be so, for instance, if the dielectric was in equilibrium under the action of a steady dc field and that field was then altered by a small perturbation (so as to ensure linearity of the ensuing response). Hence, we shall employ as our definition of the normalised autocorrelation function CM(t), strictly this is the autocovariance

)o CM(t): (M(0)-M(Q)0-(M(0))^

(M2(0))o-(M(0))J (2.8.34)

The autocorrelation function CM (t) has a clear physical meaning, namely, it describes relaxation of the average dipole moment (M) (having switched off at t = 0 a small constant external electric field E, which has been applied in the infinite past). We have for the after-effect function

b(t) = - L ( ( M 2 ( 0 ) ) o -(M(0))20)cM(t). (2.8.35)

We now can rewrite Eq. (2.8.33) as

a(co) = (/(O) '\ \-ico\ CMt)e (2.8.36)

where

«-(0) = ( ( M 2 ( 0 ) ) o - ( M ( 0 ) ) ^ ) . (2.8.37)

The basic concepts of linear response theory are best illustrated by considering the rotational diffusion model of an assembly of non-interacting electric dipoles constrained to rotate about a fixed axis due to Debye [26] which is governed by the Smoluchowski equation

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Chapter 2. Langevin Equations and Methods of Solution 207

df(0,t) _ d D dt ~de

df(8,t) uE . n£/n ' (2.8.38)

d0 kT where f(&, t) is the number density of dipoles on the unit circle, 0is the angle between a dipole and the field E(t). The Smoluchowski Eq. (2.8.38) arises from the noninertial Langevin equation

0(t) + juE(t)sm0(t) = Mt). (2.8.39) We remarked in Chapter 1, Section 1.15 that Debye obtained two solutions for the Smoluchowski equation given above. The first solution is the after-effect solution, where it is supposed that the dielectric consisting of an assembly of non-interacting dipolar molecules has been influenced for a long time by a steady external field; thus on average, the axes of the dipoles are oriented mainly in the direction of the field. The second problem considered by Debye is the behaviour when an alternating electric field E(?) = Eme,a" is applied. Debye's solution of Eq. (2.8.38) in the linear approximation is [26]

2

so that

(Mcos0) = ^EmeiM (1 + iwD)" (2.8.40)

^ ) = ^ L _ . (2.8.4!) ar'(O) 1 + icmD

We shall now introduce the concept of the correlation time.

2.9 Correlation Time

The correlation (integral relaxation) time T of an autocorrelation (relaxation) function CM(t) is defined as the area under the curve of CM(t) and is a global characteristic of a decay process [61]. We note in the context of the electric polarisation that, in general, the decay of the dipole moment will take the form of a set of decaying exponentials characterised by a set of distinct eigenvalues A.k and their corresponding amplitudes Ak. Thus, the after-effect function b(t) in Eq. (2.8.35) may be represented as

(2.9.1)

so that

whence

where

Kt) = I t V-*

^k k 3 k T ^(M2(0))o-(M(0))(

CMit) = lLkcke-K\ (2.9.2)

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208 The Langevin Equation

c t = ^ - . (2.9.3)

By definition, the correlation (integral relaxation) time is oo

t = j CM(t)dt = YJkck^1. (2.9.4)

o Thus, the relaxation times T contain contributions from all the eigenvalues.

The systems considered in this book are governed by Langevin equations which in general lead to sets of equations for the hierarchy of appropriate after-effect (or correlation) functions that may be arranged in matrix form as

X = A X . (2.9.5)

The correlation time may then be obtained by solving the characteristic equation

det(Al-A) = 0 (2.9.6)

and then determining the amplitudes using the initial conditions of the problem as described in Section 2.7 et seq. This is, in general, a slow procedure from a computational point of view because a knowledge of all the eigenvalues and their corresponding amplitudes is required. The computation simplifies considerably if the mode characterised by the lowest nonvanishing eigenvalue X\ dominates the decay process. If this is so, Eq. (2.9.4) becomes

r = A[] (2.9.7) a procedure which has been extensively used [27] to obtain approximate analytic solutions for ras X\ is proportional to the Kramers escape rate [28] (see Chapter 1). In many cases however, the approximation (2.9.7) may not be made (cf. Chapter 8, Section 8.3, where relaxation of single domain ferromagnetic particles under the influence of an applied constant field and the anisotropy field is considered). A better method of calculating T which has the added advantage that it often yields exact analytic solutions for that quantity (e.g., [29-31]) is to recall that

r = liml CM(t)e-i,dt = limCM(s) = CM(0), (2.9.8) .s-»0Jo ,s->0

where CM(s) is the Laplace transform of CMt) and CM(0) is the value of that quantity at zero frequency. The Laplace transform of Eq. (2.9.5) is

sX(s)-X(0) = AX(s) (2.9.9) which, noting the final value theorem for Laplace transforms, namely,

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Chapter 2. Langevin Equations and Methods of Solution

lim sf(s) = lim fit), .y->0 /-»«>

X(oo)-X(0) = AX(0).

209

(2.9.10)

(2.9.11) becomes

Furthermore, X(oo) = 0 (2.9.12)

because the autocorrelation functions are a set of decaying exponentials, thus

-X(0) = AX(0) (2.9.13) with solution

X(0) = -A"'X(0) (2.9.14) showing the connection between the zero frequency and zero time values of the after-effect function which is the relation required in order to calculate T.

We illustrate by referring to the dielectric relaxation of a rotator with two equivalent sites discussed in Section 2.7.2. We have by linear response theory

jucos0it)) = juflit) = juE(cos10iO)\ Cxit), (2.9.15)

where

(cos<9(0)cos<9(0)0

(cos2 0(0)

Thus

CAt)- (2.9.16)

///1(O) = //£(cos20(O))oC1(O)

and

///1(O) = //£(cos20(O))o.

Hence, the longitudinal correlation time T\\ is given by

r, = C,(0) = / , (0)/ /1(0) .

Also, Eq. (2.9.14) becomes using the system matrix A from Eq. (2.7.2.6)

'' r'~ ^ f\-a a 0 0 0 ...VY/i(0) "

(2.9.17)

(2.9.18)

(2.9.19)

/i(0)

73(0)

72p+,(0)

The relaxation time

-3a 9 3o- 0 0 ...

0 -5<7 25 5<r 0 ... /3(0)

/ 2 p + i (0 )

(2.9.20)

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210 The Langevin Equation

ii = / i ( 0 ) / / , ( 0 ) (2.9.21)

may now be extracted from the above set by simply calculating A . This is much simpler than finding the eigenvalues from the characteristic Eq. (2.9.6), which always requires one to solve a high order polynomial equation.

The correlation time may also be evaluated from the matrix continued fraction solution for C(s), Eq. (2.7.3.10), of the matrix three-term recurrence relation Eq. (2.7.3.1) (see Chapters 8-10). Having determined C(s) from Eq. (2.7.3.10), one can calculate the relaxation time Tk of the ktb component c, k (t) of Ci(t), which is defined as:

1 7 . . . c,.t(0)

a*(0)J0

j cu(t)dt = --i,t (0)

(2.9.22)

In this book, we shall use all the above methods.

2.10 Linear Response Theory Results for Systems with Dynamics Governed by One-Dimensional Fokker-Planck Equations

The linear response theory described in Section 2.8 is quite general and has a wide range of applicability in physics. For particular systems, the predictions of this theory may be formulated in a more concise form. Here, we shall summarise the principal results of linear response theory (see, e.g., [1], Chapter 7) for systems where the dynamics obey one-dimensional (ID) Fokker-Planck equations for the distribution function Wx, t) of a variable x, such that

d dt

W = LFPW . (2.10.1)

Thus, let us consider the Fokker-Planck operator LFP of a system subject to a small perturbing force F(t). On account of this, LFP may be represented as [1]

LFP - T ax

D(2),.-V(x)+B(x)F(t)_0_eV(x)-BWF(t)

dx

= L°FP(x) + Lext(x)F(t)

(2.10.2)

with

ox DW(x)e-vw"vw

ax , L°FP(x)W0(x) = 0, (2.10.3)

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Chapter 2. Langevin Equations and Methods of Solution 211

Lext(x) = -4-(Dmx)B\x)), B'(x) = ^-B, (2.10.4) dxv ; ax

where I?FP(x) is the Fokker-Planck operator in the absence of the perturbation, Wo is the equilibrium (stationary) distribution function, V is called a generalised (effective) potential, D(2)(x) is the diffusion coefficient, and B (x) denotes a dynamical quantity.

The time dependence of the average of a dynamical variable A (x) can be expressed as

t

(A)(t)= $ ®(t-t')F(t')dt', (2.10.5) —oo

where <& (t) is the pulse response function defined by

<S>(t) = ~CAB(t). (2.10.6) at

Here CABt)is the equilibrium (stationary) correlation function defined by

CAB(t) = (A[x(0)]B[x(t)])o-(A)0(B)0

: | [A(x) - (A)Q ] A ' [B(x) - B)0 Wo (*)dx, (2.10.7)

where the symbols ( ) and ( ) 0 designate the statistical averages over W and Wo, respectively, with x defined in the range x <x<x2. The step-off and step-on relaxation functions (when, on the one hand, a small constant force Fj is suddenly switched off and on the other hand switched on at time t = 0, statistical equilibrium having been achieved prior to the imposition of the stimulus in both instances) for a dynamic variable A (x) are then

(A) o # (0- (A) 0 = F,CAB(0,

^r^-(A)Q=Fl CAB(0)-CAB(t)

(2.10.8)

, (f>0). (2.10.9)

Furthermore, the spectrum of (A) (?) (ac response) is

<A)a = Fa CABQ>)-iO)\ CABt)e-imdt (2.10.10)

where A)a and Fm are the Fourier components of (A)(t) and F(t) respectively.

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212 The Langevin Equation

The main feature of the one-dimensional systems under consideration is that there exists an exact integral formula [1,32,33] for the correlation time TAB defined as the area under the curve of the normalized autocorrelation function CAB(t), viz.,

^AB=-pr~rAcAB(t)dt. (2.10.11)

The derivation of an expression for the correlation time when A ^ B is given below.

To derive an analytic expression for TAB, we shall follow a method used in [1]. The definition of CABt) [Eq. (2.10.7)] implies that we may seek the solution of the ID Fokker-Planck equation (2.10.1) in the form [1]

W(x,t) = W0(x) + w(x,t) (2.10.12) with the initial conditions

w(x,0) = [B(x)-(B)0]W0(x) (2.10.13)

and with w(x,°°) = 0. Noting that I?FPW0 = 0, we have the formal solution of Eq. (2.10.1), namely,

w(x,t) = eL°»>>'[B(x)-(B)0]W0(x), (2.10.14)

whence

j[A(x)-(A)0](L°Fpyl[B(x)-(B)0]W0(x)dx

TAB = ~- 1 ; / v / x • (2.10.15) (M)Q-(A)0(B)0

The evaluation of (LFPyl[B(x)-(B)0]W0(x) (2.10.16)

can be accomplished by taking the Laplace transform of Eq. (2.10.1) noting Eq. (2.10.13). Thus, we obtain

sw(x,s)-[B(x)-(B)0]W0(x) = L°FPw(x,s), (2.10.17)

where oo

w(x, s)= \ w(x, t)e~stdt. o

Using the final value theorem of Laplace transformation [15], i.e., in the limits—»0 sw(x,s)^> w(x,oo), we have

w(x, 0) = -(L°FP )_1 [B(x) - (B)0 ]W0(X) (2.10.18)

and Eq. (2.10.15) becomes

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Chapter 2. Langevin Equations and Methods of Solution 213

(M)^m~2[MxHA)°mXi0)dx-<2'0,9)

The quantity w(x,0) in Eq. (2.10.19) can be evaluated as follows. Utilising Eq. (2.10.1) and noting Eq. (2.10.13), we have

TAB -'

-[Bx)-(B)0]W0(X) = ± D (2) — w(x,0) + w(x,0)—V dx dx J /

Equation (2.10.20) can now be integrated to yield

fBx)dx w(x,0) = -W0(x)j

D2)(x')W0(x)

where

fB(x)=][B(y)-(B)0]W0(y)dy.

(2.10.20)

(2.10.21)

(2.10.22)

Thus, we obtain from Eqs. (2.10.19) and (2.10.21)

1 ^ TAB -

or, after integrating by parts,

J [A(x)-(A)0]W0(x)j fBix) D(2\x)W0(x)

-dx'dx

"2

f where

fAx)fBx)dx

(AB)o-(A)o(B)oi D^(x)W0(x)

fA(x)=][Ay)-(A)QW0(y)dy.

(2.10.23)

(2.10.24)

For A = B and x = - ~ , x2 = ~ , Eq. (2.10.23) reduces to Eq. (S9.14) of Risken [1]:

TAA ~ 1 * 2

J fA (x)dx

(A2)0-(A)H ° ( 2 ) ( ^ » W '

(2.10.25)

We remark that the relaxation time in integral form, Eq. (2.10.25), was first given by Szabo [32], and later reproduced by various authors in many occasions (see, e.g., Refs. [1,33,34]).

For the purpose of illustration, we may apply Eq. (2.10.25) to the noninertial rotational Brownian motion of a planar rotator in a double well potential

V (0)/(kT) = 2a sin2 0

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214 The Langevin Equation

TD—W(0,t) = 2cr—[W(0,t)sm20] + -^TW(0,t), (2.10.26)

that has been discussed in Section 2.7.2 and 2.9. The relevant noninertial Fokker Planck equation for the probability density function W of the angle 6>is [35]

4-W(0,t) = 2a-^-[W(0,t)sm20] + 4-2 dt d0 d02

Thus, the correlation time v of the equilibrium autocorrelation function (cos#(0)cos#(f))0 (which is the quantity of most interest, see Chapter 4)

is given by the exact Eq. (2.10.25), where

A = cos#, x = 0, xl=0, x2 = In, and D(2)(x) = r~^ .

Thus, we have

In (6 — tD f e-<r«»20 lC0SJCe™*2Xdx d0

2a it (2.10.27)

r T°e f e-™s20erf2(j2^Sin0)d0,

where erf (x) is the error function [15]. More examples of application of Eq. (2.10.25) to particular stochastic systems are given in Chapters 4-8.

2.11 Smallest Nonvanishing Eigenvalue: The Continued Fraction Approach

As we have seen in Section 2.9 in order to evaluate the integral relaxation time r from Eq. (2.9.5) knowledge of the eigenvalues A^ and their amplitudes ck is required. As far as a physical interpretation is concerned in many cases the relaxation time r is determined by the slowest low-frequency relaxation mode. This mode governs transitions of the Brownian particle over the barriers from one potential well into another. The characteristic frequency of this overbarrier relaxation mode is determined by the smallest nonvanishing eigenvalue k\. Thus, k\ is the reciprocal time constant associated with the long time behaviour of the relaxation function which is determined solely by the slowest low-frequency relaxation mode. The behaviour of the relaxation time and the inverse of the smallest nonvanishing eigenvalue k\ is sometimes similar. However, if different time scales are involved, the behaviour of these can be quite different. A knowledge of k\ is also of importance because other time constants such as the mean first passage time and the escape rate are mainly determined by the slowest low-frequency relaxation mode. Moreover, in many cases the influence of the other relaxation modes on the low-frequency relaxation may be ignored so that knowledge of X\

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Chapter 2. Langevin Equations and Methods of Solution 215

provides sufficient information about the low-frequency dynamics of the system.

Various methods of calculating the eigenvalues have been discussed in detail in Ref. [1]. For example, one can obtain all the eigenvalues from the system matrix A (see Section 2.7.1). On the other hand, the eigenvalues can be evaluated in terms of continued fractions. The continued fraction approach of calculating the eigenvalues has been discussed in detail by Risken [1]. This approach was further developed in Refs. 36 and 37 for the purpose of evaluating the smallest nonvanishing eigenvalue X\ in terms of continued fractions. In contrast to the previously available solution [1], the method developed in Refs. 36 and 37 does not require one to solve numerically a high order polynomial equation since Ai may be represented as a sum of products of infinite continued fractions. Besides its advantage for numerical calculations, the equation so obtained is very useful for analytical purposes, e.g., for certain problems it may be expressed in terms of known mathematical (special) functions (examples are given in Chapters 4-9).

As we have seen in several examples, the solution of the Langevin equation or the corresponding Fokker-Planck equation can be reduced to the solution of an infinite hierarchy of equations for the moments (the expectation values of the dynamical quantities of interest) describing the dynamics of the system under consideration. In general, the hierarchies of the moment equations take the form of three- (or higher-order) term differential-recurrence relations between the moments. Moreover, as we have seen in Section 2.7.3, these recurrence relations may be solved in terms of ordinary or matrix continued fractions.

2.11.1 Evaluation of X\ from a scalar three-term recurrence relation

Let us first consider the scalar three-term recurrence relation

^Cnt) = q-nCn_lt) + qnCn.t) + q+nCn+l(t), (n > 1). (2.11.1.1)

at We recall that the exact solution of Eq. (2.11.1.1) with Q ( 0 = 0 for the Laplace transform of C\(t) is given by Eq. (2.7.3.23) in terms of the infinite continued fraction Sn (s) defined by Eq. (2.7.3.24), viz.,

Sn(s) = £ _ _ . (2.11.1.2) r „ _ n In Qn+1 ( e A Hn +

T *-n an+\1n+2

V - ? n + 2 - -

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216 The Langevin Equation

In the context of the continued fraction approach, the eigenvalues Xk can be determined by inserting the separation 'ansatz' [1]

Cnt) = Cne~Xt (2.11.1.3) into Eq. (2.11.1.1). Thus, one obtains an equation for the eigenvalues, viz.,

TeA + ql+qtS2(-Jl) = 0. (2.11.1.4)

The disadvantage of Eq. (2.11.1.4) is that it may be difficult, in general, to evaluate the eigenvalues, as it involves finding the roots of a very high order polynomial equation. Moreover, there are some examples where the continued fraction S2(-X) diverges (see Chapter 6). The method then fails because the calculation gives rise to unphysical solutions. In such cases, Eq. (2.11.1.4) cannot be used for the evaluation of X.

In general, Eq. (2.11.1.4) allows one to evaluate all the eigenvalues numerically. However, if one is only interested in the evaluation of the smallest nonvanishing eigenvalue X\, Eq. (2.11.1.4) can be simplified as follows. On supposing that the continued fraction 52 may be expanded in Taylor series:

s2(-xl) = s2(0)-s'2(0)TeAl+s;(0)^^-+o((Tezl)3), (2.11.1.5)

which is subject to the condition

* ^ S 2 ' ( 0 ) « 1 (2.11.1.6) 2S2(0)

so that one may take into account the first two terms in Eq. (2.11.1.5) only, we have from Eq. (2.11.1.4)

^ [ l - ^ r S2(0)] + qi +q+ S2(0) = 0

or q1+qtS2(0)

e^ \-q;s2(0) Here S2 (0) and S2 (0) are the first and second derivatives of the continued fraction S2(s) with respect to ST£. The criterion (2.11.1.6) of applicability of Eq. (2.11.1.7) is fulfilled in the high barrier (or low temperature) limit, where xEX\«\. However, Eq. (2.11.1.7) also provides sufficient accuracy for intermediate and small barrier heights, where TEX\ < 1, since the condition (2.11.1.6) may still be satisfied as S2 (0)/[2S'2 ( 0 ) ] « 1 .

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Chapter 2. Langevin Equations and Methods of Solution 217

In order to calculate X\ from Eq. (2.11.1.7), one needs to derive an equation for S'2 (0). This may be accomplished by noting that the continued fraction Sn(s) defined by Eq. (2.11.1.2) and the derivative of Sn(s) with respect to sTe satisfy the recurrence relations:

Sn(s) = ^ (2.11.1.8)

and

5; (*)= -S2n(s)[l-q+

nS'nH(s)]/q-n. (2.11.1.9) [Equation (2.11.1.9) can be readily obtained by direct differentiation of Eq. (2.11.1.8)]. The solution of the recurrence Eq. (2.11.1.9) may be obtained by iteration and is given by

1 °° fc

<ln-\ k=0 m=0

which yields for s = 0 and n = 2

*2 (°)= — T £ f l ^ + 1 ( 0 ) 9 ; / ^ + 1 . (2.H.1.10) #1 *=1 m=\

Thus, on substituting Eq. (2.11.1.10) into Eq. (2.11.1.7), we have an equation for X\.

tA=—_ W*w> . (2.n.M1) 1 + Z II &l«»£'«»«

k=l m=\ Unlike the representation of Eq. (2.11.1.4), which always requires one to solve numerically a high order polynomial equation in A, Eq. (2.11.1.11) requires only calculation of the continued fractions 5„ (0), which can be carried out even on a programmable pocket calculator [1]. Moreover, Eq. (2.11.1.11) can be further simplified by noting that the continued fraction 5„(0) for certain problems may be expressed in terms of equilibrium averages as a ratio of known mathematical (hypergeometric) functions. This allows us to derive analytical equations for X\. The advantage of such an approach is that it can also be used for the evaluation of X\ in problems where the continued fractions 5„(0) diverge so that the continued fraction approach based on solving Eq. (2.11.1.4) is no longer applicable.

As a simple example, we shall consider here the noninertial rotational Brownian motion of a planar rotator in a double-fold cosine potential, i.e., a rotator with two equivalent sites discussed in Sections 2.7.2, 2.7.3, and 2.9. As we have seen in Section 2.7.3, the appropriate

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218 The Langevin Equation

recurrence equation for the moments f2n_l(t) = (cos(2n-l)0)(t) is given by Eq. (2.7.3.26), viz.,

TD Jf f2p+i (0 = qip+Jip-i (0 + ?2p+i/2p+i (0 + <72P+i Ap+3 ( 0 , (2.11.1.12)

where q+, q~, and qp are given by Eq. (2.7.3.27). Thus, Eq. (2.11.1.11)

becomes

qi+qlS3(0) tD\ = — (2.11.1.13)

2 (O^Vfe &=lm=l

where the continued fraction 52i+1(0) is given by

C7

m+1

2k+l (0) = - (2.11.1.14) 2£ + l + <xS2A+3(0)

The 52A:+1(0) from Eq. (2.11.1.14) may in turn be expressed in terms of modified Bessel functions of the first kind of half integer order /t+1/2(z) [15] as (see Chapter 4, Section 4.4.3 for details)

S2t+i(0) = / t + 1 / 2 ( < r ) / W < r ) . (2.11.1.15) Equation (2.11.1.13) can then be simplified after some algebra

- i - i

where the relations [15]

7t (-1)'

l-e -la

_2_

7CZ hu(z) = — s i n n z>

P=o2p + l

73/2(^)

A/2(2)

p+1/2 i°) (2.11.1.16)

= coth z •

have been used. The lowest eigenvalue Ai calculated from Eq. (2.11.1.16) agrees closely for all crwith the numerical solution A?am gained as the smallest root of the characteristic Eq. (2.9.6). As one can see in Table 2.11.1.1, X\ and ^num are in excellent agreement in the high barrier limit. Moreover, /ti and Jtfam are very close to each other for all barrier heights because the condition (2.11.1.6), which for the problem in question reads

2s;(0) « 1

still holds even for a= 0, where \xD=\, and 53' (0)72^ (0) = - 1 / 9 .

Further examples will be considered in Chapters 4, 6, and 7.

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Chapter 2. Langevin Equations and Methods of Solution 219

Table 2.11.1.1. Numerical values of A, [Eq. (2.11.1.16)] and A^m [Eq. (2.9.6)].

oil

0 1 2 3 4 5 6 7 8 9 10

r D 4 n u m

1.0 0.3237 0.0820 0.0172 0.00318 5.46310^ 8.966-10"5

1.42610"5

2.220-10"6

3.389-10"7

5.11210"8

TD\

1.0 0.32336 0.08193 0.01715 0.00318 5.463-10""4

8.966-10"5

1.42610"5

2.217-10"6

3.389-10"7

5.11210"8

2.11.2 Evaluation of A\ from a matrix three-term recurrence relation

We have demonstrated above how one may evaluate the smallest nonvanishing eigenvalue k\, if the dynamics of the system is governed by an infinite hierarchy of scalar three-term recurrence equations for the moments. Now, we shall show how the above method can be extended to multi-term recurrence equations. As we already know, a multi-term scalar recurrence relation may be converted to a three-term matrix one. Such a matrix three-term recurrence relation may be written as

Te^Cn(t) = Q-nCn_i(t) + QnCn(t) + Q+nCn+l(t), (n>\). (2.11.2.1)

at As shown in Section 2.7.2, the exact solution of Eq. (2.11.2.1) with

Co (O = 0 (2.11.2.2) for the Laplace transform of Ci (O is given by Eq. (2.7.3.11) in terms of the infinite matrix-continued fractions An (s) defined as

A„(5) = 1 .

v i - Q„ - Q : 1 Q ; + J

^ r f - Q n + 2 - " -(2.11.2.3)

In the context of matrix-continued fractions, the eigenvalues can be determined by inserting the separation 'ansatz' [1]

Cnt) = Cne~Xt (2.11.2.4) into Eq. (2.11.2.1). Thus, one can obtain an equation for the eigenvalues, viz.,

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220 The Langevin Equation

det r eAH-Q,+Qf A2(-A)Q2 1 = 0. (2.11.2.5)

Equation (2.11.2.5) allows one to evaluate numerically all the eigenvalues [2]. However, if one is interested in the evaluation of X\ only, Eq. (2.11.2.5) can be simplified just as the scalar continued fraction. In the high barrier (or low temperature) limit, where T£X\ « 1, one may take into account only the first two terms in the Taylor series expansion of A2(-/l) for X = X\.

A 2 ( - ^ ) = A 2 ( 0 ) - ^ A ' 2 ( 0 ) + o [ ( ^ r J 2 ] . (2.11.2.6)

Thus, one has from Eqs. (2.11.2.5) and (2.11.2.6)

det TeA, i - Qf A2 (0)Q2 ) + Q, + Qf A2 (0)Q2 ] = 0, (2.11.2.7)

where the prime designates the derivative of A2(s) with respect to ST£.

In order to calculate k\ from Eq. (2.11.2.7), one should obtain an equation

for A2(0)Q2. This can be accomplished by noting that An(s) and

S.'n (s) satisfy the following recurrence relations:

A„ (s) = [ V I - Q„ - Q„+ An+1 (*)Q;+1 J1 (2.11.2.8)

and

K (s) = - [I - Q;A;+ 1 (s)Q~n+l ]Aj (s), (2.11.2.9) respectively [Eq. (2.11.2.9) can be obtained by direct differentiation of Eq. (2.11.2.8)]. Equation (2.11.2.9) can be further rearranged in the form

of a recurrence equation for Q^A^, (s)Q~ :

Q;_,A; (S)Q-=-QU a-Q:\'n+1(s)Q-n+lWn(S)Q-n, the solution of which may be obtained by iteration and is given by

QfA2(s)Q2= - X fl Q;fl[ A^+1(,)Qn-_,+1. (2.11.2.10) «=2 m=\ k=l

Thus the substitution of Eq. (2.11.2.10) at s = 0 into Eq. (2.11.2.7) yields d e t ( ^ I - S ) = 0, (2.11.2.11)

where the matrix S is defined as °° n-l n-\

n=2m=l k=\

(2.11.2.12)

S = -rJ1[Q1+Q1-A2(0)Qi]

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Chapter 2. Langevin Equations and Methods of Solution 221

Here, we have made use of a theorem ([38], Sec. 14.8-7), which states that the eigenvalues // and eigenvectors y of the matrix equation

Ay = juBy

are the eigenvalues and eigenvectors of the matrix B_1A . Thus, the smallest nonvanishing eigenvalue Xy is an eigenvalue of the matrix S given by Eq. (2.11.2.12). Furthermore, if that X\«X„ (X„ are all the other eigenvalues of the matrix S), then Xx is given by

4 - ^ , (2.11.2.13) 4 Sp[D]

where the elements of the matrix D are the minors of the matrix S. The evaluation of X\ from Eq. (2.11.2.13) is much more easily accomplished than the calculation of Xx by solving numerically the high order polynomial Eq. (2.11.2.5). Equation (2.11.2.13) requires only the calculation of the matrix continued fractions An (0). An example of the application of Eq. (2.11.2.13) is given in Chapter 9.

The continued fraction approach is, in general, very useful in the evaluation of Xx. However, the condition TeXx « 1 may be breached in some cases (e.g., for relaxation of superparamagnetic particles with cubic anisotropy for small and intermediate barriers, where reXx « 1, see Chapter 9) and thus Eq. (2.11.2.13) is no longer applicable to the evaluation of Xx. Moreover, a general restriction of the matrix continued fraction approach in the calculation of X\ based on Eq. (2.11.2.13) exists [1], namely, for very small damping, the dimension of matrices to be inverted may increase considerably or the matrices involved may become badly conditioned so that the method is no longer applicable in numerical calculations.

The next topic which we shall introduce in this chapter is the effective eigenvalue.

2.12 Effective Eigenvalue

The effective eigenvalue method [39,40] may be illustrated as follows. Let us suppose that in Eq. (2.9.2) yielding the decay of the dipole moment, the autocorrelation function CM (t) may be approximated by a single exponential for all t

CM(t) = CM(0)e-U« so that

CM(t) + XefCMt) = 0. (2.12.1)

Since these equations are valid for all t it then follows that

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222 The Langevin Equation

T =X-)=-^M^-. (2.12.2) f f CM(0)

Thus, the effective relaxation time % is expressed entirely in terms of CM (0) and CM (0), which may be evaluated from the Langevin equation

for a dynamical variable M (t) in terms of the eigenvalues Ak and the corresponding amplitudes ck [see Eqs. (2.9.2) and (2.9.3) above]

thus Tef is given by

= -X* V* and CM (0) = Z* ck =l; =o

V = [Z*V*J • (2-12.3) The effective eigenvalue Xej = T~J is then given by

4/=Z*V*- (2.12.4) Just as the integral relaxation time, it is again difficult to evaluate /Lef

from this formula because a knowledge of [ck and Ak is required. On the other hand, the advantage is that Aef may be expressed in terms of equilibrium averages of the distribution function as in Eq. (2.12.2).

It is apparent that [39,40] % is the time constant associated with the initial slope of the correlation function. It contains contributions from all the eigenvalues as does the correlation time. The behaviour of the correlation time and the effective relaxation time is sometimes similar. However, if different time scales are involved, the behaviour of these can be quite different and here the effective relaxation time gives precise information about the initial decay of the correlation (relaxation) function. An example of this is the Brownian motion in a bistable potential, where the correlation time diverges exponentially from the effective relaxation time in the high barrier limit [30] (see Chapters 4,6-8). For relaxation in a uniform field [31] (see Chapter 7, Section 7.5), however, no exponentially small eigenvalue exists since escape over a barrier is not involved. Thus, the effective relaxation time provides a close approximation to the correlation time. Yet another example of the use of the effective relaxation time is relaxation under the combined action of a constant field and a bistable potential [41]. For moderate values of the constant field, Tef is a poor approximation to the correlation time in the high barrier limit, however, as the field strength increases so that the bistable nature of the potential is weakened, % provides a good approximation to the correlation time (see Chapter 8).

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Chapter 2. Langevin Equations and Methods of Solution 223

As an example, we again consider dielectric relaxation of a rotator with two equivalent sites discussed in Sec. 2.7, 2.9, and 2.10. We have from Eq. (2.7.2.3) at/? = 1

r D / 1 (0 + / i (0 = c r [ / 1 (0 - / 3 (0 ] • (2.12.5)

By supposing that fx(t) in Eq. (2.12.5) may be approximated by a single

exponential, i.e.,

^ / i ( 0 + / i (0 = 0, (2.12.6)

we have -l

V / i ( 0 )

/ i ( 0 ) = Tr l - < 7 + <7

/3(Q)

/ i (0) (2.12.7)

According to Eq. (2.7.2.11) f3(0) = I2((T) + Il(a)

/i(0) IX(C) + I0<J) Thus

V O - ^ ' ^ + W = Tj

70(CJ) + / I (g - )

l0o)-Ix(a) (2.12.8)

1,(0) + I0(a)

noting that 2I1(cr)/cr = I0(<J)-I2(<J) [15]. For the present model, Eq.

(2.12.8) may be used as an approximation of the integral relaxation time

t, Eq. (2.10.27), only for a< 1.5. In the high barrier limit, <x» 1, the

effective relaxation time r^ from Eq. (2.12.8) diverges exponentially

from 7(see Chapter 4 for details).

2.13 Evaluation of the Dynamic Susceptibility Using T, zej, and X\

As we have seen in previous sections, the three time constants, viz., the integral relaxation time r, the effective relaxation time %, and the inverse of the smallest nonvanishing eigenvalue Au are important in the interpretation of the dynamics of a stochastic system. Here, we show that they may be very useful in describing the spectrum of the generalised complex susceptibility %A (co) of a dynamical quantity A with dynamics governed by the Langevin equation. According to linear response theory [61], the generalised susceptibility ;& (CD) is defined as

XA (<») = X'A (®) - iX\ (®) = ZA \-ico\e-imCA(t)dt (2.13.1)

where

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224 The Langevin Equation

<AO)A(,)>„-(A(0)>; <12(0)>0-<A(O)>5

is the normalised autocorrelation function of the dynamical variable A, ZA = ((A2(0)> -(A(0))2)I (kT) is the static susceptibility, and the brackets ( ) 0

designate as usual the equilibrium ensemble average. Thus, in order to evaluate the susceptibility XA (<*>)> o n e m u s t calculate the spectrum (onesided Fourier transform) of the equilibrium correlation function CA(t). Furthermore, the behaviour of XA i®) in the frequency domain is completely determined by the time behaviour of CA(t).

As we have mentioned, we may formally introduce three time constants characterising the time behaviour of CA(t). These are the integral relaxation (or, in linear response, correlation) time z defined as the area under CA (t):

oo

z=\cA(t)dt, (2.13.3) o

the effective relaxation time % defined by

zef=-^— (2.13.4) ef Q(0)

(which gives precise information on the initial decay of CA t) in time domain), and the inverse of the smallest nonvanishing eigenvalue At of the Fokker-Planck operator LFP, which is usually associated with the long time behaviour of CA (t) (slowest relaxation mode) and is proportional to the Kramers escape rate. As we have seen in Sections 2.9 and 2.12, the relaxation times z and zef may equivalently be defined in the context of the eigenvalues A* of the corresponding Fokker-Planck operator LFP as follows

z = Y,kck/Ak (2.13.5)

and

z f ef

=[ztv*T. (2-13-6) where £ A c * = l .

By utilising general properties of Fourier transforms and Eqs. (2.13.3) and (2.13.4), we have from Eq. (2.13.1), equations for XA (« ) in the low- and high-frequency limits:

^ ^ - = \-i(o\cA(t)dt + ... = \-im + ... (2.13.7) XA o

for *y-» 0, and

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Chapter 2. Langevin Equations and Methods of Solution 225

Zm__CM+CM + ...a>+_ (2.13.8) XA

i(° a a** for 6>->oo. Thus, according to Eqs. (2.13.7) and (2.13.8), both the low-and high-frequency behaviour of the imaginary part of XA (°>) a r e

completely determined by T and tef, respectively. We remark that the asymptotic Eqs. (2.13.7) and (2.13.8) are quite general and hold for any system. Let us now suppose that the correlation function CA (t) from Eq. (2.13.2), which in general comprises an infinite number of decaying exponentials, e.g., Eq. (2.9.3), may be approximated by two exponentials only (see Chapter 1, Section 1.20)

C^O^A^' + a-A^e'"^, (2.13.9) where A! and % are parameters to be determined. Thus, the spectrum of XA (OJ) corresponding to Eq. (2.13.9) is a sum of two Lorentzians with characteristic frequencies Xx and % ] :

* ^ = A ' + ^ . (2.13.10) XA \ + icol\ l + i(OTw

Here, we implicitly suppose that the contribution of high-frequency "intrawell" modes to XA <°) m a v be approximated as a single Lorentzian with characteristic frequency and half-width given by the inverse of the characteristic time %. In order that Eq. (2.13.10) should obey the exact asymptotic Eqs. (2.13.7) and (2.13.8), Ai and % must satisfy the following relations

A,= eJ- , (2.13.11) 1 V - 2 + l/(^Te /)

% = 3 ^ 1 T 1 • (2-13.12)

These are the solutions (in terms of t, %, and X) of the following algebraic equations

^ - + (1-A1)TW=T, A1Al+^—^ = — . (2.13.13) \ % Tef

We remark that the parameters Aj and % can be estimated by the method described in Chapter 1, Section 1.20. One may readily verify that Eqs. (2.13.10)-(2.13.12) predict the correct behaviour of XA(®) b o t n a t l ° w

(0)—> 0) and high (a)—> °°) frequencies. Moreover, having evaluated r, %, and Xx, we may calculate Ai and % and, thus, we may predict XA (#>) m

all frequency ranges of interest as Eqs. (2.13.10)-(2.13.12) will describe XA (&) m the entire grange 0 < co< °°.

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226 The Langevin Equation

We again consider the noninertial rotational Brownian motion of a planar rotator in a double-fold cosine potential

V(0) = Ksm20. In order to study the longitudinal relaxation behaviour, it is supposed that a small uniform field E applied along the z-axis is switched off at t = 0. We have already calculated r, Au and %for this model in Sections 2.10, 2.11, and 2.12. These are given by Eqs. (2.10.27), (2.11.1.16), and (2.12.8), respectively. Having determined T, \//Lu and %, one may evaluate the complex dynamic susceptibility %A (°>) fr°m Eqs. (2.13.10) -(2.13.12). The results of such an evaluation are given in Chapter 4. Further examples are presented in Chapters 6-9.

2.14 Nonlinear Response of a Brownian Particle Subjected to a Strong External Field

A system initially in an equilibrium (stationary) state and suddenly disturbed by an external stimulus (e.g., by applying a step external field) will evolve to a new equilibrium (stationary) state. For linear response, the energy of the system arising from the external stimulus is much lower than the thermal energy. Here, we need only linear (in the external stimulus) deviations of the expectation value of the dynamical variable of interest in the stationary state in order to evaluate the generalised susceptibility and/or response functions in terms of the appropriate equilibrium (stationary) correlation functions. Linear response theory (explained in detail in previous sections) is widely used for the interpretation of nonequilibrium phenomena such as dielectric and magnetic relaxation, conductivity problems, and so on [61].

In contrast, nonlinear response theory has been much less well developed because of its inherent mathematical-physical complexity. The calculation of the nonlinear response even for systems described by a single coordinate is a difficult task. There is no longer any connection between the step-on and the step-off responses and the ac response because the response now depends on the precise nature of the stimulus. Thus, no unique response function valid for all stimuli unlike linear response exists. Nonlinear dielectric relaxation and dynamic Kerr effect are natural examples of the application of nonlinear response theory. The results that have been obtained have mainly emerged either by numerical simulations or by perturbation theory (see e.g. [42,43]). However, a few analytical and numerical solutions of particular problems dealing with nonlinear step responses exist (e.g., [44-50]).

The most important feature of the Langevin equation approach combined with the matrix continued fraction method of solution of

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Chapter 2. Langevin Equations and Methods of Solution 227

recurrence equations is that they allow us to calculate nonlinear responses in the same manner as linear responses. The reason is that the recurrence equations for the statistical averages that can be obtained from the Langevin equations have the same mathematical form both for linear and nonlinear responses. The reader can find a variety of examples in Refs. [51-58] as well as in Chapter 8, Section 8.5 and 8.6, and in Chapter 10, Section 10.3. Moreover, various analytical results of linear response theory may be generalised to the nonlinear response [47]. In the next subsection, we derive, following [47], an exact analytic equation [similar to Eq. (2.10.25)] for the nonlinear transient response relaxation time of a system governed by a one-dimensional Fokker-Planck equation just as in linear response.

2.14.1 Analytical solutions for the relaxation time of one-dimensional systems

We consider the one-dimensional Brownian movement of a particle in a potential V(x) and we assume that the relaxational dynamics of the particle obeys the Fokker-Planck equation (2.10.1). Let us suppose that at time t = 0 the value of the generalised potential V is suddenly changed from Vi to Vn (e.g., by applying a strong external field or by a change in some parameter characterising the system; see Fig. 2.14.1.1). We are interested in the relaxation of the system starting from an equilibrium (stationary) state I with the distribution function W\x), which evolves under the action of the stimulus to another equilibrium (stationary) state II with the distribution function Wn(x). Our goal is to evaluate the relaxation time TA of a dynamical variable A. This problem is intrinsically nonlinear because we assume that the changes in the magnitude of the potential are now significant. Thus, the concept of relaxation functions and relaxation times must be used rather than correlation functions and correlation times.

We define the normalised relaxation function /4(f) of a dynamical variable A by

» ( ' ) - < A ) n

fA(t) = \ (t>0),

(A\-(A)ll ' ' (2.14.1.1)

where (A) and (AY are equilibrium (stationary) averages defined as

(A\ = I Ax)Wlx)dx, (A)n = I A(x)Wu(x)dx , (2.14.1.2)

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228 The Langevin Equation

i

Vi

0

V

Vh

w, •/,.«>

W„

Figure. 2.14.1.1. Schematic representation of the transient nonlinear response,

and (A)(f) is the time-dependent average *2

(A)(0= J A(*)W(je,0<ic (2.14.1.3)

(here, x is defined in the range x\ <x<x2). The relaxation time rA defined as the area under the curve oifA ( 0 at t > 0 is then given by

oo oo

7A = J / A ( 0 * = Hm f e-stfAt)dt = /A(0), (2.14.1.4) o ^ ° o

where fA(s) is the Laplace transform oifA (t). We can now show, just as in Section 2.10, that the relaxation time, Eq. (2.14.1.4), may be written as [cf.Eq. (2.10.19)]

x2

, l . , j [A(x)-(A)u)W(x,0)dx, (2.14.1.5) 7 i ~ \ A / n x.

where "(A

oo

W(JC,S) = fW(*,0c- , 'A and W(x,0) = limW(x,s).

The quantity W(x,0) in Eq. (2.14.1.5) can be calculated analytically just as linear response (see Section 2.10). On using the final value theorem of Laplace transformation [15], namely,

limsW(x,s) = WmW(x,t) = Wll(x),

we obtain from the Fokker-Planck Eq. (2.10.1) for t > 0 [cf. Eq. (2.10.20)]

Wu(x)-Wl(x) = -^-ax

D(2\x)\ ^-W(x,0) + W(x,0)4-Vn (x) \dx ax

.(2.14.1.6)

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Chapter 2. Langevin Equations and Methods of Solution 229

The solution of Eq. (2.14.1.6) is

W(x,0) = Wu(x)\ ®y)dy , (2.14.1.7)

where

®(y) = \ [Wn(z)-Wi(z)]&. (2.14.1.8)

Thus, we obtain from Eqs. (2.14.1.5) and (2.14.1.7)

1 *trt, x / , \ ,„, , J *()0 r<=vwJ™-A)'mixi-F^** A (A) -(A) x\

or, by integration by parts,

^ = " J _ ^ £ W _ & (214.19)

where

¥ ( * ) = / [A(y)-(A)n]Wn(y)rfy. (2.14.1.10)

Equation (2.14.1.9) is an ex«c? equation for the nonlinear transient response relaxation time, which is analogous to Eq. (2.10.25) for the linear response.

2.14.2 Nonlinear transient response in the rotational Brownian motion

We show, following [47,48], how the above results may be applied to the noninertial rotational Brownian motion of a particle in an external uniaxial potential V(i9). Here, pronounced changes in V can be achieved by varying the strength of the external electric or magnetic field. The relevant Fokker-Planck equation for the distribution function W of the orientations of the particle is given by [47]

f d W d

—w+——vu s<7X kT ox

This equation can be readily obtained from Eq. (1.15.20) of Section 1.15 by using as new variable x = cos??. Here, X\ = - 1 , x2 = 1,

D(1\x) = (\-x2)l2rD), Wj(x) and Wn(jt) are the Maxwell-Boltzmann distribution functions,

Wi(jc) = e-W™/Z,, Wn(x) = e-v«x)/(kT)/Zn,

and Z\ and Z\\ are the partition functions in the states I and II. Thus, Eq. (2.14.1.9) becomes

dt dx (1-x 2 )

V , (f>0) . (2.14.2.1)

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230 The Langevin Equation

2r g i e'"'MmMz)VU)dz , , . . , , ,

a -i where

<J>(z) = [Zje-W™ -Z^e-v^)l(kT)]dy - l

z

¥(z) = f [A(JC) - (A)n ]e-v" (* ) / (* rW. - l

Equation (2.14.2.2) can be used, for example, to calculate the longitudinal relaxation time of the nonlinear dielectric and dynamic Ken-effect transient responses of systems consisting of permanently polar and polarisable molecules [55]. Here, the orientational relaxation of the molecules is governed by the Fokker-Planck equation (2.14.2.1) with uniaxial potential V given by [49-51]

VN(x)/(kT) = -aN(x2+2hNx), (2.14.2.3) where N = I for the initial state and N = II for the final state, and

h =£ 12a £ = ^ L a -^LE2

The quantities of greatest interest in the nonlinear response are the relaxation times rn (n = 1,2) of the relaxation functions fy(t) and/2(0 of the first and second Legendre polynomials [15], namely,

/j(0 = (P1(cosz?))(0-(P1(cos^))n (2.14.2.4)

and /2(f) = (P2(costf))(0-(P2(cost?) ir (2.14.2.5)

The relaxation functions fi(t) and f2(t) govern the transient dielectric relaxation and the dynamic Kerr effect, respectively. We have from Eqs. (2.14.2.2) and (2.14.2.3)

where

*(z) = f[W„(z')-WI(z')]A'=—172— (erf/[(z+/*n)V^]4«rf;[(l-/Jn)^] n ^n

' 2^% ierf/[(z+/2j )7c^]+erfi[(l-/z, )>/ojj,

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Chapter 2. Langevin Equations and Methods of Solution 231

%(z)=] [^-P^e^^dz^ 2on

ean(z'+2haz) _e<Ju(\-2h] to)l

2<7n Z n l J

*2<z)=J [P2(z')-(P2)n]e°lliZ'2+2huZ')dz' - l

-ea"ll2+2h*z) (z - hu) + g^d-z*,,) (1 + hu) 4C7TI

-e and-,,,2,) [cosh(2crII/i1I) - h u sinh(2<TII/iII)]

<?n ZII

erfi'[(z + K )<Jan ] + erfi[(l - V )y]<ru ]

ggwsinh(2q-wftAf)

W„=-<?NZN

— hN,

and

ft), 3g(7w[cosh(2(T;v^)-/ijvsinh(2<7jV/iJV)] | 3/# 3 J_

2aNZN 2 4aN 2

Here

ZN=^\^e-a»h»\trti[\ + hN)^]+ern[(\-hN)fi~]] , (N=l, ID

are the partition functions in the equilibrium states I and II and

2 f .2

erft'(x) = —r= e dt (2.14.2.7)

is the error function of imaginary argument [15]. If a strong constant field is suddenly applied at t = 0 to a system

of nonpolar and polarisable molecules, we have hx = 0, (Tj = 0, hn = 0, 0U=(T.

Thus, the relaxation time ^ is given by Eq. (2.14.2.6) for n - 2 with erfz'(vcrz) z

* (z) = 2erfi(Vc7) 2 '

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232 The Langevin Equation

Figure 2.14.2.1. Step-on response transient relaxation time [Eq. (2.14.2.8): solid line]. Bars are the experimental data from Ref. [59].

¥2(z) =

<*>>!!

4(7

4(7

ze^z2 _ea^i(4az)

edi(-Ja)

2eaJa

V^erfjXVo") - 1

2

so that

«i=-3rn

•J erfj'(wz)".

W2>nJ , I erfi(V )

l + e' ,<x(l-zz) ] - ,ff(l-z2) erf/ 24GZ) 2\ dz

1-erfz'2(V<7) J i - r

(2.14.2.8) Tolles and co-workers [59,60] have presented experimental

results for the step-on relaxation time Tq, of nonpolar polarisable zinc oxide particles as a function of an applied electric field. A comparison of the theory with these data [52,57] is given in Fig. 2.14.2.1. Here ^ [Eq. (2.14.2.8)] is plotted as a function of A = CEn, where C is the proportionality constant. The agreement of the theory with the experimental data [59,60] is good. The theory also agrees in all respects with the numerical solutions of the Smoluchowski equation obtained in [59].

Another particular case of interest is when a strong constant field is suddenly switched on at t = 0 on a system of polar and nonpolarisable molecules. This corresponds to the following values of the parameters:

6 = 0 , & = £ ^ I = ^II = 0 . Here, all the quantities in Eq. (2.14.1.12) can be expressed in terms of elementary functions, namely,

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Chapter 2. Langevin Equations and Methods of Solution 233

</>1>n=coth#-i, < / > 2 > n = l - | c o t h £ - -

<l>(z) = sinh£

z-\

lF1(z) = ^ z ( z -co th^ ) + ( l + coth^),

v'U)mv J* 2z , 2co th^ 2e~ -(l + coth#)

Thus, the relaxation times are given by

2Tr

£coth£-l 1-

Cinh(2£)

sinh2£

and

where

T2 = l-#coth# + £ 2 /3

£coth£-3 + 2 ^ M ) sinh £

(2.14.2.9)

(2.14.2.10)

_ , . . . . r cosltt-1 , Cinh(z) = dt

J t o '

is the integral hyperbolic cosine [15]. We shall apply the methods explained above to the solution of

various Langevin equations in subsequent chapters. References

1. H. Risken, The Fokker-Planck Equation, Springer Verlag, Berlin, 1984; 2nd Edition, 1989.

2. J. L. Doob, Ann. Math. 43, 351 (1942); Reprinted in Selected Papers on Noise and Stochastic Processes, N. Wax, Editor, Dover, New York, 1954.

3. N. Wiener and R. Paley, Fourier Transforms in the Complex Domain, American Mathematical Society Colloquium Publications, Vol. XIX, 1934.

4. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland, Amsterdam, 1981; 2nd Edition, 1992.

5. K. Ito, Proc. Imp. Acad. 20, 519(1944). 6. R. L. Stratonovich, Conditional Markov Processes and their Application to the

Theory of Optimal Control, Elsevier, New York, 1968. 7. C. W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin, 1985. 8. W. T. Coffey, J. Chem. Phys. 99, 3014 (1993). 9. W. T. Coffey, J. Chem. Phys. 93, 724 (1990).

Page 259: The Langevin Equation Coffey_Kalmykov_Waldron

234 The Langevin Equation

10. J. R. McConnell, Rotational Brownian Motion and Dielectric Theory, Academic Press, London, 1980.

11. W. T. Coffey, J. Chem. Phys. 95,2029 (1991) 12. G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930). 13. M. Krasnov, A. Kieslev, G. Makarenko.and E. Shikin, Mathematical Analysis for

Engineers, Vol. 2, Mir, Moscow, 1990. 14. W. T. Coffey, Yu. P. Kalmykov, E. S. Massawe, and J. T. Waldron, J. Chem. Phys.,

99,4011(1993). 15. M. Abramowitz and I. Stegun, Eds., Handbook of Mathematical Functions, Dover,

New York, 1964. 16. W. T. Coffey, Yu. P. Kalmykov and J. T. Waldron, Physica A 208,462 (1994). 17. H. S. Wall, Analytic Theory of Continued Fractions, Chelsea, New York, 1973. 18. W. B. Jones and W. J. Thorn, Continued Fractions, Encyclopedia of Mathematics

and its Applications, Addison-Wesley, Reading, MA, 1980. 19. H. Denk and M. Riederle, J. Approx. Theory 35, 355 (1982). 20. B. K. P.Scaife, ERA Report IVT 392, Electrical Research Association, Leatherhead,

Surrey, 1959. 21. B. K. P. Scaife, Complex Permittivity, The English Universities Press, London,

1971. 22. H. Frohlich, Theory of Dielectrics, 2nd Edition, Oxford University Press, London,

1958. 23. H. Jeffreys and B. S. Jeffreys, Mathematical Physics, Cambridge University Press,

Cambridge, 1950. 24. H. B. Callen and J. A. Welton, Phys. Rev. 83, 34 (1951). 25. R. F. Green and H. B. Callen, Phys. Rev. 83 1231 (1951). 26. P. Debye, Polar Molecules, Chemical Catalog, New York, 1929; reprinted by Dover

Publications, New York, 1954. 27. W. F. Brown Jr., Phys. Rev. 130, 1677(1963). 28. P. Hanggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys., 62, 251 (1990). 29. W. T. Coffey, D. S. F. Crothers, and J. T. Waldron, Physica A 203, 600 (1994). 30. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, E. S. Massawe, and J. T.

Waldron, Phys. Rev. E49, 1869 (1994). 31. J. T. Waldron, Yu. P. Kalmykov, and W. T. Coffey, Phys. Rev. E 49, 3976 (1994). 32. A. Szabo, J. Chem. Phys. 72,4620 (1980). 33. G. Moro and P. L. Nordio, Mol. Phys. 56, 255 (1985). 34. D. A. Garanin, V. V. Ischenko, and L. V. Panina, Teor. Mat. Fiz. 82, 242 (1990). 35. J. I. Lauritzen and R. Zwanzig, Jr., Adv. Mol. Relax. Interact. Processes 5, 339

(1973). 36. Yu. P. Kalmykov, Phys. Rev. E 61, 6205 (2000). 37. Yu. P. Kalmykov, Phys. Rev. E 62, 227 (2000). 38. G. A. Korn and T. M. Korn, Mathematical Handbook, McGraw-Hill, New York,

1968. 39. M. San Miguel, L. Pesquara, M. A. Rodriquez, and A. Hernandez-Machado, Phys.

Rev. A35, 208 (1987). 40. W. T. Coffey, Yu. P. Kalmykov, and E. S. Massawe, The Effective Eigenvalue

Method and its Application to Stochastic Problems in Conjunction with Nonlinear

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Chapter 2. Langevin Equations and Methods of Solution 235

Langevin Equation, in Modern Nonlinear Optics, Eds. M. W. Evans and S. Kielich, A special volume of Advances in Chemical Physics, Series Editors I. Prigogine and S. A. Rice, Wiley, New York, 85, part 2, 667 (1993).

41. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and J. T. Waldron, Phys. Rev. B 51, 15947 (1995).

42. A. Morita, Phys. Rev. A 34, 1499 (1986). 43. A. Morita and H. Watanabe, Phys. Rev. A 35, 2690 (1987). 44. L. D. Eskin, Optik. Spektrosk. 45, 1185 (1978) [Optic. Spectrosc. 45, 922 (1978)]. 45. I. B. Aizenberg and L. D. Eskin, Optik. Spektrosk. 48, 399 (1980) [Optic. Spectrosc.

48, 222 (1980)]. 46. A. Morita, J. Phys. D, 11, 1357 (1978). 47. Yu. P. Kalmykov, J. L. Dejardin, and W. T. Coffey, Phys. Rev. E, 55, 2509 (1997). 48. Yu. P. Kalmykov, Optik. Spektrosk. 84, 1000 (1998) [Optic. Spectrosc. 84, 906

(1998)]. 49. H. J. Breymayer, H. Risken, H. D. Vollmer, and W. Wonneberger, Appl. Phys. B:

Lasers Opt. 28, 335 (1982). 50. Yu. L. Raikher, V. I. Stepanov, and S. V. Burylov, J. Coll. Interface Sci. 144, 308

(1991). 51. W. T. Coffey, J. L. Dejardin, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. E 54,

6462 (1996). 52. J. L. Dejardin, P. M. Dejardin, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. E 60,

1475 (1999). 53. W. T. Coffey, J. L. Dejardin, and Yu. P. Kalmykov, Phys. Rev. B 62, 3480 (2000). 54. Yu. P. Kalmykov and S. V. Titov, J. Mag. Magn. Mater. 210, 233 (2000). 55. J. L. Dejardin and Yu. P. Kalmykov, Phys. Rev. E 62, 1211 (2000). 56. W. T. Coffey, J. L. Dejardin, and Yu. P. Kalmykov, Phys. Rev. E, 61, 4599 (2000). 57. J. L. Dejardin, Yu. P. Kalmykov, and P. M. Dejardin, Birefringence and Dielectric

Relaxation in Strong Electric Fields, Adv. Chem. Phys., Series Editors I. Prigogine and S. A. Rice, Wiley, New York, 117, 275 (2001).

58. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov, J. Mag. Magn. Mater. 241, 400 (2002).

59. W. M. Tolles, J. Appl. Phys. 46, 991 (1975). 60. W. M. Tolles, R. A. Sanders, and G. W. Fritz, J. Appl. Phys. 45, 3777 (1974). 61. R. Kubo, M. Toda, and N. Nashitsume, Statistical Physics II. Nonequilibrium

Statistical Mechanics, Springer Verlag, Berlin, 1991.

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Chapter 3

Brownian Motion of a Free Particle and a Harmonic Oscillator

3.1 Ornstein-Uhlenbeck Theory of the Brownian Motion

The formula for the mean square displacement derived by Einstein and Langevin, namely:

< ( A x ) 2 ) = ^ | f | , (3.1.1)

has (as we have discussed in detail in Chapter 2 in connection with the Wiener process) the fundamental flaw that it is not root mean square differentiable at t = 0. We have also seen that this is a direct consequence of ignoring the inertia of the particles. In 1930, Uhlenbeck and Ornstein [1] by including the inertia of the particles derived the famous formula originally given by Ornstein and Furth in 1918 [2]

<(Ax)2> = f ^ | r | - l + e - W ' » l (3.1.2) £ \™ J

which for times t » m/C, reduces to Eq. (3.1.1) and which for short times becomes

<(Ax)2> = — ' 2 -m

Thus, ((Ax)2) is now mean square differentiable. In this section, we show how Eq. (3.1.2) may be derived from the

point of view of the Ornstein-Uhlenbeck theory. In order to accomplish this we follow the method of Section 1.7 writing the Langevin equation in phase space (x, v) as

mv(t) = -Cv(t) + A(t), (3.1.3)

where this time we designate the white noise driving force as A (t) with

236

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Chapter 3. Brownian Motion of a Free Particle and a Harmonic Oscillator 231

lh)Kh) = 2D8h-h)* D = £kT. (3.1.4) It is again assumed that the particle starts off at a definite phase point (x0, v0) so that the state vector has components

x(t) = Xo+^(l-e-*) + ^-\ (l-e-^'-''AX(t')dt', p v ; mpi v '

° (3.1.5) v(r) = x(t) = v0e

_/ft +—\ e-^'^MOdt'-m-0

(here J3= £7 m). Hence

Ax = x(t)-Xo=^(l-e-P') + -L\ h-e-K^MtW. (3.1.6) p v ; mp J

Q L J

Now

^c=vMl-e^), (3.1.7)

^ 7 = 4 ( l - ^ ' ) 2+ ^ T J dt'\ dt'\\-e-^-'')][\-e-^'")]WWr)

P m P 0 0 2D r

rff'J df'\l-e ^ "\\l-e ^ ' '\d(f-

(3.1.8) /) m p 0 Q

Since

j dr"<y(/ / ,-rO[l-e" / '( '"n] = l -«"^" ' ' ) , (3-1.9) 0

we have from Eq. (3.1.8)

^ = 4 ( l - ^ ' ) 2 + 4 V f dfh-le-X'-V+e-2*'-*] ^ ^ ° (3.1.10)

^2 V ' m2/?2 mV 3 L J

This is the solution given that the collection of particles started off with the definite velocity vo- For long times, the term in t is all that is significant, so that we obtain the result of Einstein [2] and Langevin [3]:

77—2 2Dt 2kTt ... (Ax) =—-— = ——. (3.1.11)

m p2 mP

If we have a Maxwell-Boltzmann distribution of initial velocities v0, we find from Eq. (3.1.10) that

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238 The Langevin Equation

((Ax)2) = ^L(pt-l + e-p'), (f>0), (3.1.12)

mp x ' which is the Ornstein-FUrth formula [1]. We reiterate that for short times, Eq. (3.1.11) is nondifferentiable, so that in the noninertial approximation, the velocity does not exist. If inertia is included, however, ((Ax)2) is differentiable and the velocity exists. This question first emphasised by Doob [4] has been discussed in Chapter 2.

3.2 Stationary Solution of the Langevin Equation — The Wiener -Khinchine Theorem

We have illustrated the calculation of the averages from the Langevin equation for sharp initial conditions. The solution of the Langevin equation subject to a Maxwell-Boltzmann distribution of velocities is called the stationary solution. Clearly for the stationary solution

(v2) = kT/m (3.2.1) as the velocities are in thermal equilibrium. The quantity of interest now is the velocity correlation function. The stationary solution may be found by extending the lower limit of integration to - oo and discarding the term in v0 in Eq. (3.1.5). Thus, for distinct times t\ and tj, we have

h h

v(tx) = rrfx \ e'^-^MOdt', v (t2 ) = m"1 J e~ P'2'1") X(t")dt" (3.2.2)

so that

(V( ' I )V(* 2 ) ) = ^ F ) I e-^+^e^'"\X(t')Xt"))dt'dt" m * "~ (3.2.3)

m L L m

The modulus bars must be inserted in order to ensure a decaying covariance. This is the velocity correlation function of a free Brownian particle. The mean square displacement may be found using the formula

t

<(Ax)2> = 2 (t-u)(v(t)v(u))du (3.2.4) o

yielding the same result [Eq. 3.1.12)] as before. The velocity correlation function may also be computed using the

Wiener-Khinchine theorem. We now give a more detailed exposition of this than is given in Chapter 1. We follow closely the paper of Wang and

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Chapter 3. Brownian Motion of a Free Particle and a Harmonic Oscillator 239

Uhlenbeck [5]. Consider, for a very long time T', a stationary random process; that is, a process where the mechanism which causes the fluctuations does not change with the course of time. In order to achieve a general treatment, we specify the process by the random variable £(f) which is a real function of the time. We now form

T' 12

^ 7 > = l i m - L f ftt)dt. (3.2.5)

This is the time average of £(f). We may also write

£(*)<?(* +r ) =l im ^ f £(t)£(t + T)dt. (3.2.6) 1 -T'n

We also note that for a stationary process, we have equality of the ensemble averages and time averages (ergodic theorem) so that

(£(/)£(*+ *•)) = <?(')£(*+ T)- (3-2.7) Now

#(*) = — \ g(a>)eimda>, (3.2.8) lit J

—oo

where

l0) = J $i)e-imdt. (3.2.9) —oo

Hence, using the shift theorem for Fourier transforms

(&!Mt + f))=1im± J f | ( f i , y - ^ f ^ ^ | ( q ) ^ . (3.2.10) r - w -J, . J 2# ' 2#

Now, on noting that

— 7 e±ixydy = S(x) (3.2.11) IK J

—oo

and assuming that one may perform the integration over the t variables first, one obtains

(#(0£(f + r ) ) = l i m - ^ J J ^(0)t(0)l) e^SiOt +G))da>dco1 .(3.2.12) —oo —oo

Thus, with oo

J f(x)S(x + a)dx = f(-a), (3.2.13)

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240 The Langevin Equation

(£(t)Z(t + T)) = lim 1; 1 ico)Z-co)e-imd(D (3.2.14) T —»°o I J

—oo

and since (£(t)g(t + r)) is real, so that £(-a>) = £;*(a>), we have

(^(t^(t + T)) = -] ^^C6)e-imd(0, (3.2.15) 2K

where 1 / ? ,

^ ( f f l ) = U m — ( | ( a » ) (3.2.16)

is the spectral density of the random function £,t). Since <t>^(oi) is an even function of a>, we will also have

1 °° (£(*)£(*+ T)) = - J ®f(a>)cosmda>. (3.2.17)

Also by Fourier's integral theorem we have oo

*£«»)=/ frttfit + T^dT (3.2.18) —oo

so that oo

^ ( © ) = 2 j (^(t)^(t + T))cosa)TdT. (3.2.19) 0

This is the Wiener-Khinchine theorem. We illustrate its use by evaluating the velocity correlation

function of a free Brownian particle. We have x(t) = v(t), vt) = -pv(t) + X(t)lm. (3.2.20)

The velocity v ( 0 is a Markov process. On denoting the Fourier transform ofv(f)by

oo

v(0))= J v(t)eimdt, (3.2.21) —oo

we have from Eq. (3.2.20)

v(G))[fi + iG)] = X(G))/m, (3.2.22)

where Xa>) is the Fourier transform of X(t). Thus writing

Z(Q)) = (j3 + icoyl, (3.2.23)

v(co) = %(a>)%i<o)lm . (3.2.24)

%((0) is the transfer function of the system. Now

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Chapter 3. Brownian Motion of a Free Particle and a Harmonic Oscillator 241

lim J K = <$>v(co) T —>°° £

so that with Eq. (3.2.24) we have

ml m2(j32 + co2) » , M - * « ) / W ^ - * ' W , , (3.2.25)

On noting that oo

<&x (a) = J 2DST)eimdt = 2D, —oo

we obtain 2D 1 m2 01 + co

Thus, the substitution of Eq. (3.2.26) into Eq. (3.2.15) yields

, , , , ,v 1 7 2D e'^dfl) 2BkT°r COSGK J

(v(t)v(t + T)) = f — T — T = -£ f —. -d(0. x ' 2xi m2/32+co2 nm p2 + a>2

since D - fik Tm. On using

<Dv(<y) = — — 2 - ( 3 - 2 - 2 6 )

cosmx , 7t _u,i J " 0

we have

r cos/.„. , „ _ =-ax = — e

% 1 + x2 2

(v(r)v(r + T)) = (kT/m)e-P]Tl. (3.2.27) This is the velocity correlation function as obtained by the Wiener-Khinchine theorem. The method of calculation of correlation functions based on this theorem is often known as Rice's method [5].

3.3 Brownian Motion of a Harmonic Oscillator

The equation of motion of the harmonic oscillator driven by a white noise driving force A.(t) is

mx(t) + Cx(t) + m(02x(t) = A(t),

Mf)Mf + T) = 2kTCS(T) = 2DS(r). We shall now demonstrate how the correlation matrix which contains two auto and two crosscorrelation functions, namely:

C(x(t)x(t + T)) (x(t)x(t + T))') (3.3.2)

J^iWxO + T)) (x(t)x(t + T))J may also be calculated from the Wiener-Khinchine theorem. We first calculate (x(t)xt + r ) ) . To this end, we rewrite Eq. (3.3.1) as

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242 The Langevin Equation

x(t) + /?x(f) + G)2x(t) = Xt)lm .

We have as in the last section

X(co)

(3.3.3)

x(ct)) = m[co2

0 + ico/3- 6>2)

and the spectral density <£* (co) of the displacement is

®xm=-m

(3.3.4) a>2 + ico/3 - co2

Thus on recalling that the spectral density of the noise is &A(co) = 2D , we have by the Wiener-Khinchine theorem

D (x(t)x(t + T)) = J rdco

(3.3.5) Jem" U (co2-co2)2 + o)2j32

The imaginary part of this integral vanishes because it gives rise to an ,2 ^ ol odd function in the integrand. For co0 > jB 14 and r > 0, we have [5]

1 J

cos cordco -pCll ( Q \ cosfiATH sinco,T

^ 2a\ l ^Lico20-co2)2 + (o2p2 /3W2

where the damped natural frequency o\ is defined as

tf = a>20-p

2l4. Hence the position autocorrelation function is given by

kT -4( fi . (x(t)x(t + T))-

m(On

cos COT + ——sin co, t 1 2cox '

(3.3.6)

(3.3.7)

(3.3.8)

We may now use the result to calculate the remaining elements of the correlation matrix by differentiation. We have

(x(t)x(t + T)) = Ut)^^p) = -^-(x(t)x(t + T) (3.3.9) N ' \ d(t + r) I dT^ so that with Eq. (3.3.8)

(x(t)x(t + T)) = -—e-pxn s i n ^ r . (3.3.10)

x ' mcox

In order to evaluate the two remaining correlation functions, we note that by stationarity shifting the time axis from ttot-T

'd_

\dt

\stv\A/i vty u j u i n i i t n i v u i i i v u i k i u xivsj-a*. » *.*_/ •• •-

-x(t)x(t + T)\ = -l—x(t - T)x(t)\ = -—(x(t)xt + -r)) (3.3.11) / \dz J dt^

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Chapter 3. Brownian Motion of a Free Particle and a Harmonic Oscillator 243

kT

max e PT/2smco,T. (3.3.12)

The velocity autocorrelation function may likewise be evaluated. We have by stationarity,

d d2

(x(t)x(t + T)) = (x(t)x(t + T)) = Axt)x(t + T)) , (3.3.13) x dt dt whence

< i ( f ) i ( f + *)) = — e - p r l 1

m

f cosfi^T-

2o)i (3.3.14)

The correlation matrix of the harmonic oscillator in the underdamped regime is thus

m

r 1 f J_g-/Jr/2 COSQ),T-\ smear

2(0, ^ . e ^/2smo\T

—e P*12 sin a\T COy

-Prl2

0\

COS G\T-\

2o\ -smart J

.(3.3.15)

We remark that the underdamped process is no longer a Markov process, there is significant memory of previous positions. Instead xt) is regarded as [5] the projection of the two dimensional Markov process [x(t),x(t)].

3.4 Application to Dielectric Relaxation

In his first model of the phenomenon of dielectric relaxation, given in 1913, Debye [6] considers a system of molecules each carrying a permanent dipole fi, with every molecule free to rotate about a fixed axis (see Chapter 2, Section 2.6). Supposing that an external spatially uniform small dc electric field E (^=juE/(kT)« 1) had been applied to the system at t = - °° and at time t = 0 the field has been switched off, the equation of motion of the fixed axis rotator is

£0(.t) + fiEsm0(t) = Mt), t<0

0t) = Xt), t>0

where 6t) is the angle between the dipole u and the direction the field

E, and A(t) is a white-noise driving torque. One can then show that the dipole moment density satisfies the one-dimensional Smoluchowski equation:

(3.4.1)

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244 The Langevin Equation

3 r f f i O . . D £ £ < M , > o . (3.4.2) at oQ

Equation (3.4.2) together with the initial condition

/(0,O) = c ( l + c o s 0 J (3.4.3)

then yields the well-known result [13] for the mean dipole moment ,i(cos*) = | X - " " \ (3.4.4)

where TD =£/(kT) is the Debye relaxation time for rotation about a fixed axis.

Inertial effects are readily included in the above model by simply including the term Id in Eq. (3.4.1) above. Thus, that equation becomes

7(9(0 + 0t) + ME sin 0(t) = A(t) (3.4.5) for t < 0 and

W(t) + C0(t) = Mt) (3.4.6) for t > 0. Equation (3.4.6) is a Langevin equation of the same form as that for the translational Brownian motion [cf. Eq. (3.1.3)]. Consequently, all the results we have previously obtained for the translational Brownian motion can be applied to the rotational Brownian motion. In particular, the mean square angular displacement ((Ad)2) is given by

((Afl2) = ( / f c - l + e -*) , ( f>0) , (3.4.7)

where /?= pi. We note that Eq. (3.4.6) is a linear stochastic differential equation with constant coefficients. This is of central importance to what follows, where a theorem about characteristic functions of Gaussian random variables [7] is used to calculate the dipole moment autocorrelation function.

3.4.1 Theorem about Gaussian random variables

The theorem is that if X is a random variable with a Gaussian distribution, then [7] (see also Chapter 1, Section 1.6.3)

( e * ) = /*>-(<*2>-<*>2)/2. (3.4.1.1)

Consider now the function (cos6'(?1)cos<9(f2)) = ^cos<9(?1)cos[(9(?1) + A(9]^,

where A0 - 6(t2)-6(t]) . This becomes, on expanding the terms within

the angular brackets,

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-In. 2

Chapter 3. Brownian Motion of a Free Particle and a Harmonic Oscillator 245

(cosflfo )cos0(f2)) = -(cos A0) + (cos[A0 + 20(tx)])

= l R e [ ( ^ ) + (e,'Afl+2ff('l))], (3.4.1.2)

where Re denotes "real part of. Now, if 6tx) and 0(t2) are Gaussian random variables, any linear combination of them (e.g., Ad) is also a Gaussian random variable. Hence

(cos(9(f1)cos^(f2))

' iA0)-^[((Aef)-(Ae)2] i(Aff+2tf(»1))-I[([A»f2^)]2)-([A*f2fl(,1)]>2]>l (3-4.1.3)

V

Now, if the process we are considering is stationary, only those terms in Eq. (3.4.1.3) which are functions of the time difference \t2-t\\ will survive, so that Eq. (3.4.1.3) will take on the simple form

cos0tl)cos0(t2)) = -Rele^ 2^ M 'H. (3.4.1.4)

Further, if 6 is a centred Gaussian random variable (i.e., (M = 0), then

Eq. (3.4.1.4) reduces to

(cos0(r,)cos0(f2)) = - e ^ ' . (3.4.1.5)

Thus, in this case, a knowledge of ((A0))2 is sufficient to allow us to calculate orientational averages. We have given the theorem for n - 1. For any integer value of n it is simply

/ / x / x\ 1 -n2((A6>)2)/2 cosnd(tl)cosndt2)) = -e x ' . (3.4.1.6)

We reiterate that this equation is true only for centred Gaussian random variables. Thus the theorem will only hold good for those systems where 0 satisfies a linear differential equation of motion.

We may readily obtain from Eq. (3.4.7) and (3.4.1.6) the dipole autocorrelation function R(t):

, N (COS0(0)COS0(0) -kTIUPHpt-X+e-P') R(t) = X , ' A ^ = e l ; . (3.4.1.7)

(cos2 0(0)) By arguments similar to those used to derive Eq. (3.4.1.7), one may also write down the angular velocity autocorrelation function. It is

(0(O)0(O) = (^2(O)e_/" =(kTII)e-pt. (3.4.1.8)

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246 The Langevin Equation

The complex polarisability a(co) is written down with the aid of the usual linear response formula, Chapter 2, Eq. (2.8.15). It is (with id)= s)

a(s)

tf-(O)

so that [9]

a(s)

= l - s j e's'Rt)dt,

= 1 or'(O) s + yfi

1 + r- + £ + ... (s/j3) + y+l Ks/p) + y+l][(s/0) + y+2\

ST = l--—2-M[l,l + y(l + sTD),y], (3.4.1.9)

l + STD

where y=kT lIfi2), /3y=\lzD,&nd M(a,b,z) is the confluent hyper-geometric (Kummer) function defined as [15]

, w , \ * a aa + \)z2 aa + \)(a + 2) z3

M(a,b,z) = l + -z + — — + — — + v ' b b(b + l) 2! b(b + l)(b + 2) 3! This result shows that the effect of including the inertia of the dipole is to produce a denumerable set of relaxation mechanisms. The series in Eq. (3.4.1.9) may be rewritten as the continued fraction [9,10].

^ - = 1 *H .(3.4.1.10) ^ ° > sl/3 + 1

1 + 5 / ^ + ^ 2 + S/J3 + - 3y

3 + S/J3 + ... The first convergent of Eq. (3.4.1.10) yields the Debye relaxation formula

a(s) = 1 or'(O) 1 + STD '

The second convergent of Eq. (3.4.1.10) yields the Rocard equation [11]

^ = — r t — s 1 _ — . (3.4.1.H) or'(O) (s + yfi)s + /3) \ + STD + s2tD I p

This provides [9] a good approximation to the relaxation behaviour provided y< 0.05. Equations (3.4.1.9) and (3.4.1.11) have exhaustively been discussed in Chapter 2 of Molecular Dynamics [13]. It is evident that the simple Debye model, including an inertial correction, is not sufficient to explain the experimental evidence. The inertial correction embodied in Eq. (3.4.1.11) does, however, remove the unacceptable plateau in the high-frequency absorption profile (see Section 1.15.1). The

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Chapter 3. Brownian Motion of a Free Particle and a Harmonic Oscillator 247

same holds true for all the three-dimensional versions of the Debye model, including inertial effects. These are the sphere, prolate and oblate spheroid, and general ellipsoid. The far-infrared absorption and the microwave absorption linked to it cannot be satisfactorily explained by the Debye theory with inertial corrections only. As a prelude to the itinerant oscillator model of Chapter 10, which is an attempt to address the problem of the Poley absorption theoretically, we now treat the torsional oscillator model, originally discussed by Calderwood et al. [14].

3.5 Torsional Oscillator Model: Example of the Use of the Wiener Integral

The simplest model we can study that takes any account of the effect of the neighbours of a molecule on its relaxation behaviour is where we regard the molecule as a torsional oscillator. Thus, we suppose that at a time t after the switching off of a field which had been steady up to t = 0, the equation of motion of the dipole is

dt) + pdt) + a%0(!) = Wt), (O20=KII, (3.5.1)

where W(t) is the Wiener process (the symbol W being used for ease of comparison with the original literature). We use the Wiener process here in order to illustrate how this process and the Wiener integral may be used in the computation of averages, that is,

(W(t)) = 0, (w(tl)Wt2)) = c2rmn(tl,t2), W(t2)-W(tl) = £t2-tl).

If A and A'denote the time differences t[ - tj and t'i - t'j, respectively. Thus (see Chapter 1, Section 1.8.1)

(<f(A)) = 0, (<f(A)<f(A')) = c 2 IAnA' l .

The restoring torque K6(t) is used to crudely represent the effect of interaction due to the neighbours of the molecule. In reality, this restoring torque is not at all linear, because the torque on the dipole u, placed in a field E is T = (ixE so that |T| = ||i||E|sin0. Hence, Eq. (3.5.1) should only hold for small 6. The case of unrestricted 6? is considered in detail in Chapter 10. We now proceed as in the free-particle problem. We write Eq. (3.5.1) in the matrix form

X(0 = AX(0 + B[/(0,

~e 0(t)

A = "0

r^l

l

-P. , B =

"0"

1 The solution of this equation may be calculated just as the free particle. We have

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248 The Langevin Equation

0(t)

0(0

e0(0[c0(0 + A o ( 0 / A ] -2e0(t)s0(t)/ PX

- ^ 0 ( 0 ^ 0 ( 0 / ( 2 A ) e0(t)[c0(t)-J3s0(t)/^]

+ '^(0[c,(0 + ( 0 / A ] -2et(t)srt)lfr '

-ffife(f>T(0/(2A) e f ( 0 k ( 0 - ^ ( 0 / A ]

(3.5.3)

#(<**),

where we have abbreviated

eT(t) = e-P'-r)/2, sT(t) = smh[j31(t-T)/2], cr(f) = cosh [# (* -* ) /2 ] so that r= 0 on the first line in Eq. (3.5.3). We use the stationary solution of Eq. (3.5.3) rather than averaging over the initial conditions again as an example of an alternative method of calculation of the desired averages. The stationary solution is found by simply extending the lower limit of integration to -QO and setting the complementary part of the solution equal to zero in Eq. (3.5.3). The matrix elements which are useful to us are 0and 6. We have

m = \ \ e -*" ) / 2 s inh[A(f -* ) /2 ]# (dr ) A -oo

and thus

6f) = J e-^ '-T ) / 2(cosh[A(r-r) /2]-(^/A)sinh[A(?-T)/2])^(rfr) , —oo

where ft2 = ft2 - Acol. The formulae appropriate for the periodic (/?i is imaginary) and the critically damped (fit = 0) cases can be readily written down by replacing cosh(/?^/2) and /3[lsmh(filt/2), respectively, by cos 6^ and (2d)ly

l sma\t, in the periodic case, and by 1 and t/2 in the critically damped case.

We now define a function , x f ( 2 / A ) ^ ( ' - r ) / 2 s i n h [ # ( ' - T ) / 2 ] , T < t f

8'T) = \o, Thus, by the properties of the Wiener integral,

T>t.

(e(tx)e(t2))=c2\ gtxT)gh(T)dT

4c_ 2 min(ri,/2

J e-^-T)l2smh[p(tl-T)/2y^-r)nSmh[pt2 -z)/2]dT

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Chapter 3. Brownian Motion of a Free Particle and a Harmonic Oscillator 249

„2

:—^(cosh[ /6K?2-r1)/2] + ( ^ / A ) s i n h [ A K - f 1 | / 2 > -0\l2-h\/2

Similarly, one may deduce that

(^( ' i)^( '2)) = ^ f c o s h [ A ( r 2 - r J / 2 ] - | - s i i i h [ A | r 2 - r 1 | / 2 ] l e - ^ - ' > l / 2 .

and that (0(tl)) = (0(t2)) = O. Thus, for equal times, t\ = t2 -t say, we

have

(02(O) = c2l(2/3co20) and (e\t)) = c21(2/3).

If we assume as in [1] the Maxwell-Boltzmann distribution (i.e., stationarity) for the angular velocity 0(t), we must have

l(d2(t)/2 = kT/2

and thus c21(2/3) -kTII. Whence with tx = 0, t2 = t, we have

(0(O)0(t)) = — r [cosh(A' /2) + (A //3l)sinh(/3lt/2)]e-/"2

Ico0

and

(^2(0))-(^2w) = ^/(/^2). Substituting from all these equations into Eq. (3.4.1.3), we find that [14]

(cos#(O)cos0(O)o = e"rcoshfe_/?r/2 [coshCflf/2) + (>0/^)sinh(^f/2)]J,

where, in this instance, y = kT/(I(i)2). Note that the more complicated Eq. (3.4.1.3) must be used rather than the simpler Eq. (3.4.1.6) because the second term on the right of Eq. (3.4.1.3) does not vanish, due to the influence of the restoring torque. Likewise, one may show that

(cos 0 ( 0 ) ) ; ; = ^

and thus the after-effect function b (t) is

b(t) = p J ( c o s 0(0) cos 0(t))o - (cos 0(O))201

2 (3-5-4)

= e-rcosh(ye-0'/2[cosh(/3lt/2) + (/3/j3l)smh(/3lt/2)])-l.

Before calculating the complex polarisability from Eq. (3.5.4), it is instructive to make some comments on this equation. First we consider the limiting value of b (t) when 0)o is allowed to tend to zero. We find by L'Hopital's rule that

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250 The Langevin Equation

kT

co0^o 2kT

fi,-l+e-fi)

(3.5.5)

in agreement with the result for the free rotator. Returning now to Eq. (3.5.4), we consider the behaviour of b(t) for large and small values of time t. For very short times (t«f3~l), we find that

2

K 0 = ^ [ c o s h r ( l - « o ¥ ) - l ] . (3.5.6)

Thus, for short times, b (t) is independent of the friction coefficient J3. On the other hand, as t tends to infinity, we find that

lim b(t) = 0 ,

a condition which b(t) must satisfy by definition. We remark [1] that in the highly overdamped case, the displacement is again effectively a Markov process.

We have from Eq. (3.5.4), on expanding it as a Taylor series in powers of % the following expression for the after-effect function bt):

W = 7 ^ ^ " ? ' S ^ f ^ [ c o s h ( A ^ 2 ) + (^/A)sinh(Af/2)]2 'J , (3.5.7) ki n=\ y2n)-

which on introduction of the angle <f>, where sinh^ = /?, l(20)o), cosh^ = j3/(2o)0),

may be written 2 oo

K-l n=i (2n) •e~npt (20)0 l/3l f" sinh2" /3xtl2 + </>)

On use of the binomial theorem, we have

M W^-'I kT n=l

r2" (coQ In

(2ny\ft) y ( _ 1 ) m I 2 " \e2(n-m)<[>ff-[mft+n(P-A)]l

m=0 m (3.5.8)

We now recall that the complex polarisability CC(CQ) is defined by Eq. (2.8.12) of Chapter 2. Whence, with Eq. (3.5.8), we have

-r oo

a(co) = ^:r^Y. 2f_

kTn=l

y?n

(2n)\ 0^

>2n In

K-D" m=0

(2n\ e2n-

1 + ia>z„ (3.5.9)

Here rjn =mj3l + « ( / ? - $ ) . The factor in square brackets in Eq. (3.5.9)

represents the partial-fraction expansion of the Fourier transform of

e - "^ s inh 2 " (^ /2 + ) .

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Chapter 3. Brownian Motion of a Free Particle and a Harmonic Oscillator 251

One may calculate successive terms in Eq. (3.5.9) by expanding the expression in square brackets in that equation for n = 1,2,3,.... Thus for n = 1, the value of this expression is

2 ^ + 'f 2 (3.5.10) (j3 + ico)[(j3 + ico)2-tf]

while for n = 2 we must add the term

A4 f (2/? + ico? + 3/7(2/? + ico)2 + 2(5co20 + p2 )(2/3 + ico) +12co2/3

co2 (2p + ico)[(2/3 + ico)2 - fi)[(2j3 + ico)2 - A/32 ] J ' It is evident from the above considerations that the polarisability in the

underdamped case co2>]32/4 consists of a discrete set of resonant absorptions. This is of importance in connection with the itinerant oscillator model which has been extensively used to discuss the far-infrared absorption and which we shall describe in Chapter 10.

References

1. G. E. Uhlenbeck and L. S. Ornstein, Phys. Rev. 36, 823 (1930). 2. A. Einstein, in R. Fiirth, Ed., Investigations on the Theory of the Brownian

Movement, Dover, New York, 1954. 3. P. Langevin, C. R. Acad. Sci. Paris 146, 530 (1908). 4. J. L. Doob, Ann. Math., 43, 351 (1942). 5. M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945). 6. P. Debye, Polar Molecules, Chemical Catalog, New York 1929; reprinted by Dover

publications, New York, 1954. 7. H. Cramer, Random Variables and Probability Distributions, Cambridge Tracts on

Mathematics and Mathematical Physics, No. 36, Cambridge University Press, Cambridge, 1970.

8. B. K. P. Scaife, Principles of Dielectrics, Oxford University Press, Oxford, 1989. 9. R. A. Sack, Proc. Phys. Soc. B 70, 402 (1957). 10. E. P. Gross, J. Chem. Phys. 23, 1415 (1955). 11. M. Y. Rocard, /. Phys. Radium, 4, 247 (1933). 12. B. K. P. Scaife, Complex Permittivity, The English Universities Press, London,

1971. 13. M. W. Evans, G. J. Evans, W. T. Coffey, and P. Grigolini, Molecular Dynamics,

Wiley-Interscience, New York, 1982. 14. J. H. Calderwood, W. T. Coffey, A. Morita, and S. Walker, Proc. R. Soc. Lond. A

352, 275 (1976). 15 M. Abramowitz and I. Stegun, Eds., Handbook of Mathematical Functions, Dover,

New York, 1964.

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Chapter 4

Two-Dimensional Rotational Brownian Motion in iV-Fold Cosine Potentials

4.1 Introduction

A detailed numerical study of rotational Brownian motion about a fixed axis in N-fold cosine potentials in both the time and frequency domains has been given in the context of the Fokker-Planck equation by Reid [1] with particular reference to the behaviour of the dielectric dispersion and absorption spectra of polar liquids. He concluded that such a model can reproduce both relaxation and resonant behaviour when inertial effects are included. This behaviour is due to the use of a periodic cosine potential (rather than the parabolic one considered in Chapter 3). Such a potential allows the flipping of rotators to neighbouring wells, thus permitting both relaxation and oscillatory behaviour [1] in the same model. This model has also been studied (in the noninertial limit) by Lauritzen and Zwanzig [2] in connection with site models of dielectric relaxation in molecular crystals [3-6].

X M

Figure 4.1.1. Geometric scheme for the rotation of a dipole in a plane.

252

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 253

In the present chapter, the Langevin equation method is applied to this model to study the relaxation behaviour in the noninertial limit. We shall consider the rotational Brownian motion of a planar rotator with dipole moment \l in the Af-fold cosine potential

y(6>) = -y0cosM9, (4.1.1)

in the context of the Langevin equation (0 is the angle between the dipole vector u, and the z axis, see Fig. 4.1.1). The quantities of interest are the longitudinal and transverse components of the complex susceptibility and the corresponding relaxation times.

4.2 Langevin Equation for Rotation in Two Dimensions

The Langevin equations for a dipole u rotating about an axis normal to

the xz plane are [1]

10(t) + g0(t) + NVo sin N0(t) + iE(t) sin 0(t) = A(t), E||z, (4.2.1a)

Wt) + g0(t) + NVo sin N0(t) + /iE(t) cos 0t) = M0, E||x, (4.2.1b) where / is the moment of inertia of the rotator about the axis of rotation, 0 is the angle the rotator makes with the direction of the driving field E(0, £0 and A(t) are the frictional and white noise torques due to the

Brownian motion. It is assumed as usual that the random torque X (t) has the property

A(7j = 0, Mt)Mt') = 2gkTS(t-t'). (4.2.2) In order to specialise Eq. (4.2.1) to a step-on field, we write

E(t) = EU(t), (4.2.3)

where U (t) is the unit step function and E is its amplitude. We require to calculate, for this model, the statistical averages (cos 0) and (sin 0) when the inertial effects are ignored.

The problem which presents itself when treating the model using the Langevin equation in the form of Eq. (4.2.1) is that it is not apparent how that equation may be linearised to yield the solution for a small driving field. This difficulty may be circumvented by rewriting Eq. (4.2.1) as an equation of motion for

rn = e~ine (4.2.4) so that

0 = irneineln, (4.2.5)

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254 The Langevin Equation

and

6 = i[rneine-ryina]/n. (4.2.6)

The Langevin equation (4.2.1) with the field E applied along the z axis and the foregoing change of variable becomes

Irn (0 + qirn (t) - If* (t)r_n (t) + ?—±[rN+n (t) - rn_N (t)] 1 (4.2.7)

Equation (4.2.7) contains a multiplicative noise term -inrn(t)A(t) which contributes a noise-induced drift term to the average. The second order differential equation (4.2.7) reduces to a set of two first order nonlinear stochastic differential equations for the random variables £\ = rn and %2 = rn- On using these new variables together with the general expression for

skin rk=rnr (4.2.8)

in Eq. (4.2.7), we have

6 ( 0 = &( ' ) , (4.2.9)

+ nMEU(t)^l)/n ( f ) _ £(B+1)/„ (r)-j _ .nMt)^ (t)/I

(4.2.10)

2/ It is evident that Eqs. (4.2.9) and (4.2.10) are a set of nonlinear stochastic equations for the random variables ^ and £2»

ii(t) = hi[^(t)^(t)j] + gij[^(t),^(t),t]rj(t),(ij=h2), (4.2.11)

where Einstein's summation convention is assumed and *ii = Si2 = *2i=0. (4-2.12)

g22=-in^/I. (4.2.13)

By interpreting the Langevin equation for a set of N random variables

ft=£,...,&, namely:

m=hi[m,t]+8U[mu]rj(t) (4.2.14) with

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 255

r,.(o = o, r,(r)rJ.(0 = 2D^(/-r/) (4.2.15) and applying the Stratonovich rule we have already shown in Chapter 2, Section 2.5, that

i> =hi(x,t) + DgkJ(x,tfgijx]'t) . (4.2.16) axk

Thus according to Eq. (4.2.16), the average of Eq. (4.2.14) may now be expressed as equations of motion for the sharp starting values x. The jtt's in Eq. (4.2.16) are random variables with probability density function W(x,f) such that Wdxk is the probability of finding xk in the interval (Xk, Xk + dXk).

On applying the above results to the present problem [Eqs. (4.2.9)-(4.2.10), i.e., N= 2], we obtain

dxk

and thus

In 1+1/nl

x2--—x2+ 2j~ll i J 2/ L ' ' J

d2 d T.2 nNVQT , njuEU(t)f , n

or, returning to the original variables,

Td2 d T.2 nNVQT

V"+^ r-" / r" r-+~2^ [ r^ ™ 2 (4.2.17)

so that the noise-induced drift term vanishes. On averaging this equation over a large number of rotators we obtain

rd , > d . . T / .2 . nA^Vnr, . . ,n / - T < r n ) + ^ - ( r n ) - / ( r f l V „ ) + —-^[(/J^,,)-<!•„_„>] dr at i (4.2.18)

We remark that r„(0 in Eq. (4.2.7) and r„ in Eq. (4.2.18) have different meanings. Namely, rnt) in Eq.(4.2.7) is a stochastic variable while in Eq. (4.2.18), rn is the sharp (definite) value at time t. (Instead of using different symbols for the two quantities we have distinguished the sharp values at time t from the stochastic variables by deleting the time argument). However the quantity rn is itself a random variable which must be averaged over an ensemble of rotators. The symbol ( ) means such an ensemble average.

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256 The Langevin Equation

If the field E is applied along the x axis, we have a similar equation apart from some changes in the last term:

,2 <»•„ > + S—(rn ) - 7<r-fr_„ ) + —-^[<r„+n ) - </-„_„ )]

(4.2.19) injuEU(t)r, . , .-, „

~ 2 [<Vi) + <^+i)] = 0

The remaining terms in Eqs. (4.2.18) and (4.2.19) which cause difficulty are

/<r>_„> =-n2Irn62). (4.2.20)

Since the noninertial response pertains to the situation, where t » 1/g, which implicitly means that a Maxwellian distribution of angular velocities has been achieved, we may now write

(Irne2) = kT(rn) (4.2.21)

since the orientation and the angular velocity variables, when equilibrium of the angular velocities has been reached, are decoupled from each other, as far as the time behaviour of the orientations is concerned (Chapter 10). In the noninertial limit, / —> 0, we may also set

l(rn) = 0 (4.2.22) in Eqs. (4.2.18) and (4.2.19), so that we have finally for the noninertial response

^ ^ < ' i > + »2<r,> + [<rA,4<I>-<r11_ iV>] + 5^ [<r B + 1 >-<r - _ 1 >] = 0 at 2kT 2

(4.2.23) and

^D-(0 + n2(rn) + —^-[(rN+n)-(rn_N)] ^-Ufa^) + (rn+l)] = 0, at 2kT 2

(4.2.24) where TD = g/(kT) is the Debye relaxation time for planar rotators [3] and g = juE/(kT) is the dimensionless field parameter.

4.3 Longitudinal and Transverse Effective Relaxation Times in the Noninertial Limit

Equations (4.2.23) and (4.2.24) are the infinite hierarchies of differential-recurrence relations, which describe the dynamics of ensemble averages. We have mentioned in Chapter 2 that the standard approach to the

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 257

calculation of the longitudinal and transverse relaxation times is accomplished by rewriting the infinite hierarchy as a set of ordinary differential equations of the form [7]

X(f) = AX(f) + BU(r), where A is the transition matrix, B is the driving force matrix, and truncating at a given size of A. The longest relaxation time is then the reciprocal of the smallest non-vanishing root of the characteristic equation

detsI-A = 0, (4.3.1)

where s denotes the complex frequency. The disadvantage of this method is that it is, in general, impossible to obtain a closed form expression for the longest relaxation time. This difficulty may be circumvented in certain cases by means of the effective eigenvalue method.

We first consider the equation of motion for the real part of Eq. (4.2.23) for n = l , namely:

TD — (cos 6)t) + (cos 0)(t) = *-Ut)[\- (cos 26>>(0] dt 2 (4.3.2)

+—5-[<cos(W - 1)<?X0 - (oos(N + l)0(f)>] 2kT

and recall that we are only interested in the response linear in E. We therefore assume that in Eq. (4.3.2)

<cosn0)(O = (cosn<9)0 + (cosn0)l (t), (4.3.3) where the subscript "1" denotes the portion of the ensemble average which is linear in E and the subscript "0" denotes the equilibrium ensemble average in the absence of the perturbing field E, viz.,

* In

(A)o=-JA(0)ev^osNOI(kT)de, Z o

z=p< where the partition function ZQ is

In

0

On substituting this equation into Eq. (4.3.2), we find that in the linear approximation TD — <cos0)i + ( c o s ^ = —4(cos (N - W)i ~ (cos(N + \)0\ ]

dt 2kT (43A) +!#/(O(l-<cos20)o).

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258 The Langevin Equation

In the last term of Eq. (4.3.4), the ensemble average ( ) 0 is taken as that in the absence of E because that term is multiplied by £ rendering it comparable to the other terms in the equation.

Equation (4.3.4) represents a differential-recurrence relation driven by a forcing function, namely the U(t) term. In the context of the effective eigenvalue approach (Chapter 2, Section 2.12), we can write Eq. (4.3.4) as

d (cosflMO+ 4<cos0>!(f) = — £C/(O(l-<cos20)o),

dt CJ ' 2TD

where the effective eigenvalue is given by [utilising Eq. (4.3.4)] _1 d_

(cos 0)!(oo) dt Xef = ", T7T7<COS^)l(0

NVn r 1 (4-3.5) (COS 6 \ (oo) + — 5 . < C 0 S ( N + X)6\ (oo) - (C0S(7V - 1)0>, (oo)]

2ki ^(costf)^0 0)

We now have at t = °° from Eq. (4.3.3) (cos^)1(~) = (cos^)e g-(cos^)0 , (4.3.6)

where the subscript "eq" means equilibrium averaging defined as 2n In

Wea =

J A0)evocosN0KkT)+cosedg j A9)ev«™Nem\\ + Zcos9 + ...)de 0 0

«? 2K 2K J eVOcosNe/(kT)^cosOd0 j eV0™"WT)(l+<Ccose+ ) d 0

0 0

= (A)o+^((A(0)cos0)o-(A(9))o(cos0)o) + O^2).

Thus, we have with A0) = cos0

(cos 0)eq = (cos 0)o + <f ((cos2 0)o - (cos 6)l) + 0(?)

so that

(cos^)1(oo) = ^((cos2^)0-(cos^)2) + o ( f ) . (4.3.7)

The remaining term in Eq. (4.3.5) may be simplified as follows (cos(W + l)0)i (oo) - <cos(tf -1)0), (°°)

= (cos(W + \)0)eq - (cos(N - \)6)eq - (cos(N + l)0)o + (cos(N - l)0)o

= -2[(smN0sm0)eq-(smN0sm0)o~].

(4.3.8) Thus, we have

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 259

<COS(tf + \)0\ (oo) - (C0S(iV - \)0\ (oo) 3

= -2£(sin M9sin 6>(cos 6» - (cos 0)o ))0

by the same argument as in Eq. (4.3.7). The last term in the right hand side of Eq. (4.3.9) can be simplified as follows:

-2£(sin NG sin 0(cos 9 - (cos 0)o))

- * ((sin20-2sin0(cos0)o)—V(0))o . NV0

L' de Now, integrating by parts, we have

((sin 20 - 2 sin 6>(cos 6%)—V(0))Q

30

= — J (sin20-2sin0(cos0>o) J - V ^ ^ ^ W Z° o ^ (4.3.10)

2^2f(cos2^-cos^(cos^)0) /" c o s W / ( A : T )rf^ Z0 0

:2ytr((cos2(9)0-(cos(9)o).

Thus, by using the identity

%inM9sin6>(cos<9-(cos<9>0))0 =(cos2<9)0 -(cos<9>o (4.3.11) kT

and taking account of Eqs. (4.3.5)-(4.3.11), we obtain l-<cos2fl>0

; l + ( c o s 2 ^ ) 0 - 2 ( c o s ^

so that the effective longitudinal relaxation time T^ = (Xef )_1 is given by

Tef 1HCOS20)O-2(COS0)1 11 D l -(cos20)o

Here, we have evaluated X\f at the instant t = oo, because for the step-on

response (cos 0^(0) is equal to 0. Thus, the effective eigenvalue X^

defined by Eq. (4.3.5) yields the effective relaxation time t^ of the

relaxation function/(0 given by

/ (O = (cos0>e(?-(cos0>(O,

so that / (0) = (cos#)e? -(cos#)0 and /(°°) = 0 and, as usual,

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The

Of

Langevin Equation

= — f(t)\ /(O) !'=o

260

In order to calculate the transverse effective relaxation time Te[ , we consider the same problem as above, however, the step change in the field is applied parallel to the x-axis so that we need to determine the behaviour of the transverse component of the dipole moment. Therefore we consider the equation of motion for the imaginary part of Eq. (4.2.24) and we assume as before that

(sinn6)t) = (sinnd)Q + (sinnd)x, (4.3.14)

which leads one to the linearised equation for (sin d)x, i.e.,

^ ( s i n ^ + z ^ s i n f l ) , = - - ^ % + (cos20>o] , (4.3.15)

where the effective eigenvalue Xe[ is defined as

1 (sm#),(oo) <fr

NVni n (4.3.16) (sin 6\ (oo) + —£- <sin(JV + \)0\ (°°) + (sin(JV - 1)0>, (oo)]

2kT ^ ( s i n ^ ^ o o )

Noting that (sin<9)0=0, (4.3.17)

we have in the linear approximation of £ (sin0)1(oo)=(sin0>e9=£(l-(cos20>o)/2. (4.3.18)

Likewise

Thus

^fi-[(sin(N +1)0), + <sin(N - 1)0), ] = £(cos 28)Q. (4.3.19) 2kT

TXf = l + C0S2e)° (4-3.20) D ef l-(cos20>o

so that the effective transverse relaxation time zf = \X^ J is given by

Tef= 1-(COS20)O x D l + (cos20)o

Equations (4.3.13) and (4.3.21) are general formulae for the effective relaxation times in an TV-fold cosine potential. They will hold for any

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 261

potential of the form cosNO. In Chapter 5, we shall show how these formulae can be used to obtain the relaxation behaviour in a cos 6 potential with a tilt. In this chapter, we apply the effective eigenvalue approach to transverse relaxation in a cos 20 potential. (This potential is of interest in the context of the Brownian motion of a particle in a uniaxial crystalline potential created by its neighbours.) We shall also demonstrate that the effective eigenvalue cannot explain the longitudinal relaxation in the cos 20 potential.

4.4 Polarisabilities and Dielectric Relaxation Times of a Fixed Axis Rotator with Two Equivalent Sites

4.4.1 Introduction

Debye [3] in 1929 introduced a model of dielectric relaxation in a crystalline solid which assumes that a dipole can have only two oppositely directed orientations on either side of a potential barrier to rotation. He demonstrated that such a model yields relaxation behaviour similar in form to that predicted by his theory of rotational Brownian motion. However, the relaxation time depends exponentially on the height of the potential barrier. This approach was subsequently generalised and extended by a number of authors [2,4-6] to explain the dielectric relaxation behaviour of polar crystals.

The simplest site model [2] has two equivalent sites, leading to a single relaxation time proportional to eu/kT), where U is the potential barrier between the sites. According to Lauritzen and Zwanzig [2], the site model will provide a close approximation to the relaxation behaviour if the energy barrier to rotation is large and if the orientational sites are closely defined by the crystalline field. The assumptions of the site model that only two orientations are possible for the dipole will fail [2] when the ratio of the barrier energy to the temperature falls below a certain value. To analyse this, Lauritzen and Zwanzig abandoned the discrete orientation model and assumed a continuous distribution of orientations, the evolution of which is governed by the Smoluchowski equation for the distribution function W of the orientations of dipoles. To simplify their calculations, they restricted themselves to the situation where the dipole may rotate about a single axis under the influence of a potential

U(0) lkT) = 2a sin2 0, (4.4.1.1) where 0 is the angle describing the orientation of the dipole about its axis of rotation and 2o is the barrier height parameter. This potential

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262 The Langevin Equation

determines two potential minima on the sites at 0 = 0 and 0 = ;ras well as two energy barriers located at 0 = nil and 6= 3^/2. The analysis of Lauritzen and Zwanzig [2] proceeded from the Smoluchowski equation

*D -5-W,0 = 2a—[W(0,t)sm 20] + -^W(0,t) dt o0 d0

by using the method of separation of the variables in order to convert the solution of that equation to a Sturm-Liouville problem. They then integrated the Sturm-Liouville equation by the fourth-order Runge-Kutta method [2]. Hence they found the eigenvalues and eigenfunctions by trial and error. They were able to plot the transverse and longitudinal relaxation times and the corresponding complex polarisabilities as a function of the barrier height. They concluded that the longitudinal relaxation process effectively occurs by means of a single exponential decay. In the transverse process, on the other hand, they found that for very large values of the barrier height parameter two modes contribute to the relaxation. Nevertheless the relaxation times of these modes are approximately equal and the relaxation is effectively a single exponential decay. For intermediate barrier heights they found that the relaxation process may be closely represented by the sum of two-time decaying exponentials with different time constants, while for U = 0 the relaxation again occurs through a single exponential decay. In addition to this numerical treatment of the problem, they used a method based on the smallness of the lowest eigenvalue of the Sturm-Liouville equation. The method then allowed them to obtain an asymptotic expansion for the greatest longitudinal relaxation time in the limit of high potential barriers. In particular they showed that the greatest longitudinal relaxation time in this limit is

Tll(a)~TD^e2a. (4.4.1.2) OCT

In passing, one should note that this relaxation time is also of major interest in the theory of magnetic relaxation of single domain ferromagnetic particles [8] (see Chapter 7).

The analytic Eq. (4.4.1.2) is only valid for large barrier heights, which limits its range of applicability. In addition, the complex polarisabilities may only be obtained numerically. It is the purpose of this section to show how exact analytic formulae (valid for all barrier heights) for the longitudinal and transverse correlation times can be obtained from the zero frequency limit of the Laplace transforms of the corresponding decay functions as first demonstrated in [9]. These will be compared with

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 263

the effective relaxation times obtained in the previous section. In addition, we shall show how analytic formulae for the corresponding complex polarisabilities may be written down and how Eq. (4.4.1.2) may be recovered from the exact solution in the high a approximation. It will also be shown how Eq. (4.4.1.2) may be obtained from the transition state theory of Kramers [10,11] which has been introduced in Chapter 1.

In order to study the longitudinal relaxation behaviour, we suppose that a small constant field E (£ = /iE/(kT)«l) applied along the z axis is switched off at t = 0. The differential-recurrence relation for

/n(O = RerB = <cosn0>(f)-<cosn0>o (4.4.1.3)

may be obtained from Eq. (4.2.23) for N=2, (cosnd)0 is the statistical

average in the absence of the field E. We write n = p from now on. Thus

rD^fp(t) + p2fpt) = (rp[fp_2(t)-fp+2(t)]. (4.4.1.4)

By inspection of Eq. (4.4.1.3), it is obvious that f_p(t) = fp(t) (4.4.1.5)

so that the solution of Eq. (4.4.1.4) must be determined for positive p only. Equation (4.4.1.4) may further be decomposed into a set of equations for the odd and even ft (t) which are totally decoupled from each other. Here only the odd index functions /I (O =/2P+i(0 are of interest since we seek the relaxation behaviour of fi(t). The/j (t) satisfy

^ / 2 p + i ( 0 + (2/7 + l )2 /2 p + i (0 r -. (4.4.1.6)

= cr(2p + l)[f2p_l(t)-f2p+3(t)].

4.4.2 Matrix solution

The set of Eq. (4.4.1.6) may be solved numerically following the method outlined in Chapter 2, Section 2.7.2, by writing Eq. (4.4.1.6) in the matrix form

X(r) = AX(r), (4.4.2.1) where the system matrix A and the vector X(t) are given by Eqs. (2.7.2.6) and (2.7.2.4), respectively.

The relaxation modes of/i(?) may be found from Eq. (4.4.2.1) by assuming that A has a linearly independent set of P eigenvectors (Ri,..., R/>) [the value of P must be large enough to ensure convergence of the set of Eqs. (4.4.1.6)] so that (see Chapter 2, Section 7.2.2)

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264 The Langevin Equation

X(0 =

f flit))

/3(0 = b^-^% + bje'^'Rz +... + bpe'^'Rp, (4.4.2.2)

JIP+IV),

where - ^ are the eigenvalues of the the system matrix A and the bt are to be determined from the initial conditions /„(0).

The initial conditions have already been also obtained in Section 2.7.2 and are given by Eq. (2.7.2.11), viz.,

£lp+l(a) + Ip(<r) / 2 p + i (0) = - (4.4.2.3)

2 I0a) (/m(<j) is the modified Bessel function of the first kind of order m [12]) so yielding the initial value vector X(0):

f /i(0) 1

X(0) = /3(0)

= blRl+b2R2+... + bPRP. (4.4.2.4)

Thus, the decay of the longitudinal component of the dipole moment is given by

. . 2 T 7 CO

jufi(t) = M(cose)(t) = M2E kT *2/fc+lc (4.4.2.5)

t=o The first three eigenvalues ^+1 and amplitudes 02*+1 are given in Table 4.4.2.1 as a function of a.

The quantity of most interest is the area under the curve of the normalised longitudinal autocorrelation function which defines the longitudinal correlation time T\\. The longitudinal autocorrelation function Ct) of the dipole in the linear approximation in £is

C,(0 = / i (0 / / i (0 ) (4.4.2.6)

so that the correlation time Tj| is from Eq. (4.4.2.5)

Tl]=]ci(t)dt = ^ 5 > 2 * + i ^ + i - (4-4.2.7)

As shown in Chapter 2, Section 2.9, the correlation time Tn can readily be determined from the matrix equation

X(0) = -A"'X(0), (4.4.2.8)

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 265

Table 4.4.2.1. Amplitudes a2p+\ of the first three modes of the decay of the longitudinal polarisation and corresponding eigenvalues as functions of the barrier height parameter a.

a 0 1 2 3 4 5 6 7 8 9 10

KXD 1.00000

0.32370

0.08198

0.01716

0.00318

0.00055

0.00009

1.4 10~5

2.2 10"6

3.4 10"7

5.1 10"8

V D 9.00

9.61

11.62

15.19

20.31

24.83

34.52

43.05

52.00

60.99

69.80

^5TD

25.00

25.52

27.09

29.71

33.36

37.97

43.45

49.68

54.51

63.80

71.41

«i

0.500

0.719

0.842

0.900

0.929

0.945

0.955

0.962

0.967

0.971

0.974

«3

0.00000

0.00454

0.00665

0.00448

0.00239

0.00125

0.00070

0.00044

0.00030

0.00023

0.00018

«5

0.000000

0.000022

0.000175

0.000376

0.000477

0.000471

0.000410

0.000336

0.000270

0.000217

0.000176

where oo

X(0) = limfX(0<r"d/.

The relaxation time Tn may now be extracted from Eq. (4.4.2.8) by simply calculating A-1.

We remark that another time constant, the effective eigenvalue Alj given by Eq. (4.3.12), may also be defined in terms of ak and Ak as

j (4.4.2.9)

which may again be evaluated from Table 4.4.2.1. The effective relaxation time is the reciprocal time constant associated with the initial slope of the correlation function at t = 0. It contains contributions from all the eigenvalues as does the correlation time.

The comparison of 1 / (/liTD) and r„ / TD is given in Table 4.4.2.2. Evidently, the asymptotic behaviour of the longitudinal relaxation time rn

and that of the smallest nonvanishing eigenvalue X\ are the same. We shall now demonstrate how exact analytical formulae for the longitudinal complex susceptibility and correlation time may be obtained in terms of continued fractions.

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266 The Langevin Equation

Table 4.4.2.2. The longitudinal correlation time as yielded by Eq.(4.4.2.7) compared with the numerical solution of the characteristic equation [Eq.(4.3.1)] for 1 / (A]TD).

a 1 2 3 4 5 6 7 8

1/0*1*0) 3.08907

12.1980

58.2856

314.760

1830.49

11153.5

70132.9

451068

t\\ 1 ?D 3.07025

12.1006

57.9727

313.787

1827.12

11140.2

70074.9

450793

(4.4.3.1)

4.4.3 Longitudinalpolarisability and relaxation times

Following [9], let us consider the Laplace transform of the recurrence relation for/,, (t), Eq. (4.4.1.6). We will have for/? > 0

TDSf2p+l O) = */2p+l (°) + fep+lAp-l (*)

+^2p+lf2p+l(s) + 42p+l/2p+3<»>

where

a2p+i = ~(2p +1)2 + odpQ, q2p+l = cr(2/? +1), q+p+i = -a(2p +1).

(4.4.3.2) Here, we have taken into account that

/_i(0 = /i(0- (4.4.3.3) As shown in Chapter 2, Section 2.7.3, Eq. (4.4.3.1) can be solved exactly for f^s) in terms of ordinary continued fractions yielding

/ i ( * ) = %*-<?! -qtsi(s)

fm+if2P+mflq+2k-lSu+lis)

P=\ k=\ q2k+\

(4.4.3.4)

where the infinite continued fraction Sn (s) is defined by the following recurrence equation

4~P p TDs-qP-<i+pSp+2(s)

or

V * ) = , 2 op

TDs + p'+apSp+2(s)

(4.4.3.5)

(4.4.3.6)

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 267

Equation (4.4.3.4) may further be simplified if we write out the product n

Wtik-i lchk+\ explicitly. We have jt=i

k=\

-Yf_

2/7 + 1 (4.4.3.7)

Thus, Eq. (4.4.3.4) becomes

Ms)' STD +1-<7 + <JS3(S)

oo ( 1 \ P P

/i(°)+z^7/2P+i(0)n P

n (4.4.3.8)

On inserting the initial conditions, Eq. (4.4.2.3), Eq. (4.4.3.8) yields [9]

A(s) , (-1)" /,+1(<r) + /,(<r)

/,(<r)+ /„(*) n^+iw i i^ = to 1 + y

/,(0) sTD+l-er + aS3(s)[ p=l2p + lL ilKv)Ti0K.vj j t = 1

(4.4.3.9) This exact formula allows one to calculate the frequency dependence of the polarisability (X\\(D) since according to linear response theory

<*m , .-7.-.-*/i(o J ._1 ,„/i0'®) <(0)

-dt = \-ia>-/i(0) MO)

where

<(0) = //2 /,((T) + / 0 W kT 2Ua)

(4.4.3.10)

(4.4.3.11)

Thus we have

«[i(6>) 1-(7 + <TS3 (ico)

o((0) icmD +1 - a + <r53 (/«)

. ^ ( - l ) p f V i W + Wligro ,. " -ICOTD\-——^ — — M l-S2it+1("y) • D ^ 2 p + l A ^ + Zo r) Ji

(4.4.3.12)

The most significant feature of Eq. (4.4.3.9) is however that it is capable of yielding an exact expression for the correlation time which we have on setting s = 0:

11 /,(0) l-a + cxS3(0) ^2p + l

Ip+l(a) + Ip(CT)

!,((!) +I0(a) n 5 2 £ + i ( 0 ) . k=\ J

(4.4.3.13)

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268 The Langevin Equation

The continued fractions 5^(0) in Eq. (4.4.3.13) can be expressed in terms of the modified Bessel functions of the first kind of half integer order. We recall that the modified Bessel function of the first kind of order v satisfies the following recurrence equation [16]

/v_1(z)-/v+ ,(z) = (2v/z)/v(z), (4.4.3.14) which may be written as a continued fraction

- ^ - = . (4.4.3.15) 7v_,(z) 2v + z7v+i(z)//„(z)

If we now write v = k+l/2, where k is an integer, this expression becomes

WfL 1 . (4.4.3.16) /t_i/2(z) 2k + l + zIk+3/2(z)/Ik+l/2(z)

Now, the definition of S2k+l(0) given by Eq. (4.4.3.5), namely

52*+i(0)= ^ * . . . . <*>1). (4-4.3.17)

is identical to Eq. (4.4.3.15) on expansion when

S2k+l(0) = ^ ^ - . (4.4.3.18)

Equation (4.4.3.13) now becomes

l-cx + cr^^,=o2p + l

Ip+l(cr) + Ip(a) Vl/2(<7)

hl2&) (4.4.3.19)

Now, on noting that [16]

3/2 = c o r n e r - - , (4.4.3.20) Il/2(a) a

we have

T|| = *» (^.^(^W^W, ( g ) . (4.4.3.21)

This is an exact formula for the longitudinal correlation time which is valid for all a.

For large values of a, we evaluate Eq. (4.4.3.21) in the following way. The asymptotic expansion of Ip(<J) for large values of c is [16]

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 269

M<r) = V2 no

uNy (-i)knP+k + i/2) iQf i ^

tZxk\Tp-k + \l2)2af 2a (4.4.3.22)

On substituting this equation into Eq. (4.4.3.21), we obtain

h = rD > v (-DP

2a %2p + \ (4.4.3.23) TD J-a o / n _ JD71 „2a

2a 8a by the properties of the Riemann-Zeta function J3 (see Eq. (23.2.30) of Ref. [16]). This is Eq. (49) of Lauritzen and Zwanzig [2] which they derived using an asymptotic expansion of the Sturm-Liouville equation based on the smallness of the lowest eigenvalue. The limiting procedure leading to Eq. (4.4.3.23) has been carried out using the asymptotic expansions of the modified Bessel functions. Another way of proceeding to the high barrier limit which also has the advantage of allowing asymptotic correction terms of higher orders to be computed has been accomplished by Coffey et al. [17]. Their method is to write the products

Ip+yiWpiP) and Ip+l/2(a)Ip+l(a)

occurring in the summation in Eq. (4.4.3.21) as integrals using formulae given by Watson [12] [p.441 Section 13.72, Eq. (2)]. This procedure in effect allows one to replace the entire summation over p by an integral, so eliminating the p dependence. Application of the method of steepest descents as in Ref. [17] then leads to

T" "•"•- - (4.4.3.24) J^e2o 8a 4a

which is Eq. (4.4.3.23) with a small correction term. The form of the correction term agrees with the result of Visscher [18] [his Eq. (14)], who studied the escape of a particle from a single well using a refined version of the Kramers transition state method. If instead of Eq. (4.4.3.23) we use Eq. (4.4.3.24) which has the small correction term l/(4o), then the exact solution is reproduced by the asymptotic solution to a higher degree of accuracy [9]. A striking feature of the asymptotic formula (4.4.3.24) is that it deviates by a small amount from the exact solution as a -» °° .

Thus, the asymptotic behaviour of the longitudinal relaxation time f|| is governed by the smallest nonvanishing eigenvalue X\. In the context of the continued fraction approach, the smallest eigenvalue X\ for the present problem is given by Eq. (2.11.1.16), Section 2.11.1, viz.,

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270 The Langevin Equation

\ i D n

l-e -2a K-i) p 'p+l/2 (v)

p=0 2p + \

-1

(4.4.3.25)

(4.4.3.26)

Equation (4.4.3.25) can be further simplified by noting that [19]

p=0 2p + l n \ 2 2 2

where 1F^ax,a2\bx,b1,b'i\z) is a hypergeometric function [19]. Thus,

Eq. (4.4.3.26) becomes [20]

Vz> = 2a

1--2a 21 3 i , i , - . - , - ;cr

2 2 2

- l

(4.4.3.27)

\ZD =\-<J + —CT 2+0((T 3 ) . (4.4.3.28)

For a « 1, one can obtain from Eq. (4.4.3.25) or Eq. (4.4.3.27) the Taylor series expansion of X\.

_5_

16 In the opposite limit ( c » 1), on using the asymptotic expansion of Ipz), Eq. (4.4.3.22), in Eq. (4.4.3.23), one arrives at the results of Lauritzen andZwanzig [2]:

4flVD ~ 8ae-2a IK . (4.4.3.29)

We remark that according to linear response theory (Chapter 2,

Section 2.8), the quantity fy(ia))//i(0) appearing in Eq. (4.4.3.10) is the

one-sided Fourier transform of the equilibrium longitudinal dipole autocorrelation function C^(t) defined as

/ l (f)_(cosfl(0)cosfl(0)0

c„(0 = - (4.4.3.30) /i(0) (cos20(O))o

As we have shown in Chapter 2, Section 2.10, the relaxation time zj, [Eq.

(4.4.3.21)] may be equivalently presented in analytic form as [see Section 2.10, Eq. (2.10.27)]

„2<T

T„=-rDe f e-^oS2Oed2^^sin0ye ( 4 4 3 3 ! )

4o-[/1(o-) + /0(fT)]J0

The numerical calculation shows that both Eqs. (4.4.3.21) and (4.4.3.31) give the same result.

Yet another time constant of interest is the effective relaxation time if . To define if , we refer to Eq. (4.3.12) from Section 4.3, i.e.,

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 271

Tef=T l + (cos2fl)0-2(cosfl)g 11 D l -(cos20)o

In Eq. (4.4.3.32), the quantities (cosn#)0 are now such that

<cos6>>0=0

and <cos20) o=/ 1 ( (7) / / o ( (7) .

Therefore, we have [cf. Eq. (2.12.8), Section 2.12]

Tef=T /o(g) + / l W 11 DI0(a)-Il(a)'

On using the Taylor expansion of In(J) for small values of a[16], we

have

(4.4.3.32)

(4.4.3.33)

(4.4.3.34)

(4.4.3.35)

ef l + a/2 ..

For large values of a, on using the asymptotic expansion Eq for/? = 0 and 1, viz.,

I0(cr) = , 1 + — + ... 0 -JlnoY 8a

in Eq. (4.4.3.35), we obtain

1 and IAa)-

Vz 7t(J 1 + ...

8a

(4.4.3.36)

(4.4.3.22)

(4.4.3.37)

Tef -T 1 +1/(8<7)+ ... + l-3/(8cr) •

l + l/(8cr) + . . . - l + 3/(8cr) + ...

We compare the time constants r,.

= 4azD, ((7»1). (4.4.3.38)

r,f , l/zli and l / ^ " in Fig. 4.4.3.1. It is evident from Fig. 4.4.3.1 that the effective relaxation time is a poor approximation to the longitudinal correlation time for a> 1 as is also apparent on comparing Eqs. (4.4.3.23) and (4.4.3.38). This is not so for the transverse relaxation time, where the effective relaxation time provides a close approximation to the exact solution (see below). The reason for the failure of the effective relaxation time in this instance is because the lowest eigenvalue decreases exponentially with a, while the equilibrium averages entering into Eq. (4.4.3.32) do not.

Having determined lMi, Ty, and z ^ , we may also calculate the complex dynamic susceptibility from a simple analytic equation as was described in Chapter 2, Section 2.13. We recall that there, it was shown that for a system like that under consideration, the polarisability cc\\(co) may be approximated by a sum of two Lorentzians:

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272 The Langevin Equation

Figure 4.4.3.1 Tj| [Eq. (4.4.3.31); solid line], rf [Eq. (4.4.3.35); dashed line], l/A, [Eq.

(4.4.3.25); filled circles], and l/A?5 [Eq. (4.4.3.29); dotted line] vs. barrier height

parameter a.

Oj|(0) \ + icol\ 1 + ian^ (4.4.3.39)

'w where

A i = -T lTef l V||-1

ef •1/r, (4.4.3.40)

^ r 2 + l/(^Tjf)' ^ !-In the time domain, this behaviour of a\\co) is equivalent to supposing that the longitudinal correlation function C\\(t) (which in general comprises an infinite number of decaying exponentials) may be approximated by two exponentials only:

C | |(0«A1er4' + ( l-A1)<T' / % . (4.4.3.41)

The results of the numerical calculation of the normalised polarisability j||(6>) = Oj|(6>)/:77//2 from the exact continued fractions Eq. (4.4.3.12) and approximate Eqs. (4.4.3.39) and (4.4.3.40) are shown in Fig. 4.4.3.2. The parameters used in the calculation are given in the Table 4.4.3.1. It is apparent that Eq. (4.4.3.39) correctly predicts a^(co) in all frequency ranges of interest. The agreement between the exact continued fraction calculation and the approximate Eq. (4.4.3.39) is very good (the maximum relative deviation between the corresponding curves is less then 5 % in the worse cases which usually appear in the region 0.1< cor < 10). Similar (or even better) agreement exists for all other values of a.

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 213

As shown in Chapter 2, Section 2.13, Eq. (4.4.3.39) also predicts the correct behaviour of CC\\((D) at low (co-tO) and high (<y—>°o) frequencies:

^^-^l-icorn+0(co2), (©-»0) (4.4.3.42) oj[(0)

and

GW„

Fig. 4.4.3.2. X\\^) = kTc^of)l/i1 and z"\(a)) = kTc(l((0)//i2 evaluated from the exact continued fraction solution [Eq. (4.4.3.12); solid lines] for various crand compared with those calculated from the approximate Eq. (4.4.3.39) (filled circles) with numerical values of zj|, tff , and 1/Aj from Table 4.4.3.1 and with the low- and high-frequency asymptotes, Eqs. (4.4.3.42) (dotted lines) and (4.4.3.43) (dashed lines), respectively.

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274 The Langevin Equation

Table 4.4.3.1. Numerical values of l/(rDA,) [Eq. (4.4.3.27)], r n / r D [Eq . (4.4.3.31)],

T,f ITD [Eq. (4.4.3.35)], and ^ lxD [Eq. (4.4.3.40)],

1/( /1,)

-Tn/rD

ref IT

Tw ITD

CT=0.5

1.70298

1.69783

1.64026

0.134541

<7=2

12.2059

12.1006

5.61758

0.089796

a=5

1830.49

1827.12

17.7588

0.03307

<X=10

1.95622-107

1.9555-107

37.9104

0.014039

fljlfrp)

6W - + 0(CD2), (0)->°°)

•ef (4.4.3.43)

These asymptotes are also shown in Fig. 4.4.3.2. The figures demonstrate that the low and high frequency behaviours of the dielectric loss CC\\(G>) are completely determined by zj| and x^ , respectively.

4.4.4 Transverse polarisability and relaxation times

The correlation time and polarisability for the transverse relaxation may also be calculated by the methods described in Section 4.4.3. In order to study the transverse relaxation behaviour, we suppose that a small constant field E applied along the x axis is switched off at t = 0. The transverse behaviour at t > 0 is governed by

g2p+l(t) = sin(2p+ 1)0) . (4.4.4.1)

Thus, for N = 2, we have from Eq. (4.2.24)

TDg1(t) + gl(t) = -a[gl(.t) + g3(t)] (4.4.4.2)

for/? = 0and

*D S2pn (0 + (2p +1)2 82p+l (t) = o(2p +1) [g2p_x (t) - g2p+3 (f)]

(4.4.4.3) forp > 1. The initial conditions are [cf. Eq. (4.4.2.3)]

In J sin(2p + l)0e f fccsM[l + £sin0]</0

£lp(<T)-Ip+l<T) 2K

ocoslO [\ + £sm0]d0 /oW

(4.4.4.4) Similar calculations to those of the longitudinal case in Section 4.4.3 now yield the following expression for the transverse polarisability [9]

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 275

%(ft>) 1

ar[ (0) 1 + ionD + a + aS3 (ico) l + a + aS3 (ico)

(4.4.4.5) . ^ - i y / p (<r ) - / p + 1 (<7)^

£12/;+ 1 /oCo- ) -^^ ) \*[

where the continued fractions Spico) is defined by Eq. (4.4.3.6) and

^ / o ^ W l M . (4.4.4.6) ^ kT 2/0(<r)

Similarly, the transverse correlation time T± is [9]

. / ' M ^ ^ - W ^ ^ , (4.4.4.7)

The difference between Eq. (4.4.4.7) and (4.4.3.21) is striking as the first member of the leading term in Eq. (4.4.3.21) increases exponentially with barrier height a while the leading term in Eq. (4.4.4.7) does not. This behaviour is entirely due to the different symmetries of Eqs. (4.4.3.1) and (4.4.4.3). Another interesting property of Eq. (4.4.4.7) is that it is not possible to write down directly an equation for the transverse correlation time in the limit of large a using the asymptotic values of the lv [Eq.(4.4.3.22)] as was done in writing down Eq. (4.4.3.23). This procedure simply leads to an alternating series for the transverse correlation time in the limit of large a. To circumvent this difficulty we must again utilise the more refined approach [17] based on the integral representation of a product of Bessel functions which in turn allows the replacement of the p dependent sum in Eq. (4.4.4.7) by an integral. This procedure eliminates the difficulties which may be posed by a lack of uniform convergence of Eq. (4.4.4.7) for large p and a Application of the method of steepest descents [17] in the limit of large a then leads to the following formula for the transverse correlation time

r l 1 + — (4.4.4.8)

Aa\ 2a) showing explicitly that the transverse relaxation is not governed by an activation process.

Just as the longitudinal response, the relaxation time Tj_ [Eq. (4.4.4.7)] of the transverse correlation function

C (t)-8i(t)-(sineW$iaeW)o S,(0) (sin20(O))o

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276 The Langevin Equation

Figure 4.4.4.1. The transverse correlation time [Eq. (4.4.4.7); solid line] compared with the inverse of the smallest non-vanishing eigenvalue as yielded by the numerical solution of Eq. (4.4.4.3) (filled circles). The stars represent the solution rendered by the transverse effective relaxation time [Eq. (4.4.4.10)].

may be equivalently presented in integral form as (see Chapter 2, Section 2.10)

T±=-TDe

-2a r e-™s2*erfz2(V2<7cos0W, (4.4.4.9)

4c7[/0(O-)-/1(<7)]J0 V '

where erfi* (z) is the error function of imaginary argument defined by Eq. (2.14.2.7). Numerical calculation shows that Eqs. (4.4.4.7) and (4.4.4.9) yield the same result.

Let us compare the above results with the effective relaxation time. The transverse effective relaxation time is, by using Eqs. (4.3.21) and (4.4.3.34)

r*f IoW-I^cr) (4.4.4.10)

which provides quite a good approximation for all a as is evident from Fig. 4.4.4.1. On comparing Eq. (4.4.4.10) with the results for the longitudinal effective relaxation time [Eq. (4.4.3.35)], we see that

* f* f =(*z>)2- (4.4.4.11)

Therefore T?=(TD)2/r;f. (4.4.4.12)

Thus, for a» 1, we obtain, noting Eq. (4.4.3.38),

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 277

Table 4.4.4.1 Amplitudes for the first three modes of decay of the transverse polarisation as a function of the barrier height parameter a and the corresponding eigenvalues.

a 0 1 2 3 4 5 6 7 8 9 10

X\tD

1.000

2.415

4.757

8.021

12.01

14.41

20.90

24.28

29.53

33.68

37.76

A3TD

9.000

9.518

10.92

13.04

14.74

18.89

22.38

24.09

29.94

33.88

37.85

A5T0

24.00

24.52

27.11

29.86

33.94

39.60

47.06

54.34

67.26

79.50

92.65

a,

0.5000

0.2690

0.1310

0.0684

0.0415

0.0291

0.0227

0.0189

0.0163

0.0143

0.0128

a3 0.0000

0.0076

0.0202

0.0261

0.0260

0.0236

0.0208

0.0182

0.0160

0.0142

0.0128

as

0.000000

0.000026

0.000235

0.000533

0.000636

0.000504

0.000303

0.000158

0.000080

0.000043

0.000025

Tf~TD /(4<r), (4.4.4.13)

Similarly for small values of cr, we obtain

T f « T D ( l - o r ) . (4.4.4.14) It is apparent that the same arguments regarding the asymptotic

behaviour apply to Eqs. (4.4.4.8) and (4.4.4.13) as apply to Eqs. (4.4.3.23) and (4.4.3.24). The l/2crtenn in Eq. (4.4.4.8) ensures that the asymptotic solution for the transverse correlation time agrees to a high degree of accuracy with the exact solution for large o".

The values of the first three eigenvalues and their corresponding amplitudes calculated by numerical solution of the set of differential-recurrence equations (4.4.4.3) are shown in Table 4.4.4.1. These bear out the conclusions of Lauritzen and Zwanzig [2]. The most striking feature noted by them is the domination of the relaxation process for large a by the first and third modes characterised by X\ and A3 which coincide. The inverse of the smallest non-vanishing eigenvalue computed from this set is shown in Fig. 4.4.4.1.

In Table 4.4.4.2 we show the transverse correlation time compared with the effective relaxation time and the inverse of the smallest eigenvalue. It is apparent that the smallest eigenvalue slightly overestimates the transverse correlation time for intermediate a while the effective relaxation time slightly underestimates it. The results of the numerical solution of the set of Eq. (4.4.4.3) for the transverse correlation time using the matrix method (Section 4.4.2) agree completely with the analytic Eq. (4.4.4.7). Thus, the transverse complex polarisability, which is given by Eq. (4.4.4.5), can be approximated to very good accuracy by the Debye spectra

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278 The Langevin Equation

Table 4.4.4.2 The transverse correlation time [Eq. (4.4.4.7] compared with the inverse of the lowest eigenvalue obtained by the numerical solution of the characteristic equation of the set of Eqs. (4.4.4.3). The effective relaxation time is also shown.

<T

1

2

3 4

5

6

8

10

V(v0) 0.414070

0.210214

0.124672

0.083232

0.060921

0.047849

0.033862

0.026483

TIAD 0.405533

0.194062

0.110981

0.075193

0.057018

0.046141

0.033565

0.026423

xflTD

0.382753

0.178013

0.104981

0.073236

0.056310

0.045829

0.033466

0.026378

a l ( f i ) ) _ 1 (4.4.4.15) «1(0) l + ionf '

where the effective relaxation time ze[ is given by Eq. (4.4.4.10).

4.5 Comparison of the Longitudinal Relaxation Time with the Results of the Kramers Theory

It is interesting to compare the asymptotic result of Eq. (4.4.3.29) for the longitudinal correlation time with that yielded by the Kramers transition state theory [10,11] as discussed in Chapter 1, Section 1.13. We suppose that the rotator which is originally caught in a potential well [characterised by a potential V(6) ] at A will escape to a potential well at B crossing the potential barrier of height U at C. According to the Kramers theory, the probability of escape per unit time (escape rate) KAB

of a particle from a single potential well is in the high friction limit, ft » 1 (the high friction limit is consistent with the use of the Smoluchowski equation)

_ (QACOc _-u/(kT)

where

27T/3

co2A = rlV\6A)

(4.5.1)

(4.5.2)

is the squared angular frequency inside the metastable minimum and

\v"(dc)\ (4.5.3) co2r. = r u

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 279

is the squared angular frequency at the transition state, / is the moment of inertia of the rotator about an axis through its centre and

p = gll. (4.5.4)

Here 9A =0, 6C = nil so that we have from Eq. (4.4.1.1)

a)2A=(o2

c=2UII. (4.5.5)

Thus, with Eqs. (4.5.1) and (4.5.4), the prefactor in Eq. (4.5.1) is awL = _U_=2o_ ( 4 5 6 )

27tf3 7tkTTD mD

so that

MB 2a_e.2a

XT, KAB=—e-1C!. (4.5.7)

This result supposes that there is a single escape route from the well and barrier crossing by this route is irreversible. If there are two escape routes from a well, as in the present problem (over two equivalent barriers at Kl 2 and -Kl 2), the escape rate must be doubled, i.e., the escape rate from a single well of the double-well potential, Eq. (4.4.1.1), is K2 = 2/cAB . The corresponding escape rate from well B (6B = K) is 2KBA . Thus, taking account crossings and recrossings, the overall escape time re is then given by

Te = l- = _L = !l, ( 4 5 8 ) 2KAB + KBA) AK 8cr

showing that the result of the Kramers theory [10] is in agreement with the asymptotic solution, Eq. (4.4.3.23). If Visscher's [18] improved calculation of the Kramers escape rate is used in conjunction with the modification for a periodic potential of the present section, we obtain Eq. (4.4.3.24).

As shown by Praestgaard and van Kampen [22], in the application of the Kramers theory to a periodic potential having M minima, the dipole autocorrelation function must, in the high damping and high barrier limit, have the approximate form [cf. their Eq. (38) and the accompanying discussion of the assumptions which lead to that equation] [22]

Cii(t) = e-«Mn-cos(2x/M)]t _ ( 4 5 9 )

For a double-well potential, M = 2 and K2 = 2tcAB , so that Eq. (4.5.9)

becomes [23]

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280 The Langevin Equation

C||(f) = e-4'rt. (4.5.10)

Thus, the longitudinal relaxation time is

' . = 7 - = * . = ^ ( 4 - 5 - H )

4K Sex In this chapter, we have shown how exact solutions [Eqs.

(4.4.3.21) and (4.4.4.7)] may be obtained for the longitudinal and transverse dielectric relaxation times for the two dimensional rotator in a double-well potential. The crucial steps which allow one to represent the solution in closed form are first the representation of the correlation time as the zero frequency limit of the Laplace transforms of the respective autocorrelation functions and secondly the fact that these Laplace transforms satisfy three-term recurrence relations. This allows one to express the longitudinal and transverse dielectric relaxation times in terms of modified Bessel functions of half-integer order. The formula for the longitudinal relaxation time contains the previous result of Lauritzen and Zwanzig [2] as the limiting case of high potential barriers and for low potential barriers it will reduce to that of perturbation theory in a. The result for large a is in agreement with the Kramers transition state theory. An other interesting facet of the problem is that the effective eigenvalue [14] yields a close approximation to the transverse relaxation time for all a values while providing a poor approximation for the longitudinal one. The limitations of that method in relation to activation processes are thus exposed as has been discussed in [13].

We remark that the solution for the longitudinal and transverse components of the complex polarisability given here may be applied with some small changes to yield analytical formulae for the polarisability of assemblies of molecules containing rotating polar groups. The theory of dielectric relaxation of these was originally formulated by Budo [24]. The present method is far simpler than the usual approach [24,25] to the problem based on an eigenfunction expansion of the Sturm-Liouville equation as such an expansion always leads to periodic differential equations of the Hill type [25].

References

1. C. J. Reid, Mol. Phys. 49, 331 (1983). 2. J. I. Lauritzen and R. Zwanzig, Jr., Adv. Mol. Rel. Interact. Proc. 5, 339 (1973). 3. P. Debye, Polar Molecules, Chemical Catalog, New York, 1929. 4. A. H. White, J. Chem. Phys. 7, 58 (1939).

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Chapter 4. Rotational Brownian Motion in N-fold Cosine Potentials 281

5. J. D. Hoffman and H. G. Pfeiffer, /. Chem. Phys. 22, 132 (1954). 6. J. D. Hoffman, J. Chem. Phys. 23, 1331 (1955). 7. S. Roman, An Introduction to Linear Algebra, Saunders, Philadelphia, 1984. 8. W. F. Brown Jr. Phys. Rev. 130, 1677(1963). 9. W. T. Coffey, Yu. P. Kalmykov, E. S. Massawe, and J. T. Waldron, J. Chem. Phys.

99,4011 (1993) 10. H. A. Kramers, Physica 7, 284 (1940). 11. P. Hanggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251 (1990). 12. G. N. Watson, Theory of Bessel Functions, 2nd Edition, Cambridge University

Press, Cambridge, 1944. 13. M. San Miguel, L. Pesquera, M. A. Rodrigues, and A. Hernandez-Machado, Phys.

Rev. A 35, 208 (1987). 14. W. T. Coffey, Yu. P. Kalmykov and E. S. Massawe, The Effective Eigenvalue

Method and its Application to Stochastic Problems in Conjunction with the Nonlinear Langevin Equation. In: Modern Nonlinear Optics, Adv. Chem. Phys., 85, Part 2, Eds. M. W. Evans and S. Kielich, Wiley, New York, 1993.

15. M. R. Spiegel, Laplace Transforms, Schaum Publishing Co., New York, 1964. 16. M. Abramowitz and I. Stegun, Eds., Handbook of Mathematical Functions, Dover,

New York, 1964. 17. W. T. Coffey, D. S. F. Crothers, and J. T. Waldron, Physica A 203, 600 (1994). 18. P. B. Visscher, Phys. Rev. B 13, 3272 (1976). 19. A. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, More

Special Functions, vol. 3, Gordon and Breach, New York, 1990. 20. Yu. P. Kalmykov, Phys. Rev. E, 61, 6320 (2000) 21. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). 22. E. Praestgaard and N. G .van Kampen, Mol. Phys. 43, 33 (1981). 23. F. Marchesoni and J. K. Vij, Z. Phys. B 58, 187 (1985). 24. A. Budo, J. Chem. Phys. 17, 686 (1949). 25. W. T. Coffey, M. W. Evans, and P. Grigolini, Molecular Diffusion and Spectra,

Wiley, New York, 1984. (Also available in Russian, Mir, Moscow, 1987).

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Chapter 5

Brownian Motion in a Tilted Cosine Potential: Application to the Josephson Tunnelling

Junction

5.1 Introduction

The problem of the Brownian motion of a particle in a tilted periodic potential arises in a number of physical applications, for example, current-voltage characteristics of the Josephson junction [1], mobility of superionic conductors [2], a laser with injected signal [3], phase-locking techniques in radio engineering [4], dielectric relaxation of molecular crystals [5], etc. (The tilted periodic potential model currently merits attention in view of the intense interest in the effect of noise in the operation of nonlinear systems, e.g., stochastic resonance [6], and of the ever increasing areas of application of the model, e.g., to the ring-laser gyroscope [7].) A comprehensive discussion of the model is given in [8]. A concise method of numerical treatment of the model (in terms of infinite continued fractions) with a particular application to a ring-laser gyroscope has been suggested by Cresser et al. [9], and has been summarised by Risken [8]. Further development of the continued fraction approach has been given by Coffey et al. [10-13].

The main aim of the present chapter is to apply the Langevin equation method to the problem of the noninertial Brownian motion in a tilted cosine potential and to demonstrate the efficiency of the method by computing, as a particular example, the linear impedance of a point Josephson junction in the presence of noise. Hitherto, this problem has been treated in the context of the Fokker-Planck equation, e.g., in the monographs [1,14] (we refer the interested reader to these monographs where additional details and a comprehensive bibliography will be found). A related problem (although easier than the junction one) is the

282

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 283

dielectric relaxation of dipolar molecules in a co&Nd potential. This has already been considered in Chapter 4. We note that setting the tilt equal to zero in a tilted cosine potential yields the longitudinal dielectric relaxation of an assembly of single axis rotators in a cosine potential, in other words, the relaxation in a constant electric field E0 applied in the z-direction. We shall consider the Josephson junction problem following mainly the exposition of Coffey etal. [11].

5.2 Josephson Junction: Dynamic Model

The Josephson tunnelling junction is made up of two superconductors separated from each other by a thin layer of oxide [1,14]. The phase difference </) = $ - (f>r between the wave functions for the left and right superconductors is given by the Josephson equation [1]

at n (5.2.1)

Here V(t) is the potential difference across the junction; e is the charge of the electron; and h = hl27t, where h is Planck's constant. If the junction is small enough, it may be modelled (see Fig. 5.2.1) [1,8] by a resistance R in parallel with a capacitance C across which is connected a current generator Idc (representing the bias current applied to the junction). At the other end of the junction (across the resistance R) is connected a phase-dependent current generator, Ismtf), representing the Josephson supercurrent due to the Cooper pairs tunnelling through the junction. Since the junction operates at a temperature above absolute zero, there will be a white noise current Lt) superimposed on the bias current satisfying the conditions

/sin^w

Figure 5.2.1. Equivalent circuit of the Josephson junction.

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284 The Langevin Equation

L(t^)L(t2) = (2kT/R)S(tl-t2), (5.2.2)

Z(0 = 0. (5.2.3) The current-balance equation for the junction is [8]

,dV(f) , V(t) Idc + Lt) = C—— + - ^ - + 7 sin 0(f). (5.2.4)

dt R Substitution of Eq. (5.2.1) in Eq. (5.2.4) yields

^ ^ T m + ^ r m + Ism<t>(t) = Idc+L(t). (5.2.5) 2e dt 2eR dt

We may cast Eq. (5.2.5) in the form of a damped pendulum equation driven by a constant torque F0 on which is superimposed a white noise torque X [8]:

d2 , _ d

where

J-jM) + g—</>(t) + U0 sin0(0 = F0 + A(t), (5.2.6) dt dt

-IfA " 2ej ? i?U«^

2e 2e 2e We shall for the moment ignore the inertial effects. Thus, we set 7 = 0 (this assumption corresponds to ignoring the capacitance of the junction). Equation (5.2.6) may now be written as

g<j>(t) + U0sm</>(t)-F0 = A(t). (5.2.7) Equation (5.2.7) describes the noninertial dynamics of a Brownian particle moving in the tilted cosine potential [1]. In Eq. (5.2.7), by definition

U0sm</>(t)-F0=^-. (5.2.8) dip

Thus the potential energy of the mechanical system is U(<t>) = -U0cos<f>-F0</>. (5.2.9)

By analogy, the "potential energy" of the Josephson junction is given by

U(</>) = -^-(ldc</> + Icos</>). (5.2.10) 2e

A more complete discussion of Eq. (5.2.5) is given in Refs. [1] and [8], and in the papers quoted therein. The quantities of physical interest are the mean value of the voltage (V), the Josephson radiation spectrum and the junction impedance to an external high-frequency current.

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 285

5.3 Reduction of the Averaged Langevin Equation for the Junction to a Set of Differential-Recurrence Relations

In order to proceed, we change the variable in Eq. (5.2.5) by writing

rn=e~in*, (n = 0,±l,±2.. .) . (5.3.1)

Consequently, d , i d „ —<p = r dt nrn dt

(5.3.2)

and

dt' -0 = .

nr dt •rn+n2rn

dt (5.3.3)

Substituting Eqs. (5.3.2) and (5.3.3) into Eq. (5.2.5) and noting that

we have

sin^ =

1 d

r -r

dt 2r

n(t) + j-^-r"(t) + n2C <j>2(t)rn(t) 2 Rdt

= ^[r»-\t)-r»+\t)y^r»(t)[ldc+L(t)]. n L J n

On averaging Eq. (5.3.4) as described in Chapter 4, Section 4.2, we obtain an equation for the sharp values r", namely:

1 d 2..n ,-/--+ rn+nlC 0zr' +2 Rdt

nel_f „_! r " + 1 ) -

2ienl, dc jn (5.3.5) dtL Rdt ' h v ' h

We shall consider Eq. (5.3.5) in the diffusion (or noninertial) limit, where we can neglect the capacitance term. This assumption restricts the range of frequencies in which the model is applicable, viz.,

co«cop, where 0)p = slllelhC ,

is the Josephson plasma frequency [1]. This approximation can be justified if C is small enough and has been discussed elsewhere [1,14]. It comprises the so-called resistively-shunted junction (RSJ) model [1].

In the noninertial limit, the phase angle </> and its time derivative (j) are decoupled from each other as far as the time behaviour of the phase angle is concerned (this is achieved by supposing that (j) reaches the stationary state much faster than <j>). Here, one may note that according to the equipartition theorem (Chapter 1, Section 1.5.5)

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286 The Langevin Equation

2 U e so that

C m V = f. (53.6,

c\h (5.3.7)

Thus, using Eq. (5.3.7) and setting C - 0 in Eq. (5.3.5), we have

T0-rn + (n2 + inxy)rn =^(rn-1 -rn+l), (5.3.8)

dt v ' 2 v ' where

x = IdcII (5.3.9) is the ratio of bias current amplitude to supercurrent amplitude (bias or tilt parameter),

y=hI/(2ekT) (5.3.10) is the ratio of Josephson coupling energy to thermal energy (barrier height parameter), and

*b = fh\2 l

(5.3.11) 2e) kTR

is the characteristic relaxation time. We have evaluated the average of Eq. (5.3.4) by noting that

rn(t)L(t) will vanish throughout because, in the inertial Langevin equation, r" (t) and L(t) are statistically independent. However, as shown in Chapter 2, Section 2.4, this is not true for the noninertial Langevin equation

T0^rn(t) = ^\r"-\t)-rn+\t)]-l^rn(t)[ldc+L(t)], (5.3.12) at 2 L J /

which can be obtained from Eq. (5.2.5) by putting C = 0 before the transformation of variable ^—> rfl. In Eq. (5.3.12), the multiplicative noise term rn(t)L(t) contributes a noise induced drift term to the average. Indeed, Eq. (5.3.12) is a particular case of the general stochastic nonlinear equation with the white noise T(t):

£(t) = h[frt),t] + g[£(t),t]r(t), (5.3.13) which becomes on averaging (Chapter 2, Section 2.3)

= h(X,t) + Dg(X,t)^-g(X,t), (5.3.14) X=\im-[g(t + T)-X]

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 287

where ^(t + f) is a solution of Eq. (5.3.13) which at time t has the sharp value %t) = X. We may use the above results to evaluate the average of the multiplicative noise term in Eq. (5.3.12). We have (cf. Chapter 2, Section 2.4)

X = r\ g(X) = -l-^X, (5.3.15) IT0

so that

i.e.,

kT d n n „ — " £ ^ £ = * = r < (5.3.16)

,Mr') + (nUinXy)r')--"M^)-^)). (5.3,8)

(iny/I)rn(t)L(t) = -n2rn. (5.3.17) Hence Eq. (5.3.12) also yields Eq. (5.3.8) on averaging .

We note that the sharp values r" in Eq. (5.3.8) are themselves random variables with probability density function W(<p, t). Thus, on averaging Eq. (5.3.8) over W(j>, t), we obtain the system of moment equations:

^ . + („W)<r-) = f ' We remark that Eq. (5.3.18) can also be obtained from the underlying noninertial Fokker-Planck (Smoluchowski) equation for the probability distribution function W of the phase angle ^(see Chapter 2, Section 2.4).

5.4 DC Current-Voltage Characteristics

Ambegaokar and Halperin [15] computed the dc current-voltage characteristic (when the capacitance is neglected) by solving the time independent Smoluchowski Eq. (2.4.11) (Chapter 2, Section 2.4). They found (having noted that the problem is formally the same as that of the translational Brownian motion of a particle in a tilted cosine potential, the assumption of zero capacitance corresponding to ignoring inertial effects in the mechanical analogue) that

12x 2x j if IK In i if

where

and

(5.4.1)

(v)o=(V)o/IR ( 5 A 2 )

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288 The Langevin Equation

f<f>) = e-uW(kT) = er(xt+«x*) (5.4.3)

(see for details Ref. [8], p.289, and for a further discussion of mechanical analogues see Ref. [1]). They plotted these results evaluating numerically the integrals in Eq. (5.4.1) and also gave simple approximations for certain limiting cases, i.e., (a) y-> 0: (v)0 -> x, i.e., (v)Q = RIdc (Ohm's Law).

(b) y—> oo and x > 1: (v)0 = 4x2 - 1 (Ohm's Law with a correction factor).

(c) y—> oo and x < 1: (v) = 0 (pure superconduction).

(d) y» 1 and*< 1: (v)0 = 2sinh(^x)(l-x2)1/2e-2^(1-jr2)"2+xarcsinj:] .

(e)jc->0: \\m(v)Jx = \lll(y).

There are two major problems associated with the method of Ambegaokar and Halperin [15] :

(i) There is no simple analytic form for the /-V curves (ii) There is no obvious way of extending their results when the

input current contains a time dependent part. These difficulties may be overcome by using a continued fraction

method [8]. We may implement this method by recalling that, in the stationary state, we shall have from Eq. (5.3.18),

(n + )( ' - '1)o=^((rn-1)o-( '- '1 + 1)o), (5-4.4)

where the symbol ( )0 means averaging over the stationary distribution function W0((/>) which is [8]

Wo(0) = Coe kT l - ( l -e- 2^)Je kT d<j)' \e kT d<f>'

o / o (5.4.5)

Here, C0 is the normalising constant defined so that 2x \W0WW = 1. (5.4.6) o

We note that the function W0((Z>) is periodic so that it satisfies the condition W0(^) = Wo(0+2ft) [8]. Following the calculation of Risken [8], we now introduce the quantity

Thus, on dividing by (rn ' ) 0 , Eq. (5.4.4) becomes

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 289

n + iyx)Sn=yl-SnSn+l)/2 (5.4.8)

or 1/2

Sn= — . (5.4.9) n/y+ix + Sn+l/2

For n = 1, we obtain the known result [8]

S , . — • £ - _ . (5.4..0)

y + ix + 2y +ix +

3y ' + ix +. The continued fraction (5.4.9) can be expressed in terms of the

modified Bessel functions Iv(z) of the first kind of order v, where vmay be a complex number [16]. These functions satisfy the recurrence relation Eq. (4.4.3.14), which may be written as a continued fraction:

- ^ U Ml . (5.4.11) /„_,(z) v/z + Iv+l(z)/[2Iv(z)]

By inspection, the continued fraction Sn given by Eq. (5.4.9) is identical

to Eq. (5.4.11) if v = n + ixy and z = y. Hence, the Sn may be expressed

in terms of Iv(z) as follows

Sn = In+ixrir) . (5.4.12) ' n - l + i t / V / '

In order to calculate the current-voltage characteristics using Eq. (5.4.10) or (5.4.12), we first note that if we ignore capacitance effects, the current balance Eq. (5.2.4) becomes:

/TV(t) +1 sin </>(t) = 1dc+L(t). (5.4.13) Taking the averages at the stationary state and noting Eq. (5.2.3), we then obtain in the dimensionless variables

<v>0 = x-<sin^>0. (5.4.14) Equation (5.4.14) determines the dc current-voltage characteristics. We may find (sin0)o merely by extracting the imaginary part of SX from Eq. (5.4.12) for n= 1 since

5, =<e-*)0 = <cos0>o-i<sin0)o. (5.4.15) Thus, we have [cf. Eq. (5.4.1)]

v)0= x + lm[Sx = x + lm[ll+ixYy)l Iixry)]. (5.4.16)

Noting the mathematical identity

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290 The Langevin Equation

Figure 5.4.1. Current-voltage [Eq. (5.4.16)] and differential resistance-current [Eq. (5.4.18)] characteristics of a Josephson junction in the presence of noise. Curves 1-6 correspond to values of y= 0 (curves 1), 1 (2), 2 (3), 5 (4), 25 (5), and °° (6), respectively.

d

dju

l\+fi (z))

l,iz) ) zlMU)0

jl^Oh^i^dt, (5.4.17)

we can also obtain from Eqs. (5.4.16) an analytic equation for the

normalised differential resistance of the junction, R~ld (V )0 ldldc:

R-iJ-ly) = -*-(v) =1-Re AT \ ' 0 Av > ' 0 dl. dc dx 'ixAT) 0

l^0Il+ixr(f)dt (5.4.18)

[The definite integral in Eq. (5.4.18) can also be expressed in terms of the continued fractions Sn, see Section 5.7]. The results [Eqs. (5.4.16) and (5.4.18)] shown in Fig. 5.4.1 appear to be in complete agreement with those of Ambegaokar and Halperin [15] (see also [23,24], where a comparison with experiment is given). We remark that the built-in function Bessell[/7,z] of the MATHEMATICA program allows one readily to calculate the modified Bessel function of the first kind In(z) for complex n. However, the use of continued fractions S„, Eq. (5.4.9), can be much more effective in numerical calculations, especially, at very large y.

5.5 Linear Response to an Applied Alternating Current

In order to evaluate the linear impedance of the junction, one has to evaluate the linear response to a small alternating current (ac) [1,14]. Thus, we suppose that the current across the junction is now Jdc + Im e~,a*. We also suppose that the ac is small, ylm 11« 1, so that we can make a perturbation expansion in Eq. (5.3.18), namely:

(rn) = (rn)0+An(0))Ime-'m/I + ... (5.5.1)

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 291

On substituting Eq. (5.5.1) into Eq. (5.3.18) and replacing Idc with he + lm e~>a*, we obtain

^(An_l(co)-An+l(co))-(n + iyx-icoT0/n)An(co) = iy(rn)0 (5.5.2)

with ^(co) = 0. For n > 1, Eq. (5.5.2) may be rewritten as

q;An_l(co) + qn+iCOT0/n)An(CO) + q+An+l(co) = i(rn)0, (5.5.3)

where

q~n=\/2, q+n=-V2, qn=-ix-nly. (5.5.4)

As one can see, Eq. (5.5.3) has the same mathematical form as Eq. (2.7.3.22) of Chapter 2, Section 2.7.3, which has an exact analytic solution in terms of ordinary continued fractions, namely, Eq. (2.7.3.23). Thus, we may formally use this solution here. We have

A (co) = -2iSx (co) L)0 + I <r" )0 f l qUSk_C0) 1 [ n=2 k=2 <ik J

oo n

= 2iZ(-l)n(rn)0HSk(a», n=l k=l

where the infinite scalar continued fraction Sn (co) is given by

<fn

(5.5.5)

SA<*» = —

n " _kot±_ _ _ 4 i + 2 _ M + l ICOTQ _

n + 2 or

1/2 Sn(co) =

iCOT0 n 1/4 ix—y-+-+-yn y ix__to*6_ + H±± + MA

r(n+i) r ix imo in+11 1/4

y(n + 2) y ix-... (5.5.6)

Here, we have noted that

n^v^c-i)" \n+l h-i'Hk =y~i

k=2

One can readily show that Sn(0) = Sn, where the continued fraction S„ is defined by Eq. (5.4.9). On noting also Eq. (5.4.7), which may be rearranged as

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292 The Langevin Equation

(rn)0 = S„(rn-\ = SnSn_l...Sl (5.5.7)

(since (r°)0 = 1), we have oo n

~\o» = 2i'X (-1)" f l W * » • (5-5-8) n=\ k=\

In like manner, the (r~l) response defined as

(r'1) = r-\ + A_x6»Im exp(-ifl»)/7 + ... (5.5.9) may also be calculated from Eq. (5.5.8) as

A_l(co) = -A*(-co) (5.5.10)

[for n < 0, the complex conjugate Eq. (5.5.2) for -^(-co) coincides with Eq. (5.5.3)]. Here the asterisk * means the complex conjugate. Equations (5.5.8) and (5.5.10) allow us to calculate the linear ac response

( s i n ^ =(sm0-(sm0)o) = Ds(a))(lm/l)e-im, (5.5.11)

where

DS=^[AI(Q)) + A; ( - « ) ] . (5.5.12)

In particular, one may use Eq. (5.5.12) to evaluate the impedance Z(co) of the junction. In order to accomplish this we recall that the averaged current-balance equation in the presence of the ac is

Idc + Ime-im-I(sm0)-(V)/R = O. (5.5.13) We have supposed that

(sin^) = (sin^)0 + (sin^)1 and V) = (V)0 + (V)l, where the subscript "0" on the angular braces denotes the average in the absence of the ac and the subscript" 1" the portion of the average which is linear in Im. Thus, on dividing across the current-balance Eq. (5.5.13) by the supercurrent amplitude I, noting that V/IR = v , so that

x + ( / m / / ) e - ' ' ( a r - ( s in^> 0 - ( s in^) 1 -«v) 0 +<v) 1 ) -0 (5.5.14)

and recalling Eq. (5.4.14), we have

v)x=ImII)e-m-smfiv (5.5.15)

On resubstituting v = V/IR and using Eq. (5.5.11), we obtain

V)x=Za»Ime-iw,

where the linear impedance of the junction Z(*y) is given by Z(co) = Rm-iX0 = R[l-Ds(0)). (5.5.16)

Here Ra and X^ are the dynamic resistance and the reactance, respectively.

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 293

5.6 Effective Eigenvalues for the Josephson Junction

Let us suppose that a dc current ldc had been applied to the junction in the infinite past, and that at t = 0 the current Idc is incremented by a small current U (t) A/ so that the total current is / = Idc + U(t) A/, where Ut) is the unit step function and yA« 1. Now, we are only interested in the response linear in A. We therefore assume that

rn)t) = rn)eq+rn\(t) (5.6.1)

where the subscript "1" denotes the portion of the statistical average which is linear in A, and the subscript "eq" denotes the statistical average in the stationary state evaluated using the stationary distribution function W0 [Eq. (5.4.5)], where now

U (0)/(kT) = -y [cos 0 + (x + A)0]. (5.6.2)

As t —> °°, we have

lim<rB>(0 = <rB>M so that lim<i-">i(0 = 0.

On substituting Eq. (5.6.1) into Eq. (5.3.18), we obtain

[n + ir(x + A)](rn)eq+^((r^)ecl-(r"-i)eii) = 0 (5.6.3)

and nY i

(5.6.4) Equation (5.6.4) represents a three-term recurrence relation

driven by a forcing function, namely the U(t) term. In order to determine the effective eigenvalues, we shall consider its unforced part and reduce it to an eigenvalue problem, viz.,

4<r>i(0 + <r>,(0 = 0. (5.6.5) at

Here <r)1(0 = ( ^ > 1 ( 0 (5.6.6)

is a complex variable so that the effective eigenvalue X^ is also

complex, namely

A+f=A = A' + iA". (5.6.7)

The real part of the effective eigenvalue when inverted will give the effective relaxation time while the imaginary part will give the frequency of oscillation.

:(r")1+(n2 + m ^ ) < ' " ' , ) 1 = ^ ( < ^ " 1 ) i - ( ^ + 1 ) , ) - ^ ( ' - n ) e 9 A f / ( 0 .

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294 The Langevin Equation

Equation (5.6.5) yields

Kf=-(r)i(t)

(r\(t) t=0

On substituting Eq. (5.6.4) for n = 1 into Eq. (5.6.9), we obtain

1+ - l

Af - — \ + ixy+ r(r

2),(Q)^ 2 (r),(0)

Now, we have from Eq. (5.6.1)

(rn)xQ» = rn)0-rn)eq

so that on using Eqs. (5.6.10), Eq. (5.6.9) further simplifies to

^=J-(i+frr)+-2: 2rn

(r2)eq-(r\

(r)eq-(r)o

(5.6.8)

(5.6.9)

(5.6.10)

(5.6.11)

The averages (r)eq and (r ) in Eq. (5.6.11) are over the stationary

distribution W0(</>) [Eq. (5.4.5)] with the perturbed potential [Eq. (5.6.2)]. However, remembering that we are interested only in the linear response to A, we can express (rn) in a power series as

(rn)eq~(r")0+A—(r">0 (5.6.12)

r) r) The evaluation of — (r)0 and — ( r )0 may be accomplished as follows.

dA dA On substituting Eq. (5.6.12) into Eq. (5.6.3), we have

n + iy(x + A)) <rn)0 + A—(r")0 \ dA

+ l dA

~i(r"-\^4r(rn-\ = 0. (5.6.13) 2 ^ ' . '" dA

Thus, the linear approximation in A is given by the following set of equations:

f- + kj<r")0+i((r"+ 1)0-<r ' ' -1>0) = 0, (5.6.14)

< r " > ° + K^ ( r ' ! + 1 ) o "^ ( r " 1 ) o ) = ~ / < r ' ! ) o ' (5-6'i5) - + ix dA

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 295

As we have seen in Section 5.4, the solution of Eq. (5.6.14) is given in terms of continued fractions S„ by Eq. (5.4.7) or, equivalently by Eq. (5.5.7), viz.,

<Oo=n** (5.6.16) k=\

Now, on noting that Eq. (5.6.15) for drn)QldA has the same

mathematical form as Eq. (5.5.2) for ^ ( 0 ) , we have the exact solution [cf. Eq. (5.5.8)]

d dA <r>0=-i-2Z (-ir+1rK- (5.6.17)

n=l k=\

The quantity e?(r 2) 0I dA can be obtained from Eq. (5.6.15) at n = 1:

-(r2)Q=-i2(r)0-2(ix + l/y)—(r)0. dA" ' u N ' u v "dA

The substitution of Eqs. (5.6.17) and (5.6.18) into Eq. (5.6.11) yields

(5.6.18)

+ 1 ixy y T0 T 0 r0

i(r)o + ix + l/y) — (r)Q dA (r)0

= rs1 2tn

-i-\ (5.6.19)

K-irm 2 • _n=l k=\

Thus, we have expressed the effective eigenvalue A ^ in terms of the

continued fractions Sn only. Hence, on using Eq. (5.4.12), we can obtain

A*f from Eq. (5.6.19) in terms of modified Bessel functions:

x+f=x+a"=- -. (5.6.20)

2^o Z ( - 1 ) n / « + i + a r ^ ) 2 r 0 J / k y ( 0 / i + ^ ( 0 ^ «=o 0

Here, we have used the equality ([17], p.46)

fJ(-l)nI2

n+v+l(z) = jlv(t)Iv+i(t)dt. (5.6.21) n=0 0

The effective eigenvalue X~f for ( r _ 1 ) is related to X^f by

A;f=(%f)*=A'-ir. (5.6.22)

Equation (5.6.20) when evaluated at x = 0, that is at zero tilt, shows that the effective eigenvalue is given by the exact equation:

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296 The Langevin Equation

"4

100^

lOi

1-!

0.1-

0 01-

l - x = 0.1 2 - * = 0.5 3 -x=1 .0

4 -x=1 .5 5 -x = 2.5 ^ ^ ^

%^\^ ^^>^^

^^^\J^^^

^ \ — _ 2

^ 1

0.1 10 100

Figure 5.6.1. The real and imaginary parts of the effective eigenvalue vs. y.

-,+ _ , j - _ rh(r)h(r) '-o'V - 'o'V W)-i

(5.6.23)

because [17]

2j/0(f)/1(0df = /o2(z)-l.

The behaviour of the real and imaginary parts of the effective eigenvalue as a function of the barrier height ^and bias parameters x is illustrated in Figs. 5.6.1 and 5.6.2.

The representation of X in terms of the modified Bessel functions allows us to derive simple analytical equations in a number of limiting cases. In particular on substituting the Taylor expansions (z « 1) [16]

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Chapter 5: Brownian Motion in a Tilted Cosine Potential 297

IUU-

10-

< u°

1^

^ 5

^ 4 \

—"—-v? \ \

— ~ ^ ^ 2 \ \ \ \

1 ^~^~

\-r=\ 2-y=5 3 - ^=10 4 - r=20 5 - y=50

1 •

100-

10:

^ 1 -

0.1-

5 _ _ _ _

/^^\^~ /s^^S-~--~~

1- y=\ 2-y=5 3- r= io 4 - y=20 5 - r=50

Figure 5.6.2. The real and imaginary parts of the effective eigenvalue vs. x.

Iv(z)= ir (z/2) 2k

Iv(z)Iuz)- rz\v+".

2)

\2k (z/2yKr(ju+v+2k + i)

l^k\r(y + k + Y)r(ji + k + l)r(ji + v + k + l) into Eqs. (5.4.12) and (5.6.20), we have

=_rn (r/2) +0((r/2f), (5.6.24) l + ixy (l + ixy)(2 + ixy) v '

and

(A' + iA")t0 =1 + 1x7+ 3+2ixy (yy

(2 + ixy)2V2 + 0((y/2)4) . (5.6.25)

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298 The Langevin Equation

Equations (5.6.24) and (5.6.25) constitute simple analytic formulae for the quantities of interest in the limit of low (y« 1) potential barriers.

5.7 Linear Impedance Using the Effective Eigenvalues

Having determined the effective eigenvalues, we may calculate the impedance of the junction as follows. We shall suppose as in Section 5.5

that 7, = Idc + Ime~"°'. Since we are only interested in the response linear in the ac, we can again assume that

(rn) = (rn)0 + (rn\, (5.7.1) where the subscript "0" denotes the statistical averages in the absence of the ac and the subscript" 1" the portion of the statistical average which is linear in the ac. In Eq. (5.7.1), we also have

lim(r") = (rn)0 and lim(r'!)1 =0.

On substituting Eq. (5.7.1) in Eq. (5.3.18), we obtain

(n + ixr)(rn)0+^((rn+\-(r"-\) = 0 (5.7.2)

and

r 0 y < A +(n2+iru7)(r

n)l =^((r"-1>1 - < r " + ^ ) - ^ / m < f ' « V > 0 .(5.7.3) at v z /

On setting n = -1 and n = 1, respectively, in Eq. (5.7.3), we obtain the leading members of the hierarchy of differential-recurrence relations:

HW\+(\-ixY)r-\=U\-r-\y^r-\lme-im, (5.7.4) at I l

l A ^ + a + fcrKr), =^-r\)-^r)0Ime-iM. (5.7.5) at Ls l

Using Eq. (5.6.10), Eqs. (5.7.4) and (5.7.5) reduce to uncoupled ordinary differential equations of the first order:

7 ( ^ > 1 + ( ^ 1 ) i 4 < r " ' > o ' / t o (5-7.6) at I

A < r > + ^ < r > =_£2:< r > /«,-««. (5.7.7) at I

The steady state solutions of Eqs. (5.7.6) and (5.7.7) are

(r-i) = lJ^l Ls.e-i«* , (r\ = ^ !-^e-im . (5.7.8) T0Uef-io)) I T0(A+f-ia>) I

Here we have used the fact that (r~I)0=('")o=^. According to Eq. (5.5.11)

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 299

(sin<z>-(sin<z>)0> =

so that we obtain

7

2i = Ds(a»-*-e-

Ds(0)) = 2Tn

- + -

(5.7.9)

(5.7.10) A'-i(o) + A") A'-ico-A")

Thus the impedance of the junction Zco) [Eq. (5.5.16)] is given in terms of continued fractions by

2zn - + - (5.7.11)

x'-i(a+r) x'-ico-r) where A' and A" are determined from Eq. (5.6.19). On noting Eq. (5.4.12), we can also express the impedance of the junction Zco) in terms of the modified Bessel functions Iv (z), namely

R = 1 — ' l+ixy (7)

2rn / , -+-'l-ixy (7)

(5.7.12) <Ur(y)[A'-i(co-A')] I_ixrirW-K(a+A')]\

where A' and A" are determined by Eq. (5.6.20). Noting Eqs. (5.6.19) and (5.6.20), we can show that at 0)= 0, Eq. (5.7.12) reduces to the exact Eq. (5.4.18) for the differential resistance of the junction d(V)o/dIdc and, in turn, is expressed in terms of continued fractions S„, namely,

2 £_ yVM\+ixy\

Z(0) d M . Re It

:JV y(t)dt = l -2Re Z(-Dn+in5i n=\ k=\

l\xy\7) o

We now compare the impedance Za>) from Eqs. (5.7.12) and (5.6.20) with the exact solution [Eqs. (5.5.8), (5.5.12), and (5.5.16)]. The results of the calculations are shown in Figs. 5.7.1 and 5.7.2. It is apparent, by inspection, that the effective eigenvalue method gives perfect correspon-dence to the exact solution for a wide range of the parameters x and y. Thus, it allows one to represent the impedance of the junction Zcd) by the simple analytic formula of Eq. (5.7.12) which describes a resonance with natural angular frequency A" [18]. Approximate equations for the impedance of a Josephson junction have also been derived in Ref. [19]. However, these equations have a narrower range of applicability than ours because they are confined to special cases. For example, Eq. (3.11) of Ref. [19] is valid for large co, i.e., for (Ma » % and Eq. (3.14) of Ref. [19] is valid for small fluctuations only. It is of interest to compare the results obtained in this chapter with those for the noiseless case. In the absence of noise, Vystavkin et al. [19] and Auracher et al. [20] calculated the impedance analytically by finding the

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300 The Langevin Equation

differential impedance at the bias point (see also Ref. [14]). Their results are as follows: for values of the bias parameter less than the critical current (x < 1), Auracher et al. [20] obtained the normalised impedance

^ > = — ^ ( n - i V T ? ) , (5.7.13)

where Q. = d)T0 ly ; for x > 1, Vystavkin et al. [19] obtained

Z(co)

R = 1 + -

,x2 - 1 - + •

lit -S(a-Jx2-\). (x + \lx2-l)(x2-l-£l2) 2(x + Jx2-l)

(5.7.14) The above equations have a simple physical interpretation. According to Auracher et al. [20] if x < 1, the junction behaves like an inductance

S

4

3

2

1

x= 1.0

X \-r= 0.5 A 2 - y= 5.0

\ 3 - y= 20.0 \ 4-7= 50.0

\ * " / \ »7

0.1 10 100 corn

F7 1

. x=1.0

1 - r= 0.5 2-y= 5.0 3 - y= 20.0 4 - >•= 50.0

1 \ .

/ *

*\3

r T i

*\4

0.1 ftW„ 10 100

Figure 5.7.1. Comparison of the exact (solid lines) and approximate (stars) solutions for the real and imaginary parts of the normalised (R = 1) impedance vs. CO^ for x = 1.0 and various values of y.

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 301

0.5

0.0 0.1

0.6

0.4

0.2

R1 o.o If

" -0.2

-0.4--0.6

0.1

T f r r : '

/ 1 / >

l - x = 0.10 2 - x = 0.75 3-x= 1.00 4-x=1.50

10 COTn

10

100

wtti***TT*^"~—-•

\

-

\

/ j \

^ A ^ T v i V/

r=5.o l - x = 0.10 2-x = 0.75 3-*= 1.00 4-x=1.50

.^~«!SWgS •^\f-^"^

100 «T„

Figure 5.7.2. Comparison of the exact (solid lines) and approximate (.stars) solutions for the real and imaginary parts of the normalised impedance vs. an^ for y= 5 and various values of x.

h

2el (i-s) -1/2

in parallel with the resistance R, yielding an admittance

Y(0)) = R'l+i(0)L)~l,

which yields the impedance Z(co)=Y~\o)) from Eq. (5.7.13) [20]. If x> 1,

the impedance has a singularity at xs =vl + H2 . This singularity vanishes in the presence of noise. Such behaviour is evident in Fig. 5.7.3, where we have plotted the normalised impedance Zl R as a function of the bias parameter x and have compared it with the noiseless case. It is also apparent from this figure that for weak noise (large f), Eqs. (5.7.13) and (5.7.14) yield a satisfactory description of the impedance excluding the

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302 The Langevin Equation

Figure 5.7.3. Comparison of the exact (solid line), approximate [Eq. (5.7.12); stars], and noiseless [Eqs. (5.7.13) and (5.7.14); dashed line] solutions for the real and imaginary parts of the normalised impedance vs. x for y = 25 and an§ = 25.

region in the vicinity of the singular point xs. Moreover, close to the singular point xs, the noiseless solution (contrary to the noise case) possesses a negative real part which is an indication of the amplification or oscillation, which may occur if the junction is inserted in an appropriate microwave circuit [21].

5.8 Spectrum of the Josephson Radiation

Another quantity of interest is the spectrum of the Josephson radiation R(co). The exact calculation of Rco) in terms of infinite continued fractions may be accomplished just as the calculation of the junction impedance. The spectrum of the Josephson radiation [22] which is defined as

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 303

\C(t)emdt (5.8.1) R(a>) = 2RQ Re

where R^llln (5.8.2)

is the normalising constant, C(0 = (sin <£(()) sin <z>(0)0 (5.8.3)

is the stationary autocorrelation function. In Eq. (5.8.3), the symbol ( )0

means the ensemble average over the stationary distribution function W0

defined by Eq. (5.4.5). On multiplying both sides of Eq. (5.3.10) by i sin 0(0) and averaging over the stationary distribution function W0, we obtain differential-recurrence relations for the stationary averages

^ 0 T f n ( 0 + ( « 2 + ^ ) r „ ( O = ! 7 [ ^ 1 ( 0 - ^ + 1 W ] , (5.8.4) at v ' 2

where

Vsn(t) = i(sin0(O)rn(t))o. (5.8.5)

Moreover, C(t) in Eq. (5.8.3) is related to y/xt) by

C(0 = Re^(0 = k i ( 0 - ^ _ i ( 0 ] / 2 . (5.8.6)

Let us now introduce instead of y/n (f) m e t r u e correlation function Cn (t)

defined as

= /(sin0(O)rn(O>o-'(sin0(O))o(rn(O))o. (5.8.7) Then, we obtain from Eq. (5.8.4)

( n 2 + m r + „ ( » ) = Y k - , H - F „ + 1 ( » ) ] (5.8.8)

and

^-Cn(t) +—(n2 + inrx)Cn(t) = -^ l [C n _ 1 (0-C n + ] (0] . (5.8.9) (Xt Z*Q ^ ^ o

Using the one-sided Fourier transform

CnCO) = \eimCnt)dt, (5.8.10) o

we may now obtain from Eq.(5.8.9) the usual three-term recurrence relation

(-icoTo + n2+inrx)Cn(<o) = ^[Cn_x((0)-Cn+l(co)] + ToCn(O). (5.8.11)

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304 The Langevin Equation

Equation (5.8.11) has the same form as Eq. (5.5.2) in Section 5.5, whence the exact solution is of similar form, i.e.,

Cl(co) = ^-\Sl(o))Cl(0)-^-Sl(co)S2(co)C2(0) 7 \ 2

(5.8.12) +lsi(Q)S2(co)S3(co)C3(0)-... + U-\)n+[Cn(0)f]Sk((o) + ... |,

3 n k=1 )

where Sn(co) is given by Eq. (5.5.6).

The quantity C„(0) appearing in Eq. (5.8.12) can be expressed in terms of Sn as follows. We have from Eqs. (5.5.7) and (5.8.7)

C„(0) = i(sin^(0)r"(0)>0 -i(sm^(0))0(rn(0))0

= ^((r"-1)-(r»+X) + iIm(Sl)(r")0

= S1S2...Sn_l(l-SnSn+l) + ilm(S])S]S2...Sn. (5.8.13)

Now, on using Eq. (5.4.8), namely:

n + ixr)Sn=rl-SnSn+1)/2,

Eq. (5.8.13) can be rearranged as

Cn(0) = [ix + iImSl) + n/y]S]S2...Sn. (5.8.14)

and in particular

Ci(0) = [w + iIm(5,) + l/y]S1 . (5.8.15)

On substituting Eq. (5.8.14) into Eq. (5.8.12), we then have an exact

expression for Cx(co): 2 _ oo / _ i \ n + l «

Cx(o) = —&-Y* —[ix + ilmiS^ + n/^HsJ^Q) . (5.8.16)

On using Eq.(5.8.16) we can calculate the spectrum of the Josephson radiation

Rco)IRQ=2n; Re [^ (oo)] 5(co) + Re \cx (co) - C_x (co)] .. (5.8.17)

= In Im2 [5j ] 8o>) + Re \CX (co) - C\ (-co) J.

Equations (5.8.16) and (5.8.17) are useful in numerical calculations and allow us to calculate R (co) exactly.

The exact solution as expressed in terms of infinite continued fractions has the obvious drawback that the qualitative behaviour of the spectrum R(co) may not easily ascertained. We shall now demonstrate

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 305

that the main part of the spectrum R (a) may be accurately described by the single relaxation time approximation. This can be accomplished as follows.

The correlation time Tc is the global characterisation of the decay of the correlation function

Cj(0 = i<sin0(O)r(O>o -«<sin0(O)r(O)>o (5.8.18) and is defined as

oo

Tc = J [QCO/Ci (0)]</f = Ci(0) /Q(0 ) , (5.8.19) o

which is the area under the curve of the normalised autocorrelation function. Having determined the Laplace transform of the correlation function, we may easily obtain the exact expression for T by setting a>= 0 in Eq. (5.8.16). The result is

Tc =— ^ y l _ L ! _ [ / x + ; I m (5 x) + nly]T\Sl. (5.8.20)

At long times t, the time behaviour of Cx(t) may be approximated by the single exponential decay

C,(0«C!(0)exp(- / / r c) . (5.8.21) Thus, in the corresponding low frequency limit \oJTc\ < 1, we have

Ci(fl>)« C l ( 0 ) . (5.8.22)

Whence, using Eqs. (5.8.17) and (5.8.22), one may derive an approximate equation for the spectrum of the Josephson radiation, namely

^ - = 2xIm2(Sl)S(a))

[ix + ilmiSJ + My^ [ u + i l m ^ l - l / ^ ' l e \'-ia-\") + A'-i(co+A") J'

where A' and A" are the real and imaginary parts of TL7 respectively, namely:

A = 77 1 =A'+/A" .

The real part of A yields the halfwidth of the spectrum while the imaginary part yields the frequency of oscillation. The behaviour of the real and imaginary parts of the normalised reciprocal correlation time Eq. (5.8.20) is similar to that of the real and imaginary parts of the effective eigenvalue. On using the representation of the continued fractions Sn in

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306 The Langevin Equation

terms of the modified Bessel functions of complex order, we have an analytic equation for the spectrum of the Josephson radiation R(co)

Ro [ I^IA'-Hco-A")]

I_ixr(y)[A'-i(co+A")] \+ K m I Iixy(r) J 8(a»,

(5.8.24) where, according to Eq. (5.8.20), A' and A" are determined from

, , ril+ixr(r)iixAr)\ix+ii™il+ixr(r)/iur(r)+yr\ A +iA = ; .

°° (—\) r ^oT^-L^[ix + ilmIi+ixrW/IixA^+n/r\In+iXr(r)

rt=l n

(5.8.25) The single relaxation time approximation is strictly valid only in the frequency range Tca)« 1 where both Eqs. (5.8.16) and (5.8.22) lead to the same result:

Cx(a»~TcCx(Q) + O<0). However, this approximation [Eq. (5.8.21)] has a much wider range of applicability [18], so allowing us to represent the spectrum of the Josephson radiation R(co) by the simple analytic formula of Eq. (5.8.23) which describes a resonance with natural angular frequency A".

It is apparent from Eqs. (5.8.17) and (5.8.23) that the spectrum of the Josephson radiation contains two parts, a "coherent" ^function spectrum and an "incoherent" broad spectrum. The approximate equation (5.8.23) gives a good quantitative description of the main features of the spectrum of the Josephson radiation in all regions of interest: x < 1 (the so called locked region) and x > 1 (the unlocked or Josephson oscillation region). However in the noise-free (or large barrier) limit (y» 1), the approximate solution in the unlocked region gives only a qualitative description of the first harmonic contribution to R(co) and does not describe higher harmonics at all. The explanation is the use of the single relaxation time approximation for the correlation function, which leads to the appearance of the first harmonic in the spectrum only while the noise-free spectrum of the Josephson radiation consists of an infinite number of harmonics.

We have demonstrated how the Langevin equation may be applied with much success to the problem of the Brownian motion in a tilted cosine potential. We have found both exact and simple approximate

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Chapter 5. Brownian Motion in a Tilted Cosine Potential 307

analytic formulae for the impedance of the Josephson junction and the spectrum of the Josephson radiation. Our approach has the merit that we now have analytic formulae in the presence of noise, for the spectrum broadening, and the resonant frequency for all ranges of the parameters x and y. A Langevin equation of the kind used in the present chapter also arises in a number of other applications. Therefore the results obtained for the Josephson junction can be applied to analogous systems, e.g. to the ring laser gyroscope [7,9]. In the ring laser gyroscope operating in the steady state, the quantity of interest is the spectrum of the beat signal a(co) (see for example ref. [7]) defined as

oo

a0)) = 2 Re J <cos 0(0) cos 0(t))o eiMdt _o

The exact solution for the one-sided Fourier transform of the cosine correlation function spectrum of the ring laser gyroscope (Eqs. (25) and (32) of Ref. [10 (2)]) has the same mathematical form as the Josephson radiation spectrum with appropriate change of parameters. Thus, one can readily apply the above approach to this problem (see Ref. [10 (2)] for details).

Another application of the resistively-shunted junction model in the presence of noise was the calculation of the nonlinear impedance of a point Josephson junction [12,13]. The calculation proceeds by solving the Langevin equation for a Brownian particle in a tilted cosine potential in the presence of a strong ac force. The exact solution of the infinite hierarchy of equations for the moments (expectation values of the Fourier components of the phase angle), which describe the dynamics of the junction, was expressed in terms of matrix continued fractions [12,13]. This solution allowed us to evaluate the nonlinear response of the junction (nonlinear microwave impedance, for example) to an ac microwave current of arbitrary amplitude. Pronounced nonlinear effects in the resistance and the reactance were observed for large ac currents as was demonstrated by plotting the nonlinear response characteristics as a function of the model parameters. For weak ac currents, these results agree closely with the linear response solution presented in this chapter.

References

1. G. Barone and A. Paterno, Physics and Applications of the Josephson Effect, Wiley, New York, 1982.

2. W. Dieterich, P. Fulde, and I. Peschel, Adv. Phys. 29, 527 (1980). 3. W. W. Chow, M. O. Scully, and E. W. Van Stryland, Opt. Commun. 15, 6 (1975). 4. A. J. Viterbi, Principles of Coherent Communication, McGraw-Hill, New York,

1966.

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308 The Langevin Equation

5. W. T. Coffey, Development and Application of the Theory of the Brownian Motion, in Dynamical Processes in Condensed Matter, Adv. Chem. Phys. 63, 69, Ed. M. W. Evans, Wiley, New York, 1985.

6. L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223 (1998).

7. W. W. Chow, J.Gea-Banacloche, L. M. Pedrotti, V. Sanders, W. Schleich, and M. O. Scully, Rev. Mod. Phys. 57, 61 (1985).

8. H. Risken, The Fokker-Planck Equation, Springer-Verlag, Berlin, 1984; 2nd Edition, 1989.

9. J. D. Cresser, D. Hammonds, W. H. Louisell, P. Meystre, and H. Risken, Phys. Rev. A 25, 2226 (1982).

10. W. T. Coffey, Yu. P. Kalmykov, and E. S. Massawe, Phys. Rev. E 48, 77 (1993); iWd., 48, 699 (1993).

11. W. T. Coffey, Yu. P. Kalmykov, and E. S. Massawe, The Effective Eigenvalue Method and its Application to Stochastic Problems in Conjunction with the Nonlinear Langevin Equation, Adv. Chem. Phys., 85, 667, part 2, Eds: M. W. Evans and S. Kielich, Wiley, New York, 1993.

12. W. T. Coffey, J. L. Dejardin, and Yu. P. Kalmykov, Phys. Rev. E 61, 4599 (2000). 13. W. T. Coffey, J. L. Dejardin, and Yu. P. Kalmykov, Phys. Rev. B 62, 3480 (2000). 14. K. K. Likharev, Dynamics of Josephson Junctions and Circuits, Gordon and

Breach, New York, 1986. 15. V. Ambegaokar and B. I. Halperin, Phys. Rev. Lett. 22, 1364 (1969). 16. G. N. Watson, Theory ofBessel Functions, 2nd Edition, Cambridge University Press,

Cambridge, 1944, p.46. 17. A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, Vol.2,

Special Functions, Gordon and Breach, New York, 1988. 18. H. Frbhlich, Theory of Dielectrics, 2nd Edition, Oxford University Press, Oxford,

1958. 19. A. N. Vystavkin, V. N. Gubankov, L. S. Kuzmin, K. K. Likharev, V. V. Migulin,

and V. K. Semenov, Rev. Phys. Appl. 9, 79 (1974). 20. F. Auracher and T. Van Duzer, J. Appl. Phys. 44, 848 (1973). 21. C. V. Stancampiano, IEEE Trans. Electr. Dev. 27, 1934 (1980). 22. Yu. M. Ivanchenko and L. A. Zil'berman, Zh. Eksp. Teor. Fiz. 55, 2395 (1968) [Sov.

Phys. JETP, 28,1272 (1969)]. 23. M. Simmonds and W. H. Parker, Phys. Rev. Lett. 24, 876 (1970). 24. C. M. Falco, W. H. Parker, S. E. Trullinger, and P. K. Hansma, Phys. Rev. B 10,

1865 (1974).

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Chapter 6

Translational Brownian Motion in a Double-Well Potential

6.1 Introduction

The translational Brownian motion of a particle in a 2-4 potential:

V(x) = -ax2+-bx4, (6.1.1) 2 4

where a and ft are constants, -°° < x < °°, is very often used to describe noise-driven motion in a variety of bistable physical and chemical systems (e.g., [1-3]). The relaxational dynamics of the model in the high friction limit, where the inertia of the particle may be neglected, have been extensively studied either by using the Kramers transition state theory [4] or by solution of the Smoluchowski equation underlying the problem (see, e.g., [5-10] and references cited therein). The number of papers devoted to the problem is enormous and various solutions have been presented because the stochastic dynamics of barrier crossing transitions is of fundamental significance in chemical physics.

The traditional analysis of the problem has proceeded from the Smoluchowski equation by either converting the solution of that equation to a Sturm-Liouville problem (e.g., [5]) or to the solution of an infinite hierarchy of linear three-term differential-recurrence relations for statistical moments (e.g., [9,10]). Yet another method of solution was recently proposed by Perico et al. [7] who used the first passage time method to derive an integral expression for the correlation time defined, as usual, as the area under the curve of the positional autocorrelation function. On applying the Mori memory function formalism, they were also able to calculate the position correlation function and to compare it with various approximate formulae and computer simulations.

309

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310 The Langevin Equation

We, on the other hand, have formulated in Chapter 2, Section 2.7.3 an analytic method of finding exact analytic solutions of differential-recurrence relations for statistical moments. The method allows us to calculate the correlation times and the spectra of the correlation functions and of the generalised susceptibilities for dynamical variables of interest. Several problems have been already treated in Chapters 4 and 5. The interested reader may find a variety of other applications of the method in Refs. [11-18] (see also Chapter 7). However, all the exact solutions for problems considered in Chapters 4 and 5 and in Refs. [11-18] were obtained for convergent recurrence relations (when the solution is expressed in continued fraction form). The model of the Brownian particle in the double well potential (6.1.1) poses the problem of solution of divergent differential-recurrence relations for statistical moments [8,10]. Here Risken's continued fraction method [19] is no longer applicable as all the continued fractions involved diverge. Nevertheless, as we shall presently demonstrate, the continued fraction approach also succeeds in this case [20].

The purpose of this chapter is to show how the exact solution for the positional correlation time of the Brownian particle in the double well potential, Eq. (6.1.1), may be obtained directly from the Langevin equation of the process. As outlined in the preceding chapters, the analysis uses the continued fraction method for the exact solution of infinite hierarchies of differential-recurrence relations for statistical moments governing the relaxational dynamics of nonlinear stochastic systems. We shall show how the asymptotic solution of Larson and Kostin [5] may be recovered from the exact solution in the high barrier approximation. In addition, our solution will be compared with the solution of Perico et al. [7], the effective relaxation time, the inverse of the smallest nonvanishing eigenvalue and an empirical approximate equation.

6.2 Relaxation Time of the Position Correlation Function

We consider the one-dimensional translational Brownian motion of a particle in the potential given by Eq. (6.1.1) so that the underlying Langevin equation is [3]

mx(t) + Cx(t) + ax(t) + bx3(t) = f(t). (6.2.1) In Eq. (6.2.1), x(t) specifies the position of the particle at time t, m is the mass of the particle, £x is the viscous drag experienced by the particle, and f(t) is the white noise driving force so that

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Chapter 6. Translational Brownian Motion in a Double-Well Potential 311

/ W = °' (6.2.2) f(t)f(t') = 2kTCS(t-t').

Here the overbar means the statistical average over an ensemble of particles which have all started at time t with the same initial position x(t) = x and velocity x(t) = x. If the mass of the particle is ignored, Eq. (6.2.1) becomes

£x(t) + ax(t) + bx\t) = f(t). (6.2.3)

The underlying noninertial Fokker-Planck equation for the probability density function W of the position x is [3,7]

£^-W(x,t) = y - [ ( « * + bx3)w(x,t)\ + kT—-W(x,0. (6.2.4)

Let us now multiply Eq. (6.2.3) by nxn _ 1 so that it may be rewritten as

^-x»(t) + xn(t) + xn+2(t) = yxn-l(t)f(t). (6.2.5) dt C C C

Equation (6.2.5) is a Stratonovich stochastic differential equation with a multiplicative noise term

nxn~\t)ft)l£. (6.2.6)

We must now average Eq. (6.2.5) as described in Chapter 2, Section 2.3. We first recall that Eq. (6.2.5) is a particular case of the general

stochastic nonlinear equation with a white noise T(t): i(t)=h[(t),t]+g&t),t]nt) (6.2.7)

which becomes on averaging

X=h(X,t) + Dg(X,t)^-g(X,t), (6.2.8) aX

where t;(t + f) is a solution of Eq. (6.2.7) which at time t has the sharp value %t)=X. Thus, we may evaluate the average of the multiplicative noise term in Eq. (6.2.5). We have

X=x\gX) = — = ^-—-,

* ^ (6.2.9) kTCz d kTn(n~l)x(n-2)/n = kTn(n-\) 2

ygdx8 c C so that Eq. (6.2.5) becomes on averaging

C—xn + nax" + nbxn+2 = kTn(n - \)xn~2. (6.2.10) dt

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312 The Langevin Equation

We again remark that x(t) in Eq. (6.2.5) and x in Eq. (6.2.10) have different meanings. Namely, x(t) in Eq. (6.2.5) is a stochastic variable while in Eq. (6.2.10) x is the sharp (definite) value at time t.

On introducing the normalised variables as in [7]

rt) = - ^ , A = a-^-,B = b-^^,rJJ^ (6.2.11) (x2)n

0n 2kT AkT kT

(the symbol ( ) 0 denotes the equilibrium ensemble averages), Eq. (6.2.10) becomes

T — yn (t) + 2nAyn (t) + 4nByn+2(t) = n(n -1) yn'2 (t). (6.2.12) dt

Equation (6.2.12) represents a hierarchy of differential-recurrence relations which describe the time evolution of the sharp values y. Equation (6.2.12) is a previously known result (e.g., [10]) which may also be obtained from the noninertial Fokker-Planck equation (6.2.4).

Equation (6.2.12) may be recast as a hierarchy of equations for the correlation functions Cn (?) defined as

Cn(t) = (y(0)yn(t))0, (6.2.13) with y(0) the initial value of y(t). This can be accomplished by a second averaging of Eq. (6.2.12) over the equilibrium distribution function Wo defined as

W0[y(0)] = e-u[m]/Z, (6.2.14)

where U(y) is the potential (6.1.1) expressed in the normalised variables ofEq. (6.2.11) as

U(y) = V(x)/(kT) = Ay2+By\ (6.2.15)

and Z is the partition function given by

Z=]e~uwdy. - c o

Thus, the Cn (t) satisfy

TCn(t) = q-nCn_2(t) + qnCn(t) + q+nCn+2(t), (6.2.16)

where

qn=-2An, q+n=-ABn, q-=n(n-\). (6.2.17)

Equation (6.2.16) is the desired hierarchy of differential-recurrence relations for the correlation functions. The Laplace transform of Eq. (6.2.16) is given by

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Chapter 6. Translational Brownian Motion in a Double-Well Potential 313

£Cn_2(s) + (qn -sT)Cn(s) + q+nCn+2(s) = -TCn(0). (6.2.18)

The relevant quantity is the correlation time Tc of the position autocorrelation function

Ci(t) = (y(0)y(t))0. (6.2.19)

The correlation time Tc, we recall, is a global characteristic of the relaxation process involved and is defined as the area under the curve of the normalised autocorrelation function:

T =-1

Q(0)J0

jC1(.t)dt = Ci(0) (6.2.20)

because Ci(0) = 1, according to the normalisation conditions. Thus, we require a solution C^O) of Eq. (6.2.18) at s = 0, which we write down as

q-nCn_2(0) + qnCn(0) + q+nCn+2(0) = - rC n (0) . (6.2.21)

For A > 0 (corresponding to a single well potential), the exact analytic solution of Eq. (6.2.21) can be obtained in terms of continued fractions in a manner entirely analogous to that described by us in Chapter 2, Section 2.7.3. The solution is

Q(0) = --<7i

T

•qfS3 C1(0) + IC2n+1(0)n<?2ViS2i+1/<Z : 2k+\

n=\ k=\

2A + 4BS, 1 ^ (2n +l)n! LL

(6.2.22)

2k+\ „=1 \£.u-ri)ii: k=x

where Sn is the continued fraction solution of Eq. (6.2.21) with C„(0) = 0, so that

In n-\

-^n-A+2 2A + 4BSn+2

(6.2.23)

Here, we note that

n &-i _(-VnVB)n

I=T92t+i (2n + l)n! Equation (6.2.22) is the exact formal solution in terms of continued fractions.

For A < 0, which is the case of greatest interest, the recurrence Eq. (6.2.21) as well as the continued fraction (6.2.23) diverge therefore the solution cannot be directly computed from Eq. (6.2.22). Nevertheless, we shall demonstrate that the continued fraction approach, where the appropriate correlation times are first represented as continued fractions,

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314 The Langevin Equation

then as a sum of products of special functions which in turn may be expressed in integral form, may still be used to render the exact solution for the correlation time.

We shall deduce a closed form solution for the correlation time in terms of Whittaker's parabolic cylinder functions Dvx) [22]. First we note that the parabolic cylinder functions satisfy the recurrence relation [22]

xD_v(x)-D_v+1(x) = -vD_v_l(x), (6.2.24)

which may be rearranged in the form of a continued fraction as

- ^ L = \ . (6.2.25)

D_v+l(x) x + vD_v_l(x)/D_v(x)

On comparing the continued fractions (6.2.23) and (6.2.25), we obtain

(n-l)fl-o.+i)/2U)

2-J2B D_(n_l)/2(z)

where z = Aly[2B. (6.2.27)

Now, we have for the leading term of Eq. (6.2.22) 1 = \ y4^)

2A + 4BS3 2A + 4BD_2(z)/D_l(z) iJlB Here we have used Eq. (6.2.24) for v = 1 and the equality D0(z) = e~z /4

[22]. Furthermore [7]

c««»-z- j A ^ v ^ r ! ' ! 1 <6'2-29) where V(x) is the gamma function and the partition function Z is given by

Z=] e-u^dy = ^(2B)-mD_m(zV2/\

By substituting Eqs. (6.2.26), (6.2.28), and (6.2.29) into Eq. (6.2.22), we obtain the correlation time as a series:

rc = C1(0)= r f '\ , Z ( - l ) " r ( n + l/2)D.„_3/2(z)D_n_1(z),(6.2.30)

where the characteristic relaxation time T0 is given by

T0=C/y/2bkT . (6.2.31)

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Chapter 6. Translational Brownian Motion in a Double-Well Potential 315

Figure 6.2.1. Potential U (y), Eq. (6.2.15), for B = 1 and various values of A.

Here we have noted that the normalisation condition (y )0 = 1 implies

that the constants A and B are not independent and [7] ,\2

B = B(z) = -1 D_V2(z)

KD-U2(Z)) (6.2.32)

Moreover, the characteristic relaxation times r0 and rare related by

7 = rn D-y2(z) (6.2.33)

Equations (6.2.22) and (6.2.30) are meaningful in the computational sense only for A > 0 when the continued fractions and the series involved converge very rapidly. For A < 0, these continued fractions and series diverge and Eqs. (6.2.22) and (6.2.30) are purely formal solutions. Such behaviour is a consequence of the divergence of the three term recurrence equation (6.2.21) for A < 0. Thus, in order to obtain correct results for A < 0, the divergent series must be summed exactly.

First, we recall that when A<0, the potential (6.2.15) has a barrier at y = 0 (see Fig. 6.2.1), where the potential has a maximum, where the height relative to the minimum is equal to [7]

q = A2/(4B) = z2/2. (6.2.34) Thus , we have from Eq. (6.2.30) for negative A

vxD_3/2(-j2q)n=

(6.2.35)

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316 The Langevin Equation

Now, by using the integral representation of the parabolic cylinder functions [22]

Dv(x) = ^l—°\e-™-«2<2u-v-idu, ( v < 0 ) ,

we have from Eq. (6.2.35)

(6.2.36)

T = V

^ D _ 3 / 2 ( - V 2 ^ ) o o «=o n!(n +1/2)

_ "0 7n2 3/4 3^/2 °°°°

JJ' <s-fq)2-(t-fq? erf(V2jQ (6.2.37)

dsdt, D_3/2(-j2q) 0 0 V7

where we have recalled that T(n + l) = n! and the error function erf(x) has the Taylor expansion [22]

erf(jc) = - 7 = e rff = -7=2] — n „2n+l

n • (6.2.38)

We emphasise that the integral representation (6.2.37) of the exact solution is valid both for positive and negative A.

On the other hand, the correlation time Tc of the position autocorrelation function C\(t) is given by the general Eq. (2.10.25) (see Chapter 2, Section 2.10), which reads for the present problem as [7]

-|2

T = £ kTz(X\

V(x)lkT) \ xe~v^lkT)dx dx, (6.2.39)

where Z is the partition function defined above. For A < 0, we have by changing the variables in Eq. (6.2.39) in accordance with Eq. (6.2.11)

T = r(2B)l/4e-A/iSB) °r C yf^D_l/2(A/yf2B) i

,v+v J y'e-^-iy-dy' dy

^e"'2 (6.2.40)

1-W °23/4D_3/2(-V2^)Jo

1-erf (-V*)] 2ds_

Here, we have noted that

, V^/5 J ye-^-^dy = 1*1*-,?"*[erf (JBy2-AI1JB)-\\ —oo

The calculation shows that both Eqs. (6.2.37) and (6.2.40) yield the same result. In particular, for q = 0 both equations give

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Chapter 6. Translational Brownian Motion in a Double-Well Potential Z\l

Tc/To\g=0

[^fi-\x -21n(l + >/2)l = 1.01989... rV2 L J

r 2 ( l / 4 ) r ^

Let us now estimate the behaviour of the relaxation time Tc in the high barrier limit (q —> °o). We have from Eq. (6.2.37) by means of the transformation S —> u + sfq, t —> v + yjq :

erf^2(M + V^)(v+V^)] T = zg

,39/2^3/4

J. J. dudv. (6.2.41)

Recognising that the main contribution to the integral Eq. (6.2.41) is in the vicinity of u = 0 and v = 0, we can evaluate the integral in the high barrier limit as follows. On using the asymptotic (x —> °°) expansion of the parabolic cylinder functions [22], viz.,

V2^*2/4

D_J-x)~ F(v)x

and that of the error function [22]

erf(*)~l-

\-v 2x2

\Jffx

we have from Eq. (6.2.41)

V T ( 3 / 2 )

1+K-D' . m=l

(2m-l)!!

2mx2m

T ~ c qK JJ dudv = ° .-

2 ^ (6.2.42)

Equation (6.2.42) shows that in the high barrier limit, the relaxation time

is governed by the inverse of the smallest eigenvalue A[l. Indeed, the

asymptotic solution of Larson and Kostin for X\ [5] reads in our notation:

^ £ f i - A + ; Sq

Af\ K

(<7->°°) • (6.2.43)

6.3 Comparison of Characteristic Times and Evaluation of the Position Correlation Function

Let us now evaluate the smallest nonvanishing eigenvalue X\. We remark that the standard matrix method of evaluation of eigenvalues (Chapter 2, Section 2.7.2) used, for example, in Chapter 4, cannot be applied here to the calculation of X\ as it yields an incorrect result. We also note that Larson and Kostin [5] have found the asymptotic solution for A.x Eq. (6.2.43) by using a singular perturbation method. However, this solution

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318 The Langevin Equation

is valid in the high barrier limit only. On the other hand, Kalmykov [23] has obtained an approximate equation for X\ using continued fractions which allows one to estimate Ai for all barrier heights. Here, we briefly outline the results of Ref. [23]. In the context of the continued fraction approach, the smallest eigenvalue X\ is given by the approximate Eq. (2.11.1.11) of Chapter 2, Section 2.11.1, which now becomes

Zky =

1 + 2 J 11 ^2m+ll2m-l I Q2m+1

(6.3.1)

k=l m=l

where the coefficients qm,q^,qm and the continued fraction S„ are given

by Eqs. (6.2.17) and (6.2.23), respectively. Just as in the preceding section, Eq. (6.3.1) is meaningful in the computational sense only for A > 0, when the continued fractions S„ converge; for A < 0, the continued fractions Sn diverge and so Eq. (6.3.1) is a purely formal solution. Nevertheless, we shall demonstrate below that the method used above for the solution of divergent recurrence relations may still be used to render the solution for X\.

Thus, we first express the continued fraction Sn in Eq. (6.3.1) in terms of parabolic cylinder functions Dv(x), viz.,

-i-i

\TQ = 0Ql2

D_,(-V2^)n=o 2/t + l t^Tn + \)D2_n_x (-&) (6.3.2)

Here we have used Eqs. (6.2.26) and (6.2.33). Now, on using the integral representation of the parabolic cylinder functions Eq. (6.2.36), the series in Eq. (6.3.2) can be summed exactly to yield [cf. Eq. (6.2.37)]

ATo ~ f f g - ( , - ^ - ( ^ ) 2 e r f ( V 2 , Q ^ J J >]st

"l-erf(z/>/2)] .

(6.3.3) l + erf(^) J

0Jo

where we note that [22]

D_1(z) = / / 4 ^ / ^ 7 2 [ l - e r f ( z / ^ / 2 ) J . (6.3.4)

Just as the relaxation time, we can obtain from Eq. (6.3.3) an asymptotic expression in the high barrier limit (q —> °°):

4 r 0 ~ 2e-q4ql7C, (6.3.5)

which is the leading term of the asymptotic solution obtained by Larson and Kostin, Eq. (6.2.43).

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Chapter 6. Translational Brownian Motion in a Double-Well Potential 319

1.0

0.8-

>

"•r

ub

<

0.6-

0.4-

0.?-

0.0

- 0 . 2 - t — ' — • 1 • 1 • 1 • 1 •

0 1 2 3 4 5 1

Fig. 6.3.1. X\ [Eq. (6.3.3); solid line] for the 2-4 potential as a function of the barrier height parameter a compared with the asymptotic solution of Larson and Kostin [Eq. (6.2.43); dashed line] and the solution rendered by the inverse of the correlation time [Eq. (6.2.37); diamonds].

The lowest eigenvalue X\ for the potential (6.1.1) completely determines the behaviour of the correlation time Tc, which, according to Eq. (6.2.37), has the same asymptote Eq. (6.3.5). Evidently, in Table 6.3.1 and in Fig. 6.3.1, the lowest eigenvalue k\ calculated from Eq. (6.3.3) agrees closely with T~l from Eq. (6.2.37) for all q. In Fig. 6.3.1 and Table 6.3.1, ^"calculated from the asymptotic Eq. (6.2.43) is also presented. The calculation demonstrates that X\, Xf, and T~l are in agreement in the high barrier limit. Moreover, X\ and T~l are very close to each other for all barrier heights.

Table 6.3.1. Numerical values for XTQ , ^TQ , and T0/TC .

q

0 l 2 3 4 5 6 8 10

\T0 [Eq. (6.3.3)]

0.937206 0.256730 0.111443 0.048852 0.020951 0.087591 0.003583 0.000571 0.000088

4 " T 0 [Eq. (6.2.43)]

O O

0.146375 0.098999 0.048036 0.021134 0.008872 0.003624 0.000576 0.000088

T0/TC [Eq. (6.2.37)]

0.980499 0.274517 0.118609 0.051471 0.021850 0.009059 0.003682 0.000582 0.000089

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320 The Langevin Equation

Fig. 6.3.2. Correlation time [Eq. (6.2.37); solid line] as a function of the barrier height q compared with the asymptotic solution of Larson and Kostin [Eq. (6.2.43); dashed line] and the solution rendered by the effective eigenvalue [Eq. (6.3.9); dots].

Another approximation that can be used to estimate the correlation time in the low barrier limit is the inverse of the effective eigenvalue Aef(see Chapter 2, Section 2.12), which is the reciprocal time constant associated with the initial slope of the correlation function [21]. The effective eigenvalue Aef defined as

V -C, (0)7^(0) (6.3.6)

is calculated by evaluating Eq. (6.2.16) for n = 1 at t = 0, which is rC,(0) + 2AC1(0) + 45C3(0) = 0. (6.3.7)

On noting that C,(0) = 1, the effective correlation time Tef = A~f is then

Tef = --J— = . (6.3.8) f Ci(0) 2A + 4fiC3(0)

On using Eqs. (6.2.24), (6.2.29), (6.2.32), (6.2.33), and (6.3.8), we have

Tef=-TD_3/2(Z)

2A + 3V2B g-3/2(z) V25[2zD_3/2(z) + 3D_5/2(z)] 0-3/2(z)

= T-D_V2(-j2J)

(6.3.9)

D_U2(-J2q) In the low barrier limit, q < 1, Eq. (6.3.9) has the Taylor expansion

lef •2y/2 1X3/4)

r ( i /4 ) + 1-4 r 2 (3 /4)^

V2g+8T2~ r3

( 3 / 4 ) 3 + ... (6.3.10) T J(l/4) r z ( i / 4 )

It is apparent from Fig. 6.3.2 that the effective relaxation time Tef is a poor approximation to the correlation time for high potential barriers.

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Chapter 6. Translational Brownian Motion in a Double-Well Potential 321

10'i

0 2 4 6 8 10 q

Fig. 6.3.3. Exact solution for the correlation time [Eq. (6.2.37); solid line] as a function of the barrier height q compared with the solution rendered by the approximate Eq. (6.3.11) (filled circles).

Since the exact solution we have obtained is expressed in terms of special functions, and so appears rather complicated, it is still worthwhile to seek a simple empirical expression valid for all barrier heights and which can be computed more simply. Such a formula for the analogous problem of magnetisation relaxation of single domain ferromagnetic particles in a 3D double well potential has been given by Cregg et al. [24]. Here, this simple formula is [20]

Tc~T0^-(x^ + 21-^). (6.3.11)

The exact [Eq. (6.2.37)] and approximate [Eq. (6.3.11)] formulae are compared in Fig. 6.3.3. Apparently, both equations coincide within graphical accuracy. Moreover, the comparison of the exact results with the approximate solution shown in Table 6.3.1 allows us to estimate quantitatively the accuracy as better than 3% for all values of the barrier height parameter q\

Having determined the characteristic times lMi, Tc, and Tef, one may evaluate the time behaviour of the correlation function C\t) by using the method outlined in Chapter 2, Section 2.13. Thus, the correlation function Ci(t), which comprises an infinite number of decaying exponentials, may be approximated by two exponentials only, viz.,

C ^ O ^ A ^ + a - A ^ e - ' ^ . (6.3.12)

Here A, and Tw are given by (see Chapter 2, Section 2.13)

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322 The Langevin Equation

O

t/r

Fig. 6.3.4. The position correlation function as calculated from Eq. (6.3.12) for various values of q.

A 1 = -Tc/Tef-l

% - ' \Tc-\

(6.3.13) Xjc-2 + \l^Y m \-\ITef

The behaviour of C(t) is shown in Fig. 6.3.4. Thus, the spectrum of

CX((D) corresponding to Eq. (6.3.12) is a sum of two Lorentzians with

characteristic frequencies A,\ and r, w

1 + i(Ol\ l + id)Tw

(6.3.14)

One may readily verify that Eq. (6.3.14) yields the correct behaviour of Cx((0) both at low (a>—> 0) and high co—> oo) frequencies. We remark that Perico et al. [7] have derived in the high barrier limit (q » 1) a very similar approximate equation for C (t):

C, (0 = e-,IT< (A + (1 - A)e-"wl-&)] , (6.3.15)

Table 6.3.2. Numerical values of 1M, [Eq. (6.3.3)], Tc [Eq. (6.2.37), Tef [Eq. (6.3.9)], and ^ [ E q . (6.3.13)].

1 / ^ 1

TJib TJn, % / •%,

<? = ! 3.895 3.643 2.355 0.386

4 = 2 8.973 8.431 3.409 0.332

q = 4 47.73 45.77 5.191 0.240

9 = 8 1749 1716 7.716 0.147

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Chapter 6. Translational Brownian Motion in a Double-Well Potential 323

where A = (x)2well/x2)0 and (x)well is the average of x in one of the

wells (e.g, over the range 0<x<+°o). Equation (6.3.15) is valid for q » 1; in contrast Eq. (6.3.12) can be used for all q.

We remark that Eq. (6.2.18) can be formally solved for the Laplace transform Cxs) of the position autocorrelation function when the solution of that equation is expressed in terms of continued fractions. This representation of the problem however exhibits the same lack of convergence as encountered when one attempts to express its zero frequency limit C\ (0) as a continued fraction or in series form. Hence, in order to determine the exact solution for C^(s), one must again find a means of summing the series when s^O.

We have demonstrated how the continued fraction treatment of the solution of nonlinear Langevin equations may be successfully applied to the Brownian motion in the double-well potential Eq. (6.1.1). We have given an exact as well as simple approximate analytic formulae for the position correlation time Tc. Thus, we now have analytic formulae for Tc

for all ranges of the barrier height parameter q. The formula for the correlation time contains the previous result of Larson and Kostin [5] as the limiting case of high potential barriers q. The crucial steps which allow one to represent the solution in closed form are first the representation of the correlation time as the zero frequency limit of the Laplace transforms of the respective autocorrelation functions and secondly that these Laplace transforms satisfy three-term recurrence relations. These results allow one to express the correlation time as a series of Whittaker's parabolic cylinder functions, and so to sum the divergent series in order to obtain the exact analytic formula Eq. (6.2.37) for the correlation time. Our analytical results, Eqs. (6.2.37) and (6.3.12), appear to be completely equivalent to those of Perico et al. [7]. The present analysis of a nonlinear stochastic model with relaxational behaviour governed by a divergent three-term recurrence relation extends the range of the applicability of the continued fraction method. Another example of the solution of a divergent recurrence equation is given in Ref. 25.

References

1. R. S. Larson and M. D. Kostin, J. Chem. Phys. 72, 1392 (1980). 2. W. T. Coffey, M. W. Evans, and P. Grigolini, Molecular Diffusion and Spectra,

Wiley, New York, 1984; Russian translation: Mir, Moscow, 1987. 3. C. Blomberg, Physica A 86, 67 (1977).

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324 The Langevin Equation

4. H. A. Kramers, Physica 7, 284 (1940). 5. R. S. Larson and M. D. Kostin, J. Chem. Phys. 69, 4821 (1978). 6. P. Hanggi, P. Talkner, and M. Borcovec, Rev. Mod. Phys. 62, 251 (1990). 7. A. Perico, R. Pratolongo, K. F. Freed, R. W. Pastor, and A. Szabo, J. Chem. Phys.

98, 564 (1993). 8. A. Schenzel and H. Brand, Phys. Rev. A 20, 1628 (1979). 9. L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223

(1998). 10. 1.1. Fedchenia, J. Phys. A: Math. Gen. 25, 6733 (1992). 11. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and J. T. Waldron, Physica A

213,551 (1994). 12. W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, Physica A 208, 462 (1994). 13. J. T. Waldron, Yu. P. Kalmykov, and W. T. Coffey, Phys. Rev. E 49, 3976 (1994). 14. W. T. Coffey D. S. F.Crothers, Yu. P. Kalmykov, and J. T. Waldron, Phys. Rev. E,

49, 1869 (1994). 15. W. T. Coffey, D. S. F.Crothers, Yu. P. Kalmykov, and J. T. Waldron, Phys. Rev. B

51, 15947 (1995). 16. W. T. Coffey, J. L. Dejardin, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. E 54,

6462 (1996). 17. W. T. Coffey and Yu. P. Kalmykov, Adv. Chem. Phys. 113, 487 (1998). 18. J. L. Dejardin, Yu. P. Kalmykov, and P. M. Dejardin, Adv. Chem. Phys. 117, 275

(2001). 19. H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1984; 2nd Edition, 1989. 20. Yu. P. Kalmykov, W. T. Coffey, and J. T. Waldron, J. Chem. Phys. 105, 2112

(1996). 21. W. T. Coffey, Yu. P. Kalmykov, and E. S. Massawe, in Modern Nonlinear Optics,

Eds. M. W. Evans and S. Kielich, Adv. Chem. Phys., Wiley, New York, Vol. 85, Part 2, 1993, p. 667-793.

22. M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, 1964.

23. Yu. P. Kalmykov, Phys. Rev. E 61, 6320 (2000). 24. P. J. Cregg, D. S. F. Crothers, and A. W. Wickstead, J. Appl. Phys. 76, 4900 (1994). 25. Yu. P. Kalmykov and V. N. Sekistov, Radiotekh. Elektron. 46, 283 (2001) [/.

Commun. Technol. Electron. 46, 258 (2001).]

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Chapter 7

Three-Dimensional Rotational Brownian Motion in an External Potential:

Application to the Theory of Dielectric and Magnetic Relaxation

7.1 Introduction

The noninertial three-dimensional rotational Brownian motion of a particle in the presence of an external potential arises in a variety of problems. Examples are dielectric and Kerr-effect relaxation of polar fluids subjected to a constant electric field [1-6], dielectric relaxation of nematic liquid [7-10] and molecular crystals [11], magnetic relaxation of single domain ferromagnetic particles and ferrofluids (colloidal suspensions of such particles) [12-17], etc. Theories of all these phenomena have been mainly based on the Fokker-Planck equation [18,19].

The essence of the Fokker-Planck equation method when the inertial effects may be neglected is to derive from the underlying Langevin equation the particular form of the Fokker-Planck equation commonly known as the rotational diffusion or the Smoluchowski equation for the probability density function W(u,f) of orientations of a unit vector u fixed in the particle in configuration space (see Chapter 1, Sections 1.15 and 1.17). The Fokker-Planck equation can be solved by the method of separation of the variables. The separation procedure gives rise to an equation of Sturm-Liouville type [12,19]. An alternative approach to the problem is to expand VK(u,0 as a series of spherical harmonics. This procedure yields an infinite hierarchy of differential-recurrence relations which may be written in matrix form:

X(f) = AX(f). (7.1.1)

325

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326 The Langevin Equation

The general method of solution is effected by successively increasing the size of the system matrix A until convergence is attained. An alternative solution of Eq. (7.1.1) may be accomplished by using ordinary or matrix continued fractions.

The Fokker-Planck equation method is extremely lengthy to use in practice especially for rotation in three dimensions since it involves many mathematical manipulations. Thus, an alternative approach is desirable. It is the purpose of this chapter to show how hierarchies of differential-recurrence relations for the appropriate correlation (relaxation) functions arise naturally from the vector Langevin equations defined as Stratonovich stochastic equations [20], thus bypassing the problem of constructing and solving the Fokker-Planck equation entirely. Two problems are treated in detail: (i) orientation relaxation of a polar molecule governed by the vector Euler-Langevin equation and (ii) magnetisation relaxation of a single domain ferromagnetic particle described by Gilbert's equation augmented by a random field term.

7.2 Rotational Diffusion in an External Potential: The Langevin Equation Approach

We consider the three-dimensional rotational diffusion of a rigid spherical or linear molecule in an external potential V (a more general problem, viz., anisotropic rotational diffusion of a rigid asymmetric top molecule, will be considered in Section 7.6). Denoting by u(t) a unit vector through the centre of mass of the particle, we can write the equation of motion for the rate of change of u (?) and the angular velocity <o(0 of the particle [21,22]:

Figure 7.2.1. Geometry of the problem.

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Chapter 7. Three-Dimensional Rotational Brownian Motion 'ill

u(0 = w(Oxu(0, (7-2.1)

lG)(t) + g<o(t) = m(t)xE(t) + k(t) , (7.2.2)

where / is the moment of inertia of the molecule, m is the total dipole moment that may comprise the permanent dipole moment of the molecule as well as the induced dipole moment due to the polarisability and hyperpolarisability of the molecule, ga(t) is the damping torque due to Brownian movement, g is the rotational friction coefficient, and k(t) is the white noise driving torque, again due to Brownian movement, so that k(t) has the following properties

4(0 = 0, MhWjih) = UTgSgSh -t2). Here, the indices i, j = 1, 2, 3 in Kronecker's delta &• correspond to the Cartesian axes X, Y,Z of the laboratory coordinate system OXYZ (see Fig. 7.2.1). Equation (7.2.1) is a purely kinematic relation with no particular reference either to the Brownian movement or to the shape of the particle, while Eq. (7.2.2) is the Euler-Langevin equation [22]. The term m(f)xE(/) in Eq. (7.2.2) is the torque acting on the molecule. This torque can be expressed in terms of the potential energy V(u,t) of the molecule as a function of the components of the vector u, namely,

m x E = - u x — V , (7.2.3)

where

u=uxi+uYj+ uzk, — = i - — + j - — + k - — , an dux ouY duz

and wx=sinz?cos^, My=sini?sin#>, HZ=COSZ? (7.2.4)

are the Cartesian components of the unit vector u(0 (these components are not independent since ux + Uy + uz = 1), & and <p are the polar and azimuthal angles of spherical polar coordinates, respectively, and i, j , and k are the unit vectors along the axes X, Y, and Z, respectively. The potential V in Eq. (7.2.3) may have many different origins. For example, for systems of noninteracting dipolar polarisable molecules subjected to an external electric field E(0, the potential V is given by [6]

V(u,f) = - ( | i -E)- (E-<*-E)/2 ,

where ft and a are the electric dipole moment vector and the electric polarisability tensor, respectively (here the effects due to hyperpolarisability are neglected, however they may also be included in the theory by adding the corresponding terms in the potential energy). The

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328 The Langevin Equation

field E in Eq. (7.2.2) may also include externally applied ac and dc fields. On the other hand, V may be considered as an external mean field potential as in the theory of orientational relaxation in nematic liquid crystals [9]. Particular examples will be given below.

The noninertial limit (or the Debye approximation) occurs when the inertia term in Eq. (7.2.2) is neglected. In this limit, the angular velocity vector may be immediately obtained from Eqs. (7.2.2) and (7.2.3) as

co(0 = g~ Mt)-u(t)x4~V(t) du

(7.2.5)

On combining Eqs. (7.2.1) and (7.2.5), one obtains [4,10]

M ( 0 = -T-V(f) + u(0 | u(f) • 4-V(t)) + X(r)xu(0. (7.2.6) at an \ du )

Equation (7.2.6) is a vector stochastic differential equation. It contains multiplicative noise terms given by the components of the vector product X(t)xu(t). This poses an interpretation problem for this equation as, discussed in Sections 2.3 and 2.5. As far as we are concerned, here we shall use the Stratonovich definition [20,23] (see Chapter 2, Section 2.3) of the average of the multiplicative noise term, as that definition always constitutes the mathematical idealisation of the physical stochastic process of orientational relaxation in the noninertial limit. Thus, it is not necessary to transform Eq. (7.2.6) into Ito equations, so that we avoid complicated mathematical manipulations.

We recall (see Chapter 2, Section 2.5) that on taking Langevin

equations for a set of N stochastic variables§(0 = g'1(t),g2(t),—.£w(0:

l(t) = hi([$(t),t) + gumt).tWj(t), (i,j = l,...,N) (7.2.7)

with

r,.(o=o, r,.(or7.(0 = 2D<5^(r-0 (7.2.8) and interpreting them as Stratonovich equations, the averaged equations for the sharp values £.(f) = xi at time t are [18]

xi=hi(^,t) + Dgkj(x,t)^-gij(x,t) (i,j=l,...,N), (7.2.9) dxk

where ^(t+r) (t> 0) is the solution of Eq. (7.2.7) with the initial conditions ^-(f) =*,•• The proof of Eq. (7.2.9) has been given in Chapter 2, Section 2.5. Moreover, we have proved that the averaged equation for an arbitrary differentiable function/(^ ) has the following form (see Chapter 2, Section 2.5):

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Chapter 7. Three-Dimensional Rotational Brownian Motion 329

df«x) ,. nm+T))-f(x) - hm dt r->0

:^(x,0^-/(x) + D^.(x,0 d

dX: dXn g..(x,o—/(x))

oxi

(7.2.10)

(* = X, r, Z) and the

(7.2.11)

where summation over i, j , and k is understood. Here, according to Eq. (7.2.6), xk = uk

tensorial elements of gik are given by

8xx=° Sxv=uzlg gYX=-Uz/g gYY=°

gzx=uYlg gzY=-"x/S Since the methods of ordinary analysis are applicable to mathematical transformations of Stratonovich stochastic differential equations, we can readily obtain from Eqs. (7.2.1) and (7.2.5) the stochastic equation for the time rate of change of an arbitrary differentiable function/(u), namely,

d-f[u(t)] = u(t)-~f[u(t)]

gxz=-uYlg

gYZ=uxlg

£zz=0

dt'

= g-l[Ut)xu(t)]~f[u(t)]- <f dVL V-u(r) u ( 0 ~ V

an ~ / [ u ( 0 ] -dw

(7.2.12) The right-hand side of Eq. (7.2.12) comprises two terms, namely:

the deterministic drift and the noise-induced (or spurious) drift. Let us first evaluate the noise-induced drift [the first term in the right-hand side of Eq. (7.2.12)] by averaging it over an ensemble of Brownian particles, that all start at time t with the same orientation u. According to Eq. (7.2.10), it can be written as

. - i [k(t)xu(t)]~ /[u(0] dn

= kTS\gkj du,, 6Uy +Skj

3«t SXJ

du. -f + 8kj

3wt 1 duv

(7.2.13) Here, the overbar means a statistical average over an ensemble of Brownian particles that all start at time t with the same orientation u. Considering the first term on the right-hand side of Eq. (7.2.13) and taking account of Eq. (7.2.11), we have

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330 The Langevin Equation

&Tgkj duh

\UX + Uy j

d

du\

duxduz duYduz

•2u du7

/ ( u ) .

The remaining two terms can be readily obtained by means of a cyclic permutation of indices X -» Y, Y —> Z, Z -» X, leading to

r 1 [ * . ( 0 x u ( 0 ] ~ / [ u ( r ) ] = — A / ( u ) , du c

(7.2.14)

where the operator A is given by

d2

A = (uY +u2z)-^j + [u2

x + I 4 ) _ + ( I I | +uY)^j du v duv

-2 uvu yUX

duydux

+uxuz duxduz

+uYuz •+uv -+uv duYduz dux duY

-+u7 du 2 /

The operator A may equivalently be presented in the vector form as

A = U X : du

f d ^ -MX-

3u, = -!?, (7.2.15)

where L is the orbital angular momentum operator [25]. Equation (7.2.15) implies that the operator A is, in fact, the angular part of the Laplacian, which may be written in terms of the angles t? and (p as [25]

1 d2

A = — sinw—— sin $d$ I d&

+ - (7.2.16) s i n 2 •& d<p2 '

Let us now consider the deterministic drift term. On using Eq. (7.2.16), we have after some algebra

1 -?-V[u(t),t]-u(t) du

u(t)~V[u(t),t] du

~ / [ u ( 0 ] du

=—[v[u(0,*]A/[u(r)] + /[u(r)]AV[u(r),r]-A(V[u(0,r]/[u(0])]

= ±-[vu,t)Af(u) + f(u)AVu,t)-A(vu,t)f(u))]. (7.2.17)

(One may verify Eq. (7.2.17) with the help of the MATHEMATICA program.) Thus, on combining Eqs. (7.2.12), (7.2.14), and (7.2.17), we obtain the averaged stochastic equation for an arbitrary function / ( u ) :

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Chapter 7. Three-Dimensional Rotational Brownian Motion 331

2TDf(u) = -^-[V(u,t)Af(u) + f(u)AV(u,t)-AV(u,t)f)] + Af(u),

(7.2.18) where TD =g/(2kT) is the characteristic (Debye) relaxation time. Again all the quantities in Eq. (7.2.18) are, in general, functions of the sharp values u= ux, uY, uz), which are themselves random variables with probability density function W such that Wdu is the probability of finding u in the interval (u, u + du). We remark also that u in Eq. (7.2.18) and u(0 = "x(0> uyt),u^t) in Eq. (7.2.12) have different meanings, namely, u(f) in Eq. (7.2.12) are stochastic variables while u are the sharp (definite) values of u(t) = u at the averaging instant t.

Equation (7.2.18) is valid for any function / . The functions pertaining to orientational relaxation are the spherical harmonics y/m

defined as [25]

l(2/ + l ) ( / - m ) ! ,m_ m , ,„ , YLm(#,<p) = .K A V ^ / T t c o s i ? ) , r ; , _ m = ( - l ) " X , (7.2.19)

y A7rl + m)\

where the P/m(x) are the associated Legendre functions, which may be defined as [25]

P»(cosi?) = ^ U s i n z ? ) m — (cos 2 0- l ) , ' 2'/! V ' (rfcosz?)'+mV '

and the asterisk denotes the complex conjugate. The orthogonality and normalisation condition for the spherical harmonics Ylm(tf,<p) is [25]

In x

o o Thus, Eq. (7.2.18) yields

2TnY, = -1

D ,,m 2kT [v^Ylm+YLmAV-AVYLm)] + AYlm. (7.2.20)

In order to proceed, we recall the known relationships for spherical harmonics [25]

AYLm=-l(l + l)YLm,

/(2/+i)(2/1+i) a cj&fic%ZY r 7 9 9 n

'•-"'•m' "V TK hf_hl V2/2+i F , > w 122l)

where C^^p are the Clebsch-Gordan coefficients, the various

definitions of which are available, e.g., in Ref. [25]; a typical example is

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332 The Langevin Equation

rc,r = x A (c + y)\(c-y)\(2c + \) a'a^ r'a+p abcia + a)\(a-a)\(b + /3)\b-P)\

(-l)b+P+n(c + b + a-n)\(a-a + n)\

„n\(c-a + b-n)\(c + y-n)\(a-b-y+n)l' where

Kbc =. \a + b-c)\a-b + c)\-a + b + c)\

(a + b + c + l)\ and the summation index n assumes integer values for which all the factorial arguments are nonnegative. (We remark that the built-in function ClebschGordan[a, a,b,j3,c, y] of the MATHEMATICA program allows one readily to calculate these coefficients). Now, one can show that for any potential V, which can be expanded in terms of spherical harmonics as

°° R

V/(kT)=% X VR.SYR.S, R=0 S=-R

so that Eq. (7.2.20) may be presented as

. _ /(/ + 1) l y ™ 1(2* + 1X2/+ 1) Dl'm~~ 2 F/- + 8fe4/fi'SV *2/ + l)

x[j(J +1) - / ( / +1) - * ( * + l)]Cf£RfiCf£&YJtm+s ,

or, equivalently,

^D^l,m = 2 J el,m,l+r,m+,^l+r,m+.'i ' (7.2.22)

where

W + 1 ) * x i ^mV(2/ + l)(2/ + 2r + l) el,mMr,m+s = Z drOds0 + ( _ 1 i o

(7.2.23) V V K2/ + r + l) -/?(/? + !) „fi,0 *,_,

2 J V«,.« / ^ „ = t-/,0,/+r,0L '/,m,/+r,-m-.v

j £ JTC(2R + \)

Here, we have used the symmetry property of the Clebsch-Gordan coefficients [25], viz.,

In order to obtain equations for the moments (*/,„,) governing the

relaxation dynamics of the system, we must also average Eq. (7.2.22)

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Chapter 7. Three-Dimensional Rotational Brownian Motion 333

over the distribution function W of the sharp values <p and z?. Thus, we obtain

*z> JtYi,m)(.t) = X e,„J+rjn+s(r/+r,m+,)(r). (7.2.25)

However, if the system is in equilibrium, all averages are either constant or zero. Thus, first we require from Eq. (7.2.25) a set of differential-recurrence equations for equilibrium correlation functions.

In equilibrium, the system has the Boltzmann distribution function Wo given by

W0(#,(p) = Z-le-v^mT\ (7.2.26) where Z is the partition function. The equilibrium averages satisfy

Z-ir,s el,m,l+r,m+.i yi+r,m+s /0

=®> (7.2.27)

where ( ) designates the equilibrium average over the distribution Wo:

2K K

(Y'A= I J YLm(#,<p)Wo(#,<P)sin#d#d<p. 0 0

On subtracting Eq. (7.2.27) from Eq. (7.2.22), then multiplying the equation so obtained by an appropriate function F[$Q),(p(Q)], and

averaging the resulting equation over W0 at the instant t = 0, we obtain

^ / . m (0 = Er,.v el,m,l+r,m+,Cl+r,m+s (t) • (7.2.28)

Here clm(t) are the equilibrium correlation functions, namely:

clm(t) = (F[m),<p(0)]Yhm[m,<Pit)])0 -(^[^0),^0)])0(y/im[iS(0),^(0)])0. (7.2.29)

We shall illustrate the above results by considering two examples: (A) dielectric relaxation of an ensemble of noninteracting polar linear molecules in the presence of a strong uniform electric field Eo applied along the z axis and (B) dielectric relaxation of a nematic liquid crystal with uniaxial crystalline anisotropy originally described by Martin, Meier and Saupe [7]. In these problems, Eq. (7.2.28) can be considerably simplified due to the form of the external potential.

In example A pertaining to the Brownian motion of linear polar molecules in the presence of a strong constant electric field E0, the potential energy is given by

V = - (fi • E0) = -juE0 cos t? = -juE0 V4/r/3y10.

Thus, using the explicit representation for the Clebsch-Gordan coefficients [25], we have from Eqs. (7.2.23) and (7.2.28)

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334 The Langevin Equation

2V„,m(0 + "(« + l)crt,m(0

J 2 2 n -m

— c. . . - (n + l)2 -m2

«-i,m (0 - 4nA c, \ (2n + 3)(2n +1)

n+l,m (0, (7.2.30)

(2n-l)(2/t + l)

where £ = juE0 l(kT). In longitudinal relaxation, the relevant quantities are the

equilibrium correlation functions defined as

/„(*) = (cosiX0)Pn cosW))0-(cosW))QPn (cos^0))>o, (7.2.31)

where Pn(x) is the Legendre polynomial of order n [24,45]. On noting that [25]

YnS)(#,<p) = yl(2n + l)/(4x)Pn (cosz?) (7.2.32)

[Pn(x) = P°(x)l we may obtain from Eq. (7.2.30) a differential-

recurrence equation for fn(t):

£n(n +1) -2TDfn(t) + n(n + l)fn(t) = -

2n + l - [ /„ - i (0- / B + i (0] . (7.2.33)

Equation (7.2.33) governs the longitudinal relaxation of the system. Similarly, we can obtain a dynamic equation for the transverse correlation functions gn(t) defined as:

gnt) = (cos<p(0)sini(0)cos <p(t)P* (costf(f)) (7.2.34)

(here, the equilibrium averages \Ynl) = 0). This equation is

2TDgn(t) + n(n + \)gnt) = -^—\n + \f gn.xt)-n2gn+l(t)]. (7.2.35) 2n + 1 L J

(B) As a second example, we take the uniaxial potential

V=-Kcos2& = -(4K/3)y[xT5Y20-K/3. (7.2.36)

According to Eqs. (7.2.23) and (7.2.28), we have

*iA,m(0 + n(n + \) n(n + l)-3m _ i --o-(2n-l)(2n + 3)

C«,m(0

n + l (n2-m2)[(n-l)2-m2]

2n-\ (2n-3)(2/i + l) Ln-2,m (0 (7.2.37)

2n + 31

[(» + 2)2-m2][(n + l ) 2 - m 2 ]

I (2n + 5)(2n + l) Ln+2,m (0

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Chapter 7. Three-Dimensional Rotational Brownian Motion 335

where o = KIkT). (7.2.38)

Now, one can derive differential-recurrence relations for the longitudinal, fn(t) - cn0(t), and transverse, gn(t) = cnl(t), correlation functions:

tDJtm + nn + \) ann + \)

(2n-l)(2« + 3) /„(')

an(n2-l) an(n + l)(n + 2) f

(7.2.39)

(2n-l)(2/i + l) and

on(n + lY

nn + \)

•Sn-iif)-

(2n + l)(2n + 3)

n(n + l ) - 3

' (2n- l ) (2n + 3)_

an2n + l)

8n(t) (7.2.40)

-8n+2(t)-(2n - l)(2/i +1) (2n + l)(2n + 3)

On solving Eqs. (7.2.33), (7.2.35), (7.2.39), and (7.2.40) for the Laplace transforms of/i and gi and then using linear response theory (Chapter 2, Section 2.8), we obtain below (see Sections 7.4 and 7.5) exact analytic solutions for the longitudinal and transverse complex susceptibilities and correlation times for both problems (A) and (B).

7.3 Gilbert's Equation Augmented by a Random Field Term

Fine single domain ferromagnetic particles are characterised by thermal instability of the magnetisation due to thermal agitation [26] (see Chapter 1, Sections 1.16 and 1.17). This results in the phenomenon of superpara-magnetism (so-called) [27] because each fine particle behaves as an enormous paramagnet of magnetic moment 104-105 Bohr magnetons. The thermal fluctuations and relaxation of the magnetisation of single domain particles currently merit attention in view of their importance in the context of information storage and rock magnetism, as well as in connection with the observation of magnetisation reversal in isolated ferromagnetic nanoparticles and nanowires.

The pioneering theory of thermal fluctuations of the magnetisation of a single domain particle due to Neel [26,27] was further developed by Brown [12] using the theory of Brownian motion. Brown took the Langevin equation as Gilbert's equation [28] augmented by a random field describing the dynamic behaviour of the magnetisation M which incorporated the collision damping incurred by the precessional motion (see Chapter 1, Section 1.17)

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336 The Langevin Equation

M(f) =y[M(t)x[U(t)+h(t)-7]M(t)]]. (7.3.1) Here h (t) is a random Gaussian field having the properties of white noise and H (t) is the magnetic field defined as [13,29]

dm Msdu where V(M,t) is the free energy per unit volume expressed as a function of the components of the magnetisation M=Mxi+MY j+M zk,

MX=MS sint^cosp, MY=MS sin&sm<p, MZ=MS cosz?. (7.3.3) u is the unit vector directed along M, Ms is the saturation magnetisation, which is determined by the material and the temperature. Brown derived from Gilbert's Eq. (7.3.1) the Fokker-Planck equation for the distribution function W(M,t) of the orientations of the magnetisation M [13,29] [Eq. (1.17.17) of Section 1.7]

—W = LFPW = — dt FP

2TN

a lu du 3u J 3u du

+ AW\

(7.3.4) where LFP is the Fokker-Planck operator, a = yrjMs is the dimensionless dissipation (damping) parameter, and

tN=/3Ms(l + a2)/2ya) (7.3.5)

is the characteristic (diffusion) time (the definition of all the remaining quantities are given in Chapter 1, Section 1.17). It is assumed that the magnetisation is always uniform inside the particle and that only the orientation and not the magnitude of the magnetisation is subject to variations. Moreover, magnetic interactions between particles and memory effects are ignored. A detailed discussion of the assumptions made in the derivation of the Fokker-Planck and Gilbert equations is given elsewhere (e.g., [14,15,30]).

The Fokker-Planck equation (7.3.4) can be solved by expanding W in terms of a complete set of appropriate functions, usually as a series of spherical harmonics 7;m [29]. This procedure yields an infinite hierarchy of differential-recurrence equations for the moments [the expectation values of the spherical harmonics Yt^t)]. Just as the rotational diffusion problem considered in Section 7.2, Gilbert's equation (7.3.1) can also be reduced to the moment system for (Yi,m)(t) by an appropriate transformation of variables and by direct averaging (without recourse to the Fokker-Planck equation) of the stochastic equation so obtained (e.g., [31-35]).

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Chapter 7. Three-Dimensional Rotational Brownian Motion 337

7.3.1 Langevin equation approach

Here, we obtain the hierarchy of differential-recurrence relations for the averages governing relaxation dynamics of single domain ferromagnetic particles directly from Gilbert's equation. The method we shall describe is analogous to that used in Section 7.2. Following [13,14,36], let us first transform Gilbert's equation (7.3.1) to the Landau-Lifshitz form [37]. Transposing the TJ term, we have

M+ p][MxM~]=y[Mx(U + h)] . (7.3.1.1)

On cross-multiplying vectorially by M in Eq. (7.3.1.1) and using the triple vector product formula

[ [ M x M ] x M ] = M ^ M - M ( M - M ) , (7.3.1.2)

we obtain

[ M x M ] =-p7AffM+y[[Mx(H + h ) ] x M ] . (7.3.1.3)

We remark that according to Eq. (7.3.1), (M • M) = 0. The substitution of Eq. (7.3.1.3) into Eq. (7.3.1.1) yields

(l+7272Ms2)M=j'[Mx(H + h) ]+7 2 7[[Mx(H + h ) ] x M ] . (7.3.1.4)

Finally, we have an explicit solution for M [12,14]

M =M*stf-1[Mx(H + h) ] - fc [Mx[Mx(H + h) ] ] , (7.3.1.5)

where b=j3/(2tN). It is apparent that Eq. (7.3.1.5) is similar to that for rotational Brownian motion of a dipolar molecule, namely, Eq. (7.2.6).

Following [38], we shall use a spherical coordinate system [25]

as shown in Fig. 7.3.1.1. In the basis (e r,e l,,e?,):

M=(M s ,0 ,0) , M=(0, Ms&,Ms sin0$) (7.3.1.6)

Fig. 7.3.1.1. Spherical coordinate system

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338 The Langevin Equation

and

H 1 dV

Ms du 0,

dV 1 dV (7.3.1.7)

Ms d& sinz?3#>y

Thus, Eq. (7.3.1.5) is equivalent to two stochastic equations for ??and <p.

Mt) = bMs\htt)-or\t)\

-b —vm)Mt)A-61>

9(t):

1

bM<

sin t?(0

asm&t)d(p

\hp(t) + cr\t)~\

d

Vm),<P(t)J] (7.3.1.8)

h-vW)Mi),i\+X vmt)Mt\n

(7.3.1.9)

smi^t)ys\wdt)dq) add

where the components he(t), hft), of the random field h(t) in the spherical coordinate system are expressed in terms of the components hx(t),hYt), hz(t), in the Cartesian coordinate system as [24]

h#(t) = hx(t) cos &(t) cos p(t) + hY(t) cos &(t) sin q>(t)-hz(t) sin •&,

h9(t) = -hx (t) sin <p(t) + hy(t) cos <pt), (7.3.1.10)

so that( according to Eqs. (1.17.5) and (1.17.6) of Chapter 1)

MO=yo=o, M O M O = V')V')=2rf/rld(t-1'). (7.3.1.H) We shall again use the Stratonovich definition [20] of the stochastic differential equations (7.3.1.9) and (7.3.1.10) as that definition always constitutes the mathematical idealisation of magnetic relaxation of superparamagnetic particles. Thus, one can obtain from Eqs. (7.3.1.8) and (7.3.1.9) the stochastic differential equation for any function/(r?, (p) of the angles z? and (p.

d(p d#

=-b —ym\<pi),i\- 1

orsin$/)<V y\m\¥s\t\ dv

sin^O^sint^O^

1 a v[m,<p(t),t]+-4-y[mMtit]\^-jm)Mt)] add

bMs

sin$/)L

dp

+fcMs[^(0-«%(0]^/[^0,^]+-^^(0+a%(0]|-/[*),^(0]

or, equivalently, in vector notation

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Chapter 7. Three-Dimensional Rotational Brownian Motion 339

f[W), <P(t)] = -bMsh(t) • [u(0 x V/MO, <p(t)]] + a"1 Y / W ) , ?(f)]

- W / M O , ?(*)] • W[t?(0,?(0, ' ] + a"1 [u(0 x VV[tf(0, P(0, *]]. (7.3.1.12)

Here, V is the orientation space gradient operator defined as [67] „ s a i a a V = — = u x — = -e# h e„ — ,

Sip 3u sintf 6<p r dtf where <5<p is an infinitesimal rotation vector. On averaging Eq. (7.3.1.12), we have

r-sO T

,t+T

-Urn- J wW),#/V]+cT\u(t')xVVW),<p(t'),t']]• VfW),<p(t')W

1 t+T

-bMs]im- f W)\W)xVf[m^t')+a'mW\(iKt%dt'. (7.3.1.13) r

Here, the overbar means the statistical average over an ensemble of particles which all have at time t the same magnetisation, i.e., the sharp values ??and <p of the stochastic variables (processes) &(t) and (p(t) at the averaging instant t.

The right-hand side of Eq. (7.3.1.13) comprises two terms, namely: the deterministic drift and noise-induced drift. Evaluation of the deterministic drift term [the first term in Eq. (7.3.1.13)] yields

t+T _

i

= -bvV(&,<p,t) + a-l[u(t')xVV(d,<p,t)]-Vf(#,<p). (7.3.1.14)

Let us evaluate the noise induced drift term

i t+T

-bhm- J vvm')Mt'),t] + a~\u(t')xVVW),<pitV]]- Wm'\<p(t')]dt'

< t+T

-Mf s l im-J W)\W)xY[+alv)fm'\<P(t')W• (7.3.1.15) r-*0r

t In order to accomplish this, we remark that a rotation of the vector u through the infinitesimal angle 8(p(t') = <p(?') - (p transforms the function / ( 0 , p ) into f[tKt'),<p(t')] so that [25]

fWt'),<p(t')] = e*>(,yvf(#,<p)~l + ap(t')-V)f(#,<p). (7.3.1.16)

By recalling that the infinitesimal rotation vector &p(t') is related to the

angular velocity a(t') as <o = Stp/St (see Sec. 7.6), one can obtain (o(t')

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340 The Langevin Equation

for the present problem by comparing Eq. (7.3.5) with the kinematic equation M(f') =(o(t')xM(t') so that

CXT) = h'Ms [u(0 x (H(0+h(0 ) ] - tf"1 (H(0+h(0) • (7.3.1.17)

Thus, according to Eq. (7.3.1.17) t'

<Sp(0 = /i'Ms J[u(Ox(H(0+h(0)]-«r" ' H(t")+h(t'))dt". (7.3.1.18)

Substitution of Eqs. (7.3.1.16) and (7.3.1.18) in Eq. (7.3.1.15) yields i t+t

-6M5lim- J tyty[\tt)xV(l+ap(tyV)]+alVl+S&t')-V)f(A<pW r - i O f

t^at u(0xvj J -flTVO• V+[u(r')xh(0] • v/(tf,##' ^ '

-or 1 t+T ('

ttr'Mjlim-J h(/)-V If -flT,h(0-V+[u(Oxh(0]-V/(i51fMf' T - J O T rff'

= Af(0,p)/(2TN), (7.3.1.19)

where A is the angular part of the Laplacian defined by Eq. (7.2.16). Here, we have noted the properties of the random field h, Eq. (7.3.1.11).

On combining Eqs. (7.3.1.14) and (7.3.1.19), we have the averaged equation

2TNm <p) = A/X# <p) - fiVV(#, <p, t)+cfl [u(0 x VV(t>, <p, 0] • V/(i?, q>),

(7.3.1.20) In applications, the relevant quantities are averages involving the spherical harmonics 7/,m(#0), which are defined by Eq. (7.2.19). Equation (7.3.1.20) accordingly yields for YUm($,(p)

2TNYlm=AYlm-p(vV + a-1[nxVV])-VYK ll,m

= Mljn-P dVdYlm+l dVdY^

3t? 3tf sin2??3^ 3#> +-asintf

dVdYLm dVdYLn

= AY,m +£[VAYljn +YLmAV-AVY,m)] + P orsint?

d<p d# d# d<p

'dVdYLm dvdY^

d<p d& 3tf 3#> (7.3.1.21)

For «->«>, Eq. (7.3.1.21) reduces to Eq. (7.2.20). Next, we express the right-hand side of Eq. (7.3.1.21) in terms of the orbital angular

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Chapter 7. Three-Dimensional Rotational Brownian Motion 341

momentum operator L = -/V [25] [cf. Eq. (7.2.15)]. We recall first that

Lz, L±, Z? are defined as [25]

Z?=-A, 4 = - i — , U = e±i(p ± — -fzcotz?—

3*9 d(p) (7.3.1.22)

One may now radically simplify the solution of the problem by using the theory of angular momentum [25] because the action of the angular momentum operators on Y^m is [25]

Lz*l,m - m*l,m '

4 y ; j m = ^ ( / + l)-m(m±l)F / j m ± 1 , (7.3.1.23)

Thus, we obtain from Eqs. (7.3.1.21) and (7.3.1.23)

^/,m=f[£l(vi'^)-vz?y/,m-y^£1v]-ii?y/im

- ^ ^ y u [ ( 4 v + ) ( ^ ^ ) - ( ^ V + ) ( 4 ^ ) ] (7-3.1.24)

+ -i [(Lzv_)(L_Yltm) - (Lv_ )(4r/>m)], where we have used the following representation for the expansion of V in terms of spherical harmonics

V =V+ +V_, V+ = I ; £ v w r w , V_ = f) 2 W*, s . (7.3.1.25) /?=1S=0 R=IS=-R

Furthermore, the functions Fy1 and Y^\ may also be considered as

operators, which act on Yt<m in accordance with the rule (m > 0)

_ l&r(2/ + l ) ( l - M ) i " | (2L+l)(L+m-l)!

where AL = 2 and f im = 1, if the indices / and m are of the same order of evenness and £/,m=0 otherwise. Equation (7.3.1.26) is the solution of the recurrence equation

8^(2/-1) (2/+ 1) (2/ + l)(/ + m)(/ + m - l ) -l W / , m ^ 3 ( Z ± m ) ( / ± m _ l )

F ' - > ^ + ^ 2 / _ 3 ) ( / ± m ) ( / ± m _ l )F i " ^ - 2 . '

(\m +1| < / -1 ) , which follows from the known relation [25]

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342 The Langevin Equation

& y l±m + l)l±m + 2) _ (/ + m- l ) ( / + m)

V 3 '•" l'm \ (2/ + l)(2/ + 3) /+1'm±1 \ (2/ - l ) (2/ + l) M 'm±1 '

(7.3.1.27) By using Eqs. (7.3.1.23) and (7.3.1.26) and Eq. (7.1.21), we can also obtain

y - i f v _ . 8*(2/ + l ) ( / -m)! « (ZLj^ jL + m - ^

(7.3.1.28)

\ 3(l + m)\ L=m+eltm\ [L-m)\

(here m > 0, AL = 2). Noting Eqs. (7.2.24), (7.3.1.26)-(7.3.1.29), we can

transform Eq. (7.3.1.24) to [35]:

TN ~ r^ / ,m = 2^e/,m/,m+.s*V,m+.s- ' ( 7 . 3 . 1 . 3 0 )

where

c/,m r,m±., - — r ¥ r t + ( - 1 ) T ) |

-r(r + l)-/(/ + l)] 2V27TT c/,o/,o^/,m/,-mT.

- f[r(r+l)-r(r + l ) - / ( / + l)] v Z j V r , ± . s 1 _ / T — — H o / , 0 1 - /

i (2r + l)(r-s)\ -J ( L + £ - l ) ! Li0

«V (' + *)! , A J ( L - + 1) ! '0/'° AL=2

m>/(Z, + s)(L-s + l ) C ^ _ m T , ±sj( l + m)(l±m + l ) C $ g r , - » *

(7.3.1.31) [heres>0].

The quantities F/m in Eq. (7.3.1.30) are again functions of the sharp values $ and (p, these are random variables with the distribution (probability density) function W. Therefore, we must also average Eq. (7.3.1.30) over W. Thus, we may derive the moment system for the averaged spherical harmonics, namely

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Chapter 7. Three-Dimensional Rotational Brownian Motion 343

*A(Y,„)(t)= X W-m'^/wX') • (7.3.1.32) " ' l',m

where the angular brackets ( ) designate averaging over W. Just as in dielectric problems (see Section 7.2), one can readily derive from Eq. (7.3.1.30) a hierarchy for the appropriate equilibrium correlation

functions clm(t):

tN~TCl,m(t) = Z-i e/,m,/+r,m+.vC/+r,m+.s'(?). ( 7 . 3 . 1 . 3 3 ) d t s,r

where Cimt) are the equilibrium correlation functions given by Eq. (7.2.29).'

Equations (7.3.1.32) and (7.3.1.33) constitute the general solution of the problem and allow one to evaluate the matrix elements el my^ for a given magnetocrystalline anisotropy (see Chapter 9).

7.3.2. Fokker-Planck equation approach

In Section 7.3.1, we have derived an equation for the matrix elements ei,m,i',m using the Langevin equation. Here, we show how the same results can be obtained from the Fokker-Planck equation (7.3.4). Thus, we seek the solution of Eq. (7.3.4) as:

W(0,qf,t) = V(0,q>,ty¥\0,p,t), (7.3.2.1) where ^fycpj) is given by

W,p,O = Z/z.m(Ol'/,IB(0,P)- (7.3.2.2) l,m

The normalisation condition for W(&, <p, t) is

j jW(#,<p,t)sm&diM<p = Z\flm(t)\ = 1 . (7.3.2.3) 0 0 l,m

Equation (7.3.2.1) differs from that used by other authors (for example, [14,15,29]), where W itself was sought in the form of Eq. (7.3.2.2). The representation (7.3.2.1) has the advantage that it is unnecessary to apply additional conditions to the distribution function in order that it should be physically meaningful (e.g., W should be positive and real). Moreover, the direct quantum-mechanical analogy is obvious because the function W is now similar to the probability density |*P |2 (¥ is the wave function), which obeys the continuity equation ([39], p.75):

| - | ¥ | 2 + d i v j = 0,

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344 The Langevin Equation

where j is the probability current density . On using Eqs. (7.3.4), (7.3.2.1) and (7.3.2.2), we can now obtain

the moment system by means of the transformation

- ( ^ > ( 0 = f J^WsintWiM^ at oo

Inn

= z /v(o/;v«jji//,m^(i'/>^v)sin^^^ l',l",m',m' 0 0

v J(2Z' + l)(2/* + l) r o rm~

/'./"./"m'.m'VV 4 ; r l 2 / + 1 )

x J \YljnLFPYr^sintWdrfp = £</ , .„ , , / > m ( r r v ) (0 , (7.3.2.4) 0 0 /',m'

where

2;r;r

0 0

are the matrix elements of the Fokker-Planck operator LFP and Inn

(1/.m>(0= J J r/im(i?,p)W(i?,p,OsiniWiWp 0 0

_ l ( a + l ) ( y + l ) ,,0 ,,„• . < 7 ' 3 ' 2 ' 6 )

~~ Z-i 4 A.Tt(7]"+ \\ l>°<1'0 l,m,l\mJl,m JI ,m V ) JI ,m "VJ-

m',m"

We express the Fokker-Planck operator LFP in Eq. (7.3.2.5) as before in

terms of the angular momentum operators Lz, L+, Z?. We obtain [cf. Eq.

(7.3.1.24)]

^fci(W(WMW&M 4T, (7.3.2.7)

+y_ I [(4v)(^v)-(^-)(4^v)]. where the V+ are given by Eq. (7.3.1.25). Furthermore, by using Eqs. (7.3.1.23), (7.3.1.27) - (7.3.1.29), and [25]

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Chapter 7. Three-Dimensional Rotational Brownian Motion 345

2?" , (2/ + l)(2/ ' + l) r o . . . J \Y^YrAn-sinMMcp = J 4^(2r+l) C^''Am'W-

we have the matrix elements of the Fokker-Planck operator [35], namely

0 0 ZTN

+ H ) ^ j * £ V H 2V27 qo / ' °q- ' ' - -mT>v

+5A/(/'±m + S ) ( / ' + m - . + l ) C ; ^ m T S ± 1 ] ^ (7.3.2.8)

By means of the known relation [25]

W3,m, = ( / 3 ±^) (^3+^+ l )w^ T i _ (/1+m1)(/1±m1+l) , ^

comparing Eqs. (7.3.1.31) and (7.3.2.8), one may prove that dlWJ,m=ei,m,l',m'/TN- (7.3.2.9)

Equation (7.3.2.9) demonstrates clearly that both of the moment systems given by Eqs. (7.3.1.32) (from the Langevin equation) and (7.3.2.4) (from the Fokker-Planck equation) are in complete agreement. This result also proves the equivalence of the Langevin and the Fokker-Planck equation approaches.

In order to apply these general results, we shall consider, as a simple example, the free energy per unit volume of a uniaxial particle subjected to a constant dc magnetic field H0 is (see Section 1.16)

PV = -^co^-acos^ = - ^ Y l f i - f ^ Y w - ^ , (7.3.2. .10) y J • i n 5

where a = vK/(kT) and t, = vMsH0 lkT) (7.3.2.11)

are the dimensionless barrier height and external field parameters, K is the anisotropy constant Then on using Eqs. (7.3.2.8) and (7.3.2.10) together with the explicit representation for the Clebsch-Gordan coefficients [25], we obtain:

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346 The Langevin Equation

dt Y +

n(n +1) itm n(n +1) - 3m — - + — a-

2a (2w-l)(2n + 3)

= a n + \ (n2 - m 2 ) [ ( n - l ) 2 - m 2 L

2n-Y (2n-3)(2n + l) n—2,m

/ [Qi + 2)2-m2][(tt + l ) 2 - m 2 ],

(2n + 5)(2n + l) n+2,m

(« + l ) 2 - m 2

Ln+l,m' '(2n + 3)(2n + l) (7.3.2.12)

One may derive from Eq. (7.3.2.12), just as in Section 7.2, an infinite hierarchy of differential-recurrence relations for the appropriate equilibrium correlation functions.

In particular, one can obtain from Eq. (7.3.2.12) the hierarchy for the longitudinal correlation functions fnf) defined by Eq. (7.2.32):

rN-fn(tH n(n +1) ann +1)

(2/i-l)(2n + 3) /.W = |^ [ /„ - , (0 - / . . + , ( 0 ]

2(2n +1)

^ _ 1 ) fn-2(t)-a:(n+^n + 2:!fn+2(t). (7.3.2.13) (2n - l)(2n +1)"" ' " (2n + l)(2n + 3)

It is apparent from Eq. (7.3.2.13) that the gyromagnetic terms no longer influence the longitudinal relaxation. Moreover, for £=0 or <r=0, Eq. (7.3.2.13) reduces to Eqs. (7.2.33) and (7.2.39), respectively. Thus, both dielectric and magnetic longitudinal relaxation in uniaxial potentials are governed by the same equations.

Similarly, we have the hierarchy for the transverse correlation

functions gn (t) = (cos (0)P/ [cos #(0)]eim*Pxn [cos tf(0l)

dt gn(t) +

n(n +1) i£ n(n + l ) - 3

2 la (2n-l)(2n + 3)

(n + \)(n + \ i<7S

2n + l l 2 a£, « n - l ( 0 ~ n

2n + \ —+ 2 ag)

o

8n+l(t) (7.3.2.14)

+a nn + \f

(2n-l)(2n + l) «n-2(0-

n2(« + l)

(2M + l)(2w + 3) Sn+lO)

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Chapter 7. Three-Dimensional Rotational Brownian Motion 347

In Chapter 8, we shall show how Eqs. (7.3.2.13) and (7.3.2.14) can be solved exactly in terms of matrix continued fractions.

Thus, we have demonstrated how the infinite hierarchies of differential-recurrence relations governing the orientation relaxation of a Brownian particle in the presence of an arbitrary external potential may be obtained directly from the vector nonlinear Langevin equations, namely: the Euler-Langevin equation for a polar molecule and Gilbert's equation for a single domain ferromagnetic particle. Such a procedure eliminates the excessive step in the theory of constructing and solving the corresponding Fokker-Planck equation entirely.

7.4 Brownian Rotation in the Uniaxial Potential

Here, we comprehensively treat rotational Brownian motion in the uniaxial potential

V(&) = Ksin2d. (7.4.1) Formally speaking, the longitudinal relaxation is mathematically identical for both magnetic and dielectric relaxation. In general, the transverse relaxation is complicated to some extent by the gyromagnetic term in the Landau-Lifshitz-Gilbert equation which will give rise to ferromagnetic resonance. However, omitting this term the underlying Langevin equation will be mathematically identical to that used in the theory of dielectric relaxation of nematic liquid crystals with uniaxial physical properties given by Martin et al. [7]. That theory bears a close resemblance to the theory of magnetic relaxation of single domain ferromagnetic particles [13,41,42] and the dynamic Kerr effect [3,6]. Following [43,44], we will consider the dielectric problem here.

7.4.1 Longitudinal relaxation

As shown in Section 7.2, the longitudinal dielectric relaxation in the uniaxial potential (7.4.1) is governed by Eq. (7.2.39), viz.,

2a d n(n + l) rD—fn(t) + dt " 2

1 (2n-l)(2/i + 3) /„(')

<7n-\)n(n + l) an(n + \)(n + 2) r

(7.4.1.1)

(2n + l)(2n -1) In + l)(2n + 3)

for the equilibrium correlation functions fn(t) = (cos*?(0)Pn[cos*?(0])0-

By inspection of Eq. (7.4.1.1), it is obvious that it decomposes into sets for even- and odd-index /•(*). Here, only the odd-index f2n+i(t) are of interest since we seek/i(0. The odd-index/2„+i(f) satisfy

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348 The Langevin Equation

— / 2 B + I ( 0 = at

(2n + l)(n + l) 2<7

+ 4<T(n + l)(2n + l)

7D(4n + 3)

(4n + l)(4n + 5)

2« + 3

- 1 / 2 B + I ( 0

4n + l /2„-i(0- /2«+3(0

(7.4.1.2)

2(4n + 5)' The set of Eqs. (7.4.1.2) may be solved numerically by

transforming it into the matrix equation

X(f) = AX(f), where

A =

?. — G 5 24 — G 35

0

2 -G 5

, 4 6 G 15 40

G 33

0

20 — G 21

i< 1 0

15 G 39

0

0

210

143 G 0

X(0 =

—j

(7.4.1.3)

/ i (0 '

/3(0

v/2«+i(0.

(7.4.1.4) In Eq. (7.4.1.3) n is taken large enough (equal to P say) to ensure convergence of the set of equations above. The lowest eigenvalue, which corresponds to the reciprocal of the longest relaxation time, is then the smallest root of the characteristic equation

det(Al-A) = 0. (7.4.1.5) We recall (see Chapter 2, Section 2.7.2) that the relaxation modes of f(t) may be found from Eq. (7.4.1.3) by assuming that A has a linearly independent set of P eigenvectors (R,,...,RP) so that

(7.4.1.6) (7.4.1.7)

and R ' A R = B

X(0 = ReB 'R - 1X(0), where R is the matrix consisting of the n eigenvectors of A,

B o 0

v •••

0

0

0

0

-4 eB' =

0

0

0 0

0 (7.4.1.8)

and X(0) is the matrix of the initial values of/2„+i(0- The solution of Eq. (7.4.1.3) may then be exhibited as

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Chapter 7. Three-Dimensional Rotational Brownian Motion 349

X(0 = V - i V R , + V ~ ^ ' R 2 + - + bpe'^^'Rp, (7.4.1.9) where the bt are to be determined from the initial conditions. The initial value vector X(0) is determined as follows. The initial value of f„(t) is

1 2 / ' 2 /„ (0) = I xPn (x)eax dx/j eax dx. (7.4.1.10)

- l / - l

The functions /„(0) may be expressed in terms of the confluent hypergeometric functions M (a, b, z) [45] as shown in Section 7.4.2.

Equation (7.4.1.3) may be solved to any desired degree of accuracy to yield fx(i) in the form

^£= IX+i^+" /fX+, • (7-4.1.11)

The quantity of most interest is the correlation time tj| which is the area under the curve of the longitudinal autocorrelation function. The longitudinal autocorrelation function is

q | (0 = / 1 (0/ / 1 (0) = (cos^0)cos^(0)0/(cos2z?(0))o (7.4.1.12)

so that the correlation time T\\ (Chapter 2, Section 2.9) is oo

TJ, = Jc„(r)rfr = Z^k+A /HkA2k+l • (7.4.1.13) 0

Another quantity which we shall require is the effective relaxation time defined as (Chapter 2, Section 2.12)

tf =Z*A2*+i / Z * 4 A + I 4 * + I • (7.4.1.14) We recall that the effective relaxation time is the time constant associated with the initial slope of the correlation function. It contains contributions from all the eigenvalues as does the correlation time. The behaviour of the correlation time and the effective relaxation time is sometimes similar. However, we shall see explicitly below that if different time scales are involved, the behaviour of these times can be quite different.

7.4.2 Susceptibility and relaxation times

On taking the Laplace transform of Eq. (7.4.1.1), we have

cTnfn-2(s) + (qn-sTD)fn(s) + q+fn+2s) = TDfn0),n>\), (7.4.2.1) where

n(n +1) In i ?2

(2n-l)(2/i + 3) (7.4.2.2)

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350 The Langevin Equation

a(n-l)n(n + l) crn(n + l)(n + 2)

(2n + l)(2n-l) (2n + l)(2n + 3) As shown in Chapter 2, Section 2.7.3, Eq. (7.4.2.1) can be solved exactly

for f(s) in terms of ordinary continued fractions to yield

/,(*) = ^ 7 T / I ( 0 ) + S f2n+l(0)Uq+2k-lSLl(S)\ , (7.4.2.4)

where the infinite continued fraction SJ](s) is defined by the following recurrence equation

Si (*) = <fn [*DS 'In- ti4+2 (*)]"' (7-4.2.5) or, equivalently,

2an-\)

Sl(s) = — — ^ = 1 — . (7.4.2.6) 2rDs n 2<T , 2a(n + 2) ,„

n(n + l) (2n-l)(2/i + 3) (2n + l)(2n + 3) "+

Equation (7.4.2.4) may further be simplified if we write the product

f l («2*-i7 42*+i) explicitly. We have k=\

rrQot i „(4« + 3)r(n + l/2) T T ^ L «v— ; I ; (7.4.2.7) liq-k+l 3r(n + 2)r(l/2)

where T(z) is the gamma function [45]. So, Eq. (7.4.2.4) reduces to

Tn r £ , (4n + 3)r(« + l/2) " H

/.(*)=——[/i(0)+y(-i)72n+i(0)- * - ^ — r ^ n ^ L + i W 1 G(<r,sy l ti 3r(n+2)r(i/2) i l 2*+1

(7.4.2.8) where

G(cr,s) = rt-D+l-2o-[l-5|(s)]/5. (7.4.2.9) The initial values /2n+1(0) given by Eq. (7.4.1.10) can be

expressed in terms of the equilibrium averages as /2 n + i (0 )=-^- r [2(n + l)(P2„+2)o + (2n + lXP2n)0]- (7.4.2.10)

4n + 3 Here, we have used the equality [45]

(2n + l)xPn(x) = (n + l)Pn+l(x) + nPn_l(x). (7.4.2.11)

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Chapter 7. Three-Dimensional Rotational Brownian Motion 351

Now, the equilibrium averages (P„)0 satisfy Eq. (7.4.1.1) with the time derivative set equal to zero, viz.,

n(n + l) 1-

2a

(2n - l)(2n + 3) (Pn)o

= cr(n-l)n(n + l) <7n(n + l)(n + 2)

(2n + l)(2n -1) ^ ""2;° (2n + l)(2n + 3) ^ " + 2 ; ° ' Hence, on noting Eq. (7.4.2.6), we have

(Pn)o/(Pn-2)0=Sl(0),

where SJj(O) is given by Eq. (7.4.2.6) at s = 0, namely:

(7.4.2.12)

(7.4.2.13)

2cr(n-l) 2cr 2a(n + 2) -l- i

Sl+2(0) .(7.4.2.14) 5 (0) = - ^ 1 +-4 n 2 - l L (2n-l)(2n + 3) (2n + l)(2n + 3)

Thus, according to Eqs. (7.4.2.10) and (7.4.2.13), the initial conditions fin+i (0) m a v ^ e expressed as

/ M ( 0 ) = ^ 4 ( 0 ) . J » ( 0 ) 4« + 3

2n + 2

2n + T 2n+2 (0) + l , (7.4.2.15)

(n > 1), so yielding the initial conditions entirely in terms of Sn (0).

The continued fraction SJJ(0) can be expressed as a ratio of confluent hypergeometric (Kummer) functions M(a,b,z) defined as [45]

v ; b b(b + \) 2! b(b + l)(b + 2) 3! + ... (7.4.2.16)

This may be accomplished by noting that Eq. (7.4.2.14) can be rearranged to yield after simple algebra

•5jJ(0) = . (7.4.2.17) 1

1 + -

2<r(n-l) (2ra + l ) (2n-l)

1 — 2cr(n + 2) -[l-SJL(O)]

(2n + l)(2n + 3) On comparing Eq. (7.4.2.17) with the continued fraction ([46], p.347)

Ma + \,b + \,z) 1 M(a,b,z) z(b-a)

b(b + l)

, (7.4.2.18)

z(a + Y) M(a + 2,b + 3,z) 1 + -(b + l)(b + 2) M(a + \,b + 2,z)

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352 The Langevin Equation

5ii (0) = ; ; 2 ZZT ::,7.. „ * - ' • (7.4.2.22)

we can see that Eq. (7.4.2.17) is identical to Eq. (7.4.2.18) if z = -a, a = n/2, and b = n - 1 / 2 . Thus we can write

5U(Q) = 1 _ M ( l + n/2,,z + l/2,-cr) . (7.4.2.19) M(nl2,n-\l2,-a)

On using the Kummer transformation ([45], Eq. (13.1.27))

M(a,b,z) = ezM(b-a,b,-z) (7.4.2.20) and the recurrence relation

CL7

M(a,b-l,z)-M(a,b,z) = —Af(a + l,fc + l,z), (7.4.2.21) b(b-l)

we can rearrange Eq. (7.4.2.19) as follows: 2 ( n - l ) g M((rc + l)/2,n + 3/2,<T)

(4n 2 - l ) M((n-l)/2,n-l/2,a) Thus, on using Eq. (7.4.2.22) in Eq. (7.4.2.15), we obtain

o*r(n + 3/2)M(n + 3/2,2n + 5/2,a) /2n+i(0) = ~ , l—\—7 T2-1- (7.4.2.23) J2n+lK 2r(2« + 5 /2)M(l /2 ,3 /2 , a)

Equation (7.4.2.8) allows one also to calculate the longitudinal susceptibility o^(ct)) - a\co) -ia"(co) since according to linear response theory (Chapter 2, Section 2.8)

^ ) = 1 _^ lM, (7.4.2.24) <(0) /i(0)

where

oftO) = SNoW) = M2N0M3/2,5/2,CT)

^ kT 3kT M\l2,3l2,a)

is the static susceptibility, fj. is the dipole moment of the molecule, and N0

is the number of polar molecules per unit volume. (We remark that the susceptibility, Eq. (7.4.2.24), differs from the polarisability introduced in Chapter 2, Section 2.8 by the factor N0 only). Thus, we have

^(O)) ian^

<(0) ~ ion +1 - 2(j |"l - 53(ifl))l/5 L J (7.4.2.25)

The most significant feature of Eq. (7.4.2.8) is, however, that it yields an exact expression for the longitudinal correlation time. On setting s = 0 in Eq. (7.4.2.8), we have,

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Chapter 7. Three-Dimensional Rotational Brownian Motion 353

T» = /l(0) = *o

MO) l -2o [ l -5 3 (0 ) ] /5

x 1 + V r nH/2n+l(0)(4" + 3) r(" + 1/2)fTcll

U /,(0) 3r(n + 2)r(i/2) ! i 5 " + 1 (0)

(7.4.2.26)

On using Eq. (7.4.2.22) and properties of Kummer's functions [45] M(0,b,z) = \,

(7.4.2.27) bM(a,b,z)-bM(a-l,b,z)-zM(a,b + l,z) = 0,

we can express the leading term of Eq. (7.4.2.26) as

1-2o_ 5

M<0)]| = 1-2cxM(l,7/2,<7)

- H

= M(1,5/2,<7). (7.4.2.28) 5 M(l,5/2,<7).

On taking into account Eqs. (7.4.2.22), (7.4.2.23), and (7.4.2.28), Eq. (7.4.2.27) may further be simplified to

3/2 oo

IK) n (n + 3 /4) r (n + 3 /2 ) r (n + l/2)

TD M (3/2,5/2,(7)^ ' (n + l ) r 2 (2n + 5/2)

xM(n + 3/2,2n + 5/2,a)M(n + \,2n + 5l2,o), (7.4.2.29) which is the exact solution in terms of known functions. Equation (7.4.2.29) may be used to calculate the relaxation time for all values of a (see Table 3 of [43]). We remark that all the Kummer functions appearing in Eq. (7.4.2.29) may be expressed in terms of the more familiar error functions of real and imaginary arguments erf (z) and erfi (z) [47].

As shown in Ref. [48] (see below Section 7.4.3), the series in Eq. (7.4.2.29) can be summed analytically to yield

3e°

TD <j2M(3/2,5/2,(j)

cosh[(7(l-z2)]-l

1 - z 2 dz. (7.4.2.30)

Equation (7.4.2.30) is the exact solution for the relaxation time Tj| rendered in integral form. Equation (7.4.2.30) can readily be obtained fromEq. (2.10.25) of Section 2.10 with %\ =-l , ;c2 = - l , and

D(2)(X) = (1-X2)/(2TD)

[by recalling that Mid))//^O) in Eq. (7.4.2.24) is the one-sided Fourier

transform of the autocorrelation function C^(t), Eq. (7.4.1.12)].

An approximation that has been used to estimate the longitudinal relaxation time is the effective relaxation time [49] (see Chapter 2, Section 2.12). The effective relaxation time can be found by evaluating Eq. (7.4.1.1) for n=\ aW = 0. We have

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354 The Langevin Equation

Figure 7.4.2.1. Exact solution for the longitudinal correlation time [solid line: Eq. (7.4.2.30)] compared with the solution rendered by the effective eigenvalue [dashed line: Eq. (7.4.2.32)] and the inverse of the smallest eigenvalue [filled circles: Eq. (7.4.2.34)].

TD/,(0) + [ l -2a /5 ] / 1 (0 ) = - (2a/5) / 3 (0) . The effective relaxation time is then

*f

rD/,(0) i-

2(7 | 2<T/3(0)

5 5 /,(0)

-i-i

2cr 12cT M(5/2,9/2,cr)

5 + 175 M(3/2,5/2,(7)

-i-i

(7.4.2.31)

(7.4.2.32) M(3/2,5/2,(7)

M(l/2,5/2,(7) Here we have used the recurrence Eqs. (7.4.2.21) and (7.4.2.27) for Kummer's functions. It is apparent from Fig. 7.4.2.1 that the effective eigenvalue approach is inadequate when applied to the longitudinal relaxation, as noted in [48], because the effective relaxation time cannot reproduce the behaviour of the correlation time in the large a limit.

In the context of the continued fraction method, the smallest eigenvalue is given by the approximate Eq. (2.11.1.11) of Chapter 2, Section 2.11.1, viz.,

TDA ~ — q1+qtS3(0)

1 + Z f l S22m+1(0)<72m-i/<7:

(7.4.2.33)

2m+l k=\m=\

where qu qm, and q^ are given by Eqs. (7.4.2.2) and (7.4.2.3). Thus we obtain from Eqs. (7.4.2.2), (7.4.2.3), and (7.4.2.33) [50]:

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Chapter 7. Three-Dimensional Rotational Brownian Motion 355

4 = 3^Tn

8M(l,5/2,cr)^

f O2)" r(2n + l)M2(n + l,2n+5/2,(7)

(n + l)r(2n + 3/2)F(2n+5/2)

T-l

, (7.4.2.34)

which is the solution in terms of known functions. For a « 1, one can determine from Eq. (7.4.2.34) the Taylor series expansion of X:

ArTD=l-f + ±S+0(S). (7.4.2.35)

On the other hand, on using the asymptotic expansion of Kummer's functions [45]

M(a,b,z) = ezz' - ^ - f c l ^ J y r(fr-a + n ) r ( l - a + n) Z ^ ± + 0(z-s)\, (7.4.2.36)

a)z"n! l ')

one obtains for a » 1 Brown's asymptotic formula [13]

4 -2cr 3 / V g

(7.4.2.37)

It should be noted that for practical calculations of both the longitudinal relaxation time and the smallest eigenvalue, one may use the simple empirical equation [51,52]

4»* - i ( 2a .3/2 N

TD(ea-l) 7^(1 + 0") J (7.4.2.38)

which is valid for all values of a [51,52].

7.4.3 Integral form and asymptotic expansions

In order to write our series solution for rn / TD, Eq. (7.4.2.29), in integral form, we note the formula from Bateman [53], [Vol. 1: Section 6.15.3, Eq. (18)]

M(a,b,z)M(a,b,-z) = T\b)z 1-6

J /6_1(zsechOsechfe(*"2a)'rfr, Ta)T(b-a)_

Re[a]>0, Re[b-a]>0, (7.4.3.1) where Ib-i(z) is the modified Bessel function of the first kind and of order b - 1. In order to apply the above formula to the series, Eq. (7.4.2.29), we note that the Kummer transformation Eq. (7.4.2.20), namely

M(a,b,z) = ezM(b-a,b,-z), on taking a = n + 3/2,b = 2n + 5/2, becomes

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356 The Langevin Equation

M(n + 3/2,2n + 5/2,a)M(n + l,2n + 5/2,cr) (7.4.3.2)

= eaM(n + 3/2,2n + 5/2,a)M(n + 3J2,2n + 5/2, -a) so casting Eq. (7.4.2.29) into a form suitable for conversion to an integral. Thus, that equation becomes

f[l _ 3/2 y(-g-2)"(tt + 3/4)r(n + 3/2)r(rc + l/2)

TD ~ Af (3/2,5/2, a) & (n + \)T(2n + 5/2)

e a c r - 3 / 2 - 2 „ r ( 2 n + 5 / 2 ) -x -J—L- e ' sech t L„,?h (asecht)dt

T(n + l)r(n + 3/2) £ 2n+3/2

3eCT<7~3/2 ^ , (- l )"(n + 3/4)r(n + l /2)r r , . ,cosh(f/2), = - - V ^—^ l-J—- '—*• \ I2n. 3/2 (crsech t) dt.

M (3/2,5/2, a)^ T(n + 2) J 2n+3/2 coshr (7.4.3.3)

On using the change of variable sech? = sin#, Eq. (7.4.3.3) reduces to *i. 3ea K? /T+sin? 1 xR^-^W' ^7-4-3-4)

J X T c i n /J TD criUM (3/2,5/2,a) J0 V 2sin<9

where

FW=£(-i)-(^3/4)n„+1/2) ffsine, (7A3.5) n=0 r (« + 2)

The series, Eq. (7.4.3.5), can be summed to yield

F(0) = - 7 ==L=r C osh(crs in6O-l l . (7.4.3.6) V2o-sin6>L J

Thus

—!L = — cosh((7sin#)-l d # . (7.4.3.7) rD 2cr2M(3/2,5/2,0") J

0 sin<9 L J

By means of the transformation 0 -> arcsin(l - z 2 ) , Eq. (7.4.3.7) can be reduced to Eq. (7.4.2.30).

Now, we may apply the method of steepest descents to obtain the asymptotic expansion of the exact solution. In order to apply this method, we note that the exact solution Eq. (7.4.3.7) has no singularity at 0=0 and has a saddle point at 6- Ttl 2. Since the saddle point is at 6= KI 2, it will be convenient to replace 0by Kl 2- dm Eq. (7.4.3.7) so that 6= 0 is now the saddle. Thus,

ii y J e°™oGe)dd, (7.4.3.8) TD 2a2M(3/2,5/2,(7) J

0

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Chapter 7. Three-Dimensional Rotational Brownian Motion 357

where

Gm = ^^-(l-e-^f. 2 cos 6

In the limit a » \ , let us now write

•J,

(7.4.3.9)

(7.4.3.10) TD 2a2M'(3/2,5/2, a)

where, in accordance with the method of steepest descents,

J=\ [G(0) + ^G"(0) /2 + . . . ] / ^ 2 / 2 ^ / 2 4 + - U . (7.4.3.11) o

On noting the asymptotic form of M(a,b,z), Eq. (7.4.2.36), and on neglecting exponentially small terms, we have

V^ a e a

-3/2 r 1 1 + —+ ... v o

(7.4.3.12)

Equation (7.4.3.12) is compared with the exact solution and Brown's asymptotic formula [13]

TD4nea

h~^2- C7A3.13)

in Table 7.4.3.1. It is apparent that Eq. (7.4.3.12) reproduces the asymptote far more accurately than Brown's asymptotic formula Eq. (7.4.3.13) for a > 2.5 (see Table 7.4.3.1). If the a~2 term is included in the asymptotic expansion, using the method of steepest descents, one finds after a tedious calculation:

4x e'er-3'2 i + i +

2al (7.4.3.14)

Table 7.4.3.1. Numerical values for the relaxation time.

a 1 2 3 4 5 10 18 26 38 50

Eq. (7.4.2.30) 1.528 2.460 4.198 7.606 14.59 691.02 8.084 105

1.362 10" 1.238 1014

1.326 101"

Eq. (7.4.3.12) 4.818 3.473 4.568 7.560 14.12 679.02 8.043 105

1.359 10* 1.237 1014

1.326 10'y

Eq. (7.4.3.13) 2.409 2.315 3.426 6.048 11.76 617.3 7.620 105

1.308 10" 1.205 1014

1.300 101"

Eq. (7.4.3.14) 8.432 4.341 5.139 8.127 14.82 688.3 8.078 105

1.362 10" 1.238 1014

1.326 10'"

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358 The Langevin Equation

Equation (7.4.3.14) provides an even closer approximation to the asymptotic behaviour for large a. Equation (7.4.3.14) should be compared with the asymptotic expansion of the inverse of the smallest eigenvalue which is [13]

- — e a ' 1 + —+ —=- + .... ATD a Aa2 (7.4.3.15)

7.4.4 Transverse response

The set of equations governing the relaxation of the transverse component of the dipole moment following the removal of a small electric field applied in the transverse direction is given by Eq. (7.2.40), viz.,

^D—8i(t) = qIgi-2(t) + %8iO) + qt8i+2(t), ( /£1), (7.4.4.1)

at where

1(1 + 1) /(/ + l ) - 3 ,nAA^ q, =— - + <J— , (7.4.4.2)

2 (2/-l)(2/ + 3) /(/ + 1)2

+ cr/2(/ + l) , n t t ^ qt =<T - 1 q+= i ! _ . (7.4.4.3)

(2/- l)(2/ + l) (2/+ 1)(2/+ 3) Just as the longitudinal response, Eq. (7.4.4.1) can be solved exactly for the Laplace transform g^s) in terms of ordinary continued fractions to yield

ft(j)= ^ I T T U W + Z ^ C O ) ] ! ^ ^ ^ , (7.4.4.4)

TDs~q\ -qisi(s) [ n=\ k=\ fe+i J where the infinite continued fraction SJ~(s) is defined by

S^(s) = qj [TDS -qt- ql S^+2(s)j

or, equivalently, 2a(l + l)

i - i

2TDS il 2 q 1 " 3 / [ / ( / + 1)] I 2al S1 (s) 1(1 + 1) (2/-1)(2/+ 3) (2/+ 1)(2/+ 3) /+2

On noting that

.(7.4.4.5)

T]&-I= (-i);a+3/4)r(/+i/2) ( ? 4 4 6 )

H q-lk+l (2/ +1)(/ +1)21X5/2) H / +1) '

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Chapter 7. Three-Dimensional Rotational Brownian Motion 359

Eq. (7.4.4.4) may be further simplified as

STD +1 + cr/5 + 2aSl (s)/\5

r (0) l Y (-1)'g2m(Q)(/ + 3/4)r(/ + l/2) ' ± .

*l(0) + £ r(5/2X/+i)2(2/ + i)r(/ + i) i r 2 i + l W (7.4.4.7)

= fp y (-D/g2/+1(0)(/ + 3/4)r(/ + l / 2 ) T l T , 1

o-V^itS (/ + i)2(2/ + i)r(/ + i) J i 2*+1

The initial conditions may be calculated just as the longitudinal case. We have

g,(0) = (/f(cosi?)/f(costf)cos2^ =^F11(cos^)P/(cosz?)) / 2 . (7.4.4.8)

We now make use of the recurrence relation [47]

(2v + 1)P? (x)P (x) = v(v +1) [Pv+1 (x) - Pv_, (x)] (7.4.4.9)

so that with m = 1 Eq. (7.4.4.8) reduces to

/(/ + !)

2(2/+ 1)1

Equation (7.4.4.10) may also be computed in terms of SJ'(O) and, in turn,

in terms of Kummer functions [see Eq. (7.4.2.22)]. We have

(2/ + l)(2/ + 2)ri-4+2(0)l W 0 ) = ^ ^ J4(0)4-2(0)...s!!(0)

ftw=S[<PM>»"(l?+i>J- (7-4A10)

(/ + l)o-'r(/ + 3/2)M(/+ 1/2,2/+ 5/2,(T) (7.4.4.11)

2r(2/ + 5/2)M(l/2,3/2,cx) In particular,

M(l/2,5/2,c7) 3M (1/2,3/2,(7)"

The exact solution for the Laplace transform of the transverse after-effect function allows one to calculate the transverse susceptibility cc±(co) since

^(0) = (l/3)[l-(/>2>0] = (l/3)[l-5J!(0)] = ^ ^ . . ( 7 . 4 . 4 . 1 2 )

^ = \-ico]e-imC\t)dt = \-ico~^^-, (7.4.4.13) «1(0) J g,(0)

where

a> (Q) _ M2N08i<f» = M2N0 M(l/2,5/2,(7) 1 &T 3W M (1/2,3/2,(7)

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360 The Langevin Equation

and

CLt) = gx (t) (sin z?(0) cos q>(0) sin z?(0 cos (0)0

Si(0) (sin2t?(0)cos2

* ° > ) o (7.4.4.14)

Thus, we have

4(0) o - ^ S gl(0)(z+i)2(2/+i)r(/+i) [ i which again indicates that ccL(co) is made up of an infinite number of Debye-type relaxation mechanisms.

The exact solution for the transverse relaxation time r± is obtained by setting s = 0 in Eq. (7.4.4.7). We have

= | i ( 0 ) _ J ^ _ y (-iyg 2 M(0Xi + 3/4)IX7 + l/2) , ^ -, f [ ^ + 1 ( 0 ) . (7.4.4.16)

At first glance, it would seem apparent that this equation could be easily represented as a series of products of Kummer functions just as in the longitudinal relaxation. Unfortunately, it is not at all obvious how to do this. Thus, it is best to seek a simple analytic formula for the effective

relaxation time ze[ according to Eq. (7.4.4.1) at n = 1 as

ef gi(0)

*b*i(0) 1 | a | 2a g3(0)

5 15 gl(0) (7.4.4.17)

On using Eq. (7.4.4.11), we may express Eq. (7.4.4.17) in terms of the Kummer functions to yield [cf. Eq. (7.4.2.32)]

Figure 7.4.4.1.Transverse correlation time TL ITD as a function of a computed from the

exact solution [Eq. (7.4.4.16); solid line] and from Eq. (7.4.4.18) (dashing).

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Chapter 7. Three-Dimensional Rotational Brownian Motion 361

Table 7.4.4.1. Numerical values for the transverse correlation time computed from the effective eigenvalue Eq. (7.4.4.18) and the exact solution Eq. (7.4.4.16).

a

0.2 0.4 0.8 1.0 2.0 4.0 8.0 10.0

r±/TD,Eq. (7.4.4.16)

0.9601 0.9205 0.8429 0.8051 0.6306 0.3722 0.1538 0.1155

T/ /TD,Eq. (7.4.4.18)

0.9598 0.9194 0.8386 0.7987 0.6122 0.3466 0.1482 0.1134

r ef *j_ 1 + 0/5 +

8<72M(3/2,9/2,<r) -i-i

= 2 3M(l/2,3/2,cr) l

M(l/2,5/2,a) 175M(l/2,5/2,<7) (7.4.4.18)

It is apparent from Fig. 7 AAA and Table 7.4.4.1 that the effective eigenvalue Eq. (7.4.4.18) yields a close approximation to the exact solution for r±_ for all a, unlike Zj| rendered by that method.

7.4.5 Complex susceptibilities

In Figs. 7.4.5.1. and 7.4.5.2, we plot the real and imaginary parts of the longitudinal and transverse components of the normalised susceptibility XYco) = arGJ)IG=x'r((0)-ix'Y((0) 0HI.-L). where G = jU2N0/(3kT). The Debye spectra

* » = » rf(.,.fi^£, (7.4,.D 11 I + ICOT,, 1 + lCOt,

2.5"-

' -2

1.5-

" t——-,

- 2 - 1 log10(urD)

Figure 7.4.5.1. Real, x[\> and imaginary, ^*, parts of the normalised longitudinal

susceptibility as functions of log^coTo). The circles are the exact solution Eq. (7.4.2.25)

and the dashed lines are the single relaxation time approximation Eq. (7.4.5.1). Large

dashing is a= 0, the small dashing is <T= 5, and middle dashing is <7= 8.

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362 The Langevin Equation

1 * .

xi Xl \

-i.

. . /_ \ x

o i l o g 1 0 ( « r D )

Figure 7.4.5.2. Real, %'L, and imaginary, %"L, parts of the normalised transverse

susceptibility as functions of log10(fiWb). The circles are the exact solution Eq. (7.4.4.15)

and the dashed lines are the single relaxation time approximation Eq. (7.4.5.1). Large

dashing is a= 0, the small dashing is <J= 5, and middle dashing is <7= 8.

l-

0.8-

0.6-

* H 0 .4 /

0 . 2 /

/.

*s

v \ i i

0.5 1 1.5

X\\

\

U - J S 3 -

0.3-

0 .25 0 . 2

0.15-

0.1-

0 .05

.'''' ' * —• —

/,' ' *

?' f

-». ^

N.

^ \ ^ - s . *»

\ \

S v \ N \ i i :

XI

0 . 1 0 .2 0 .3 0 .4 0 .5 0 .6 0 .7

X'l

Figure 7.4.5.3 Cole-Cole plots [Eqs. (7.4.2.25) and (7.4.4.15)]: a= 2 is small dashing, a = 3 is middle dashing and <r= 4 is large dashing.

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364 The Langevin Equation

Table 7.4.5.3 Amplitudes A\k+l and corresponding eigenvalues jjt+i of the first three

decay modes of the longitudinal relaxation as a function of a.

a

0

1

2

3

4

5

7

10

A" 0.333

0.428

0.528

0.619

0.696

0.755

0.832

0.889

A11

0

0.00098

0.00365

0.00676

0.0088

0.00916

0.00674

0.00289

4 0

9.17 10"7

0.000146

0.000066

0.000171

0.000319

0.000624

0.000812

*o4 i

0.653

0.404

0.236

0.130

0.068

0.016

0.00144

TD4 6

5.81

5.77

5.91

6.23

6.74

8.37

12.3

TD^5

15

14.8

14.8

14.8

15.

15.4

16.5

19.2

Table 7.4.5.4. The same as in Table 7.4.5.3 for the transverse response.

a

0

1

2

3

4

5

7

10

Ax 0.333

0.285

0.231

0.179

0.135

0.100

0.0575

0.0308

At 0.

0.00077

0.0035

0.0079

0.0130

0.0174

0.0225

0.0222

A! 0.

5.92 10"7

0.000010

0.000049

0.000134

0.000263

0.000535

0.0006

*D*t 1.00

1.24

1.57

2.00

2.54

3.20

4.85

7.95

TDA^

6.00

5.88

5.89

6.04

6.3

6.66

7.66

9.7

TDA^

15.0

14.8

14.8

14.9

15.2

15.6

16.8

20.

More insight into the decay modes governing the relaxation process may be gained by calculating the dielectric response using Eq. (7.4.1.11) and its counterpart for the transverse relaxation:

oo

si(0=*5 Si<0) y , i

k=0

This calculation requires a knowledge of the set of eigenvalues Ak and the corresponding amplitudes Ak of the relaxation process. The first three eigenvalues Ak and amplitudes Ak were calculated using the method described in Section 7.4.1. We remark that in our notation Xk differs by a factor of two from that used in [7]. The eigenvalues and amplitudes

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Chapter 7. Three-Dimensional Rotational Brownian Motion 363

Table 7.4.5.1. Numerical values of the real and imaginary parts of the normalised longitudinal susceptibility for a = 5 calculated from the exact Eq. (7.4.2.25) and the approximate Eq. (7.4.5.1).

log10(fiMb)

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

X\\ Eq. (7.4.2.25)

2.24 1.89 0.74 0.128 0.0382 0.0245 0.00937 0.00139 0.000148

X\\ Eq. (7.4.5.1)

2.25 1.93 0.788 0.114 0.0119 0.0012 0.00012 0.000012 0.0000012

Z\\

Eq. (7.4.2.25) 0.327 0.868 1.05 0.465 0.157 0.0592 0.0285 0.0108 0.00352

Z\\

Eq. (7.4.5.1) 0.311 0.841 1.09 0.499 0.165 0.0524 0.0166 0.00525 0.00166

Table 7.4.5.2. The same as in Table 7.4.5.1, however, for the real and imaginary parts of the normalised transverse susceptibility.

logio(onb)

-2.0 -1.5 -1.0 -0.5 0.0 0.5 1.0 1.5 2.0

Xl Eq. (7.4.4.15)

0.354 0.354 0.353 0.35 0.323 0.181 0.035 0.00404 0.000414

ZL

Eq. (7.4.5.1) 0.354 0.354 0.353 0.351 0.33 0.206 0.0434 0.00488 0.000494

Eq. (7.4.4.15) 0.00109 0.00345 0.0109 0.0342 0.0995 0.176 0.104 0.0364 0.0117

xl Eq. (7.4.5.1)

0.000945 0.00299 0.00944 0.0297 0.0882 0.174 0.116 0.0413 0.0132

(where T\\ and t± are given by Eqs. (7.4.2.30) and (7.4.4.16), respectively) are shown in both figures for comparison. Equation (7.4.5.1) is obtained from Eqs. (7.4.2.24) and (7.4.4.13) by assuming that the decay of both fx(t) and gi(t) is a single exponential:

f1(t) = fl(0)e-'lTK gl(t) = gl(0)e-"^. (7.4.5.2) It is apparent both from Figs. 7.4.5.1 and 7.4.5.2 and Tables

7.4.5.1. and 7.4.5.2 that there is no practical difference between the exact Eqs. (7.4.2.25) and (7.4.4.15) and the Debye Eq. (7.4.5.2) at low frequencies (COTD < 1). Thus both longitudinal and transverse dielectric relaxation can be approximately described as a single exponential decay for all values of a. This conclusion is borne out by the semicircular shape of the Cole-Cole plots shown in Fig. 7.4.5.3. The deviations at high frequencies from the single relaxation time approximation shown in Tables 7.4.5.1, are mainly due to the contribution of the high-frequency modes.

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Chapter 7. Three-Dimensional Rotational Brownian Motion 365

shown in Tables 7.4.5.3 and 7.4.5.4 again reinforce our conclusion that both relaxation processes are effectively described by a single relaxation mechanism. It is obvious from Table 7.4.5.3 that for the longitudinal relaxation, the contributions of the modes characterised by the eigenvalues A3 and As are small. This conclusion is in agreement with Martin et al. [7]. A different situation obtains in the transverse relaxation. Here the modes characterised by A\ and AT, are near degenerate for large <y, while the mode characterised by A5 has almost zero contribution (see Table 7.4.5.4). Thus the relaxation may again be described by a single mechanism.

In order to take into account the contribution of the high-frequency modes to the longitudinal response, we use the approach of Section 2.13 of Chapter 2. We recall that in order to accomplish this for the present system, one must represent the susceptibility a^(a>) as a sum

of two Lorentzians :

5 ^ = A > + ! - A . , (7.4.5.3)

«jj(0) l + ico/Ai \ + ioruw

where

A, = T| | /7|l _ 1 , , % = -^±- (7.4.5.4)

Here the integral relaxation time zj|, the effective relaxation time T^ , and the smallest eigenvalue Ax are given by Eqs. (7.4.2.30), (7.4.2.32), and (7.4.2.34), respectively. In the time domain, this behaviour of Zui®) 1S

equivalent to supposing that the longitudinal correlation function C^(t) (which in general comprises an infinite number of decaying exponentials) may be approximated by two exponentials only.

The results of the numerical calculation from the exact continued fraction Eq. (7.4.2.25) and approximate Eqs. (7.4.5.3) and (7.4.5.4) are shown in Fig. 7.4.5.4. The parameters used in the calculation are given in Table 7.4.5.5. It is apparent that the two-mode Eq. (7.4.5.3) correctly predicts ^(fiJ) in all frequency ranges of interest. Similar behaviour, which seems to be a characteristic of the simple uniaxial potential, has been observed in the two-dimensional version of the model (Chapter 4). The agreement between the exact continued fraction calculation and the approximate Eq. (7.4.5.3) is very good (the maximum relative deviation between the corresponding curves is less then 5 % in the worst cases which usually appear in the region 0.1< (OT\\ < 10).

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The Langevin Equation

10"

10"'

3 10~2

lCf3

10"4

l l l l l i n

10"' an.,

Fig. 7.4.5.4. %"\(.a>) and X^®) evaluated from the exact continued fraction solution [Eq. (7.4.2.25); solid lines) for various a and compared with those calculated from Eq. (7.4.5.3) with numerical values of T^ , T^ , and \IXX from Table 7.4.5.5 (filled circles) and with the exact low (dotted lines) and high (dashed lines) frequency asymptotes Eqs. (2.13.7) and (2.13.8), respectively.

Table 7.4.5.5. Numerical values of V(TM [Eq. (7.4.2.34)], r,, lzD [Eq. (7.4.2.30)], T? ltD [Eq. (7.4.2.32)], and % / rD [Eq. (7.4.5.4)].

1/(W0 t/TD

ref IT

T\yl To

o=\ 1.5326

1.5280

1.5040

0.24495

£7=5

14.793

14.589

6.4842

0.15934

(7= 10

693.95

691.02

16.644

0.07202

£7=20

5.072-106

5.068106

36.876

0.02902

Page 392: The Langevin Equation Coffey_Kalmykov_Waldron

Chapter 7. Three-Dimensional Rotational Brownian Motion 367

7.5 Brownian Rotation in a Uniform DC External Field

7.5.1 Introduction

The application of a strong direct current (dc) bias electric field E0 to a polar fluid comprised of permanent dipoles results in a transition from the state of free thermal rotation of the molecules to a state of partial orientation with hindered rotation. This change in the character of the molecular motion under the influence of the bias field has a marked effect on the dielectric properties of the fluid insofar as dispersion and absorption of electromagnetic waves will be observed at the characteristic frequencies of rotation of the molecule in the field E0.

A similar problem arises in the theory of magnetic relaxation of ferrofluids. The similarity of the problems of dielectric relaxation of a polar fluid and magnetic relaxation of a ferrofluid is not surprising because, from a physical point of view, the rotational Brownian motion of single domain ferromagnetic particles (magnetic dipoles) in a constant magnetic field H0 [14,15,54], where the Neel relaxation mechanism (that is reorientation of the magnetisation within the particle) is blocked, is similar to that of polar molecules (electric dipoles) in a constant electric field E0 [4,5].

It is the purpose of this section to demonstrate how the linear response of an assembly of noninteracting polar molecules for ac fields applied parallel and perpendicular to the bias field may be calculated exactly. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a polar molecule may be described by the Langevin equation, Eq. (7.2.6), in which the inertial effects are neglected. The problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the correlation functions Eqs. (7.2.33) and (7.2.35). The longitudinal and transverse susceptibilities may be written down from the Laplace transform of the corresponding correlation functions using linear response theory. The Laplace transform in both cases is presented in terms of infinite continued fractions in the frequency ft? and the bias field parameter £ defined as

Z = juE0/(kT), (7.5.1.1)

where jU is the permanent dipole moment of a polar molecule. On proceeding to the limit of zero frequency in the Laplace

transforms of the appropriate after-effect functions, we derive expressions for the dielectric correlation times in terms of continued fractions in £, only [5]. These correlation times are global characteristics of the orientational relaxation of dipolar particles and may be compared with

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368 The Langevin Equation

those extracted from experimental spectra of the dielectric or magnetic susceptibilities.

7.5.2 Longitudinal response

We have shown in Section 7.2 that the equation governing the longitudinal relaxation for the present problem is Eq. (7.2.33), namely:

*b/„(0 = <Zn"/„-i(0 + onfnt) + q+Jn+1(t), (n > 1), (7.5.2.1) where fn(t) is defined by Eq. (7.2.32) and

*„=-^±l>, -=M!±1), q>J-^±A. (7.5.2.2) 2 " 2(2n + l) 2(2n + l)

We are interested in the decay of the orientational correlation function / i ( 0 . As shown in Section 2.7.3 of Chapter 2, Eq. (7.5.2.1) can be

solved exactly for the Laplace transform f\(s) in terms of ordinary continued fractions to yield

n „+

/,w= 9i-Ws) -—T-^-T-V-, /i(0)+z/„(0)n -*b*-?i -<i\S\s) [ „=2 k=2 qk

(7.5.2.3)

where the infinite continued fraction SJJ (5) is defined by the following recurrence equation

Sl(s) = q-n[TDs-qn-q+

nSl+1(s)Jl

or, equivalently,

Sl(s) = 2n + l

1 + 2rns

- + - ?:+i(*) (7.5.2.4) n(n + l) 2n + l

Equation (7.5.2.3) may further be simplified if we write out the product explicitly:

ll92*-i /92*+i=(-1) T T T 7 T -

Thus, we obtain

/ .W^sfto 00

9 H -1-1 "

n=2 n(n + l ) t = 2

2 r n ^ .xn+1 „ _ 2n + l

J (7.5.2.5)

/Z(-ir+7„(0) # „=i /i(n + l ) t = ,

nto-Equation (7.5.2.5) is the exact solution of the problem.

The initial values /„(0) are evaluated as follows. We note that

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Chapter 7. Three-Dimensional Rotational Brownian Motion 369

fn(0) = (PlPn)o-(Pi)o(Pn)o n + l n (7.5.2.6)

= ^^(P««)o + ^i(Pn-i)0-(pi)o(Pn)o>

where we have used Eq. (7.4.2.11). Now, on using Eq. (10.2.36) of [45], namely

ezcos,=^/(2z)^(2n + l)In+1/2(z)Pn(cos^), (7.5.2.7) n=0

where /n+1/2(z) are the modified Bessel functions [45], we have K

\ Pn(cost?)/COSI?sinz&/tf

( P ) =-9 = / n + 1 / 2 ( ^ . (7.5.2.8)

J ^ ' s i n i W t ? ' " ^ o

Here, we have used Eq. (7.5.2.7) and the orthogonality property of Legendre polynomials. Thus, the /„(0) may be expressed in terms of the modified Bessel functions as

'n-l/2

2n + l 71/2(<f) 2n + l 71/2(£) 71/2(£) 71/2(<f) ' In particular,

/](0) = - ^ Z 2 ( | ) + I - 4 1 ^ = l + - c o t h V . (7.5.2.10) J*J 3 71/2(£) 3 72/2(<f) £2

Here we have used the fact that ([45], Eq. (10.2.13))

£ ^ = c o t h £ - I = L(£), (7.5.2.11)

= T2" + 1 r- = 1 - T L ( ^ ) ' (7.5.2.12) Wfl = 3 , 3cothcf 3

where L(£) is known as the Langevin function. The longitudinal complex susceptibility is given by

aj[(0) MO)

where

(7.5.2.13)

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370 The Langevin Equation

«i[(0)= M2N0fl(0)_M2N0'

1 + -1 -coth2£ (7.5.2.14)

kT kT According to linear response theory (Chapter 2, Section 2.8), the quantity f^ico)//j(0) appearing in Eq. (7.5.2.13) is the one-sided Fourier transform of the equilibrium longitudinal dipole autocorrelation function CM) defined as

CM = fY(t) = (cosfl(0)cosi9(r))0-(cost9(0));

/i(0) (cos2tf(0)\ -(costf(O))* (7.5.2.15)

The correlation time Ty of CM) is, as before, determined from the s = 0

limit of Eq. (7.5.2.5). We have from Eq. (7.5.2.4):

^(0) = - (7.5.2.16) 2n + l + <fSJ|+1(0)

Thus, on comparing Eq. (7.5.2.16) with Eq. (4.4.3.17) of Chapter 4, we obtain

Also we have sho)=ik+mtf)/ik_m&.

f l 5|(0) = /n+1/2(<f)//1/2(<f). k=\

(7.5.2.17)

(7.5.2.18)

Thus, according to Eq. Eq. (7.5.2.5), (7.5.2.17), and (7.5.2.18), we have

TJI=JC,|(0<* = / i (0) \\o> _ 2Tr

MO) ^(l + r2-coth2#)nt K-D «+l 1n+\

^ 3 / 2 ^ ) , " /n-l/2(^) 2n + l / 3 / 2 ( ^ ) (7.5.2.19)

/„+ i /2(#) n + l/B+i/2(#) " + 1 'i/2(£)_ For £ » 1, Eq. (7.5.2.19) has the following asymptotic behaviour [5]

T | | -T D /^ . (7.5.2.20)

In the opposite limit of small £ , we have

T | r r D [ l - ( l / 9 ) f 2 + ( l /90)£ 4 - . . . ] . (7.5.2.21)

For the Brownian particle in the uniform field E0, the dynamics of the system are described by a single variable Fokker-Planck equation [1-3] [see Section 1.15, Eq. (1.15.21)] so that the integral relaxation time Zji may be expressed in closed form as (see Chapter 2, Section 2.10)

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Chapter 7. Three-Dimensional Rotational Brownian Motion 371

Figure 7.5.2.1. rl as a function of £ The numerical solution of the set of Eq. (7.5.2.1) for Afl (small dashing) compared with Tj, from Eq. (7.5.2.22) (bold line) and rf from Eq. (7.5.2.23) (large dashing). The asymptotic Eq. (7.5.2.20) is represented by the dotted line.

Tu = 2Tn

Z((cos2tf)0-<costf>o): \z'-cos#)0)e

4z'dz -fc

\-zl rdz

(7.5.2.22) where Z = £/(4;rsinh£) is the partition function. Equation (7.5.2.22) yields the same results as the continued fraction solution Eq. (7.5.2.19) and may serve as an independent check of numerical calculations.

The effective eigenvalue method when applied to this problem yields according to Eq. (7.5.2.1) at n = 1, and Eqs. (7.5.2.9)- (7.5.2.12) [4,5]

r* / i (0)

W,(0) 1 + £/2(Q)

3/i(0) £

L(#) -£L(£)-2. (7.5.2.23)

Table 7.5.2.1. Numerical values for longitudinal relaxation

4 I 2 3 4 5 6 7 8 10

TJ!/TD>Eq. (7.5.2.22)

0.089 0.6842 0.4844 0.3457 0.2593 0.2053 0.1697 0.1447 0.1119

(r^r'.Eq. (7.5.2.1)

0.9092 0.7155 0.5297 0.3907 0.2946 0.2291 0.1841 0.1528 0.1142

Tf/TD,Eq. (7.5.2.23)

0.8815 0.6476 0.4518 0.3259 0.2488 0.1998 0.1666 0.1429 0.1111

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372 The Langevin Equation

In Table 7.5.2.1. and Fig. 7.5.2.1, we show zj|, Z[l, and x^ vs. £ In order to ensure convergence for values of <f up to 10, a system matrix A of size 18x18 was used in the evaluation of X\ from Eq. (7.5.2.1) rearranged in matrix form Eq. (7.4.1.3).

7.5.3 Transverse response

The appropriate recurrence equation for the transverse relaxation functions

gn(t) = (cos(p(0)P? [cos iXO)] cos <pt)P\ [cos tf(0])o, (7.5.3.1)

is Eq. (7.2.35), namely:

^ T f t W = U „ - i W + ?„«„(0 + „ + i ( 0 . (n>l), (7.5.3.2) at

where

n(n + l) _ £(n + l)2 + fo2 , „ , ~ „,

an= ; — . ? » = ^ r - 7 : . an = - ^ — 7 7 - (7-5.3.3) 2 2(2« + l) 2(2n + l)

We are interested in the decay of the transverse correlation function gj(0- Just as the longitudinal response, Eq. (7.5.3.2) can be solved exactly for g^s) in terms of ordinary continued fractions to yield

ft(0)+Z*-(0)fl^=4^1. <7-5-3-4) n=2 k=l %

where the infinite continued fraction 5J| (s) is defined by the following

recurrence equation

5>> = ^ + g i , , (7-5-3-5> *DS-an-<lnSn+\(S)

8i(s) = TDS-Q\ -qtSiis)

or, equivalently,

*n(s)=-— ,r :v .T~r ; 'w" .„ , . . ,„• <7-5-3-6) 2rD5/[

Thus, on noting that

; ! ,_ ,_ £(/t + l)/[fl(2n + l)]

2rD5/[n(« + !)] + ! + €nS£+1(s)/[(n + l)(2n +1)]

A + - B+i 4(2n + l) 11^-1^2/ t+l =( - ! ) —7- ~2> t=2 3 « 2 ( n + l )

we have 9 T °° 9 M 4-1 n

£ „=i n (n + l) t=1

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Chapter 7. Three-Dimensional Rotational Brownian Motion 373

The initial values gn (0) are evaluated by noting that

? n ( 0 ) = /p 11 (cos^)P n

1 (cosz?)cosV) = (Pi (cosz?)P„](cost?)) 12

and by using Eq. (7.4.4.9). Thus we obtain

which, with the help of Eq. (7.5.2.8), reduces to

tt(» + l ) [ / „ _ 1 / 2 ( # ) - / n + 3 / 2 ( c f ) ] = nn + \) Ia+U2(£)

2(2n + l)/1/2(<?) 2£ 7 1 / 2 (£) '

Now g1(0) = Iy2(Z)/ZlU2(Z)), (7.5.3.10)

thus

M l = TD y ( - i ) " + l ( 2 n + 1 ) 7 " ^ ( ^ ) r T o i w (v.5.3.11) g,(0) #/3/2(#)S n(n + l) U *

The transverse complex susceptibility (X±(co) and the correlation time r±

are given by

g ^ = 1 - f f l > M ^ = 1 - . ^ ^ f : ( - i r i ( ^ + ^ i / 2 ^ f [ ^ ( t o )

tfJO) g,(0) £/3/2(<?);£ n(n + l) LI (7.5.3.12)

and

liW = _ ^ _ £ ( _ i r i ( 2 i » + l)/.+1/2(g)Agx(0) ( 7 5 3 1 3 )

where

On using the asymptotic expansion for the modified Bessel functions [24], we may also deduce that in the limit of large £ the relaxation time T± from Eq. (7.5.3.13) has the following asymptotic behaviour [5]

T±~2TD/£ (7.5.3.15) while in the limit of small £ we have

£ka,i_J_£2+JZ_^_.... (7.5.3.16) TD 12 2160

The corresponding effective relaxation time ze[ is according to Eqs. (7.5.2.11), (7.5.2.12), (7.5.3.2), and (7.5.3.9) [4,5]:

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374 The Langevin Equation

Figure 7.5.3.1. Transverse correlation time T± as a function of £ The numerical solution

of the set of equations Eq. (7.5.3.2) for A^1 (small dashing) compared with TL as

calculated from the exact Eq. (7.5.3.13) (bold line) and tf from Eq. (7.5.3.17) (large

dashing). The asymptotic Eq. (7.5.3.15) is the dotted line.

.ef= g l (0 ) x g,(0)

= Tr 1 + &2(0) ' = 2r r US)

S-ug) (7.5.3.17)

6^(0)

In Table 7.5.3.1. and Fig. 7.5.3.1, we show the transverse

relaxation time rL, Te[ vs. £ In order to ensure convergence for values of <f up to 10, a system matrix A of size 18x18 was used in the evaluation of X\ from Eq. (7.5.3.2) rearranged in matrix form Eq. (7.4.13) and the first 8 terms were taken in the infinite summation of Eq. (7.5.3.13). In

order to ensure convergence of 5^(0), it was sufficient to assume that this quantity was zero for A; > 16.

Table 7.5.3.1. Numerical values for transverse relaxation.

4

I 2 3 4 5 6 7 8 9 10

Eq. (7.5.3.13)

0.924 0.759 0.598 0.476 0.390 0.328 0.283 0.248 0.221 0.199

Mr1, Eq. (7.5.3.2)

0.931 0.775 0.615 0.489 0.398 0.333 0.286 0.250 0.222 0.200

rflrD, Eq. (7.5.3.17)

0.911 0.735 0.577 0.462 0.381 0.323 0.279 0.255 0.219 0.198

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Chapter 7. Three-Dimensional Rotational Brownian Motion 375

7.5.4 Comparison with experimental data

In Figs. 7.5.4.1 and 7.5.4.2, we plot the real and imaginary parts of the longitudinal and transverse components of the normalised complex susceptibility

Zr(co) = ar(co)/G (y=\\,±),

where G = jU2N0 /(3kT). The Debye spectra

01,(0)70

" l + lO)T„

ZH*> = *V»G 1 + iCOT

(7.5.4.1) i

Z\\

0 . 8

4 A . —""-" -?-

pcs^a - 1 0 1

log10(«rD)

Figure 7.5.4.1. Real, %!,, and imaginary, Xn > parts of the normalised longitudinal

susceptibility as functions of logj0(<WTD). The circles are the exact solution Eq.

(7.5.2.13) and the dashed lines are the single relaxation time approximation Eq. (7.5.4.1).

Curves 1, 2 and 3 are x\ f° r ^ = 0, 3 and 6, respectively. Curves 4, 5 and 6 are x"\ f° r

4 = 0, 3, and 6 respectively.

lOg10(O»rD)

Figure 7.5.4.2. Real, X'L . a nd imaginary, %]_, parts of the normalised transverse

susceptibility as functions of \ogwcoTD). The circles are the exact solution, Eq.

(7.5.3.12), and the dashed lines are the single relaxation time approximation Eq. (7.5.4.1).

Curves 1, 2 and 3 are %'L f° r f = 0, 3 and 6, respectively. Curves 4, 5 and 6 are xl f° r

4 = 0, 3 and 6 respectively.

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376 The Langevin Equation

fc (kHz) 3

i—i—s—J—s—§—*> E (kV/cm)

Figure 7.5.4.3. Comparison of the theoretical bias field dependence (dashed lines) of the critical relaxation frequency fc with experimental data (circles) of Block and Hayes [55] for dilute solutions of polar macromolecules PBLG with molecular weig hts 4.6xl05 (1), 2.6xl05(2)andl.lxl05(3).

where the relaxation times zf| and T± are given by the exact Eqs. (7.5.2.19) and (7.5.3.13), respectively, are shown in both figures for comparison. Equation (7.5.4.1) is obtained by assuming that the decay of/i(f) and gi(t) is a single exponential given by Eq. (7.4.5.2). It is apparent from Figs. 7.5.4.1 and 7.5.4.2 that there is no practical difference between the exact equations Eqs. (7.5.2.13) and (7.5.3.12) and the Debye equation (7.5.4.1). Thus both longitudinal and transverse dielectric relaxation can be effectively described as a single exponential for all values of £

Our results for the longitudinal susceptibility [Eqs. (7.5.2.13)] and relaxation time [Eq. (7.5.2.19)] are in qualitative agreement with available experimental data [55,56]. (These data were obtained for the longitudinal component of the susceptibility with the strong dc field applied parallel to a weak ac probe field). As observed by Block and Hayes [55] for dilute solutions of macromolecules and Fannin et al. [56] for ferrofluids, with increasing £ both the loss J^ai) and the relaxation time \ decrease compared with those in the isotropic case. Fig. 7.5.4.1 shows clearly that these observations are in qualitative agreement with our results.

In Fig. 7.5.4.3, we compare the theoretical and experimental critical relaxation frequency

/c=(2*T,,) -1

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Chapter 7. Three-Dimensional Rotational Brownian Motion 311

i i

0 . 8 - \ i \

• s h 0 .6- ',

\ 0.4- \ .

\ • 0.2 \

?-- — ? - __..» 10 20 30 40

tf (kA/m) Figure 7.5.4.4. Comparison of the theoretical (dashed line) and experimental (circles) normalised longitudinal relaxation time T^/TD against the bias magnetic field H for a ferrofluid. Experimental data from Fannin et al. [56].

where ?|| is given by Eq. (7.5.2.19), for dilute solutions of polar macromolecules of poly- y -benzyl-L-glutamate (PBLG) in a non-polar solvent at 298 K [55]. The data were fitted using a standard least-squares method. This system is very suitable for comparison with the theory since the dipole moments of the PBLG molecules are large. They are 23.3, 14.7 and 4.8 x 10"27 Cm for the PBLG molecules with molecular weights 4.6, 2.6 and l.lxlO5, respectively. Thus, the values of the energy of a dipole in a dc bias field are comparable with the thermal energy (£ ~ 1). The values of ju so obtained are greater than the originally reported data but are in agreement with the results of Ullman [1] who suggested that this difference results from a poor estimate of the internal field in the context of a simple model for the evaluation of dipole moments from static dielectric measurements used in [55]. Our calculation for PBLG with molecular weight 4.6x105 is also in accordance with the results of Ullman [1] based on the numerical solution of Eq. (7.5.2.1).

We remark that experiments on the polarisation induced by a weak ac field superimposed on a strong dc field may be realised in practice in a ferrofluid as a large value of £ can be achieved with a moderate constant magnetic field due to the large value of the magnetic dipole moment m (10 4 - 105 Bohr magnetons) of single domain ferromagnetic particles (Chapter 1). In Fig. 7.5.4.4, we compare the theoretical and experimental relaxation time obtained from the experimental data of Fannin et al. [56] for the imaginary part X\\ (#>) of the longitudinal magnetic susceptibility. The value of the average magnetic dipole moment m of particles is found from the least-squares fit to be 4.29 x 105

Bohr magnetons.

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378 The Langevin Equation

In summary, the rotational diffusion theory has been generalised in this section in order to account for the effect of a constant electric field on the dielectric properties of polar media. The theory may be applied to ferrofluids with blocked Neel mechanism by a simple change of notation. We remark in the context of dielectric relaxation that the area of applicability of these results is restricted to the low frequency range, as defined by the inequality anr< 1 (y= ||, _L), because the theory does not include the effects of molecular inertia. A consistent treatment of inertial effects must be carried out using the kinetic equation for the probability density function in phase space (Chapter 10).

7.6 Anisotropic Noninertial Rotational Diffusion of an Asymmetric top in an External Potential

In Section 7.2, the Langevin equation method was applied to the rotational Brownian motion of linear and spherical top molecules in an external potential. Here, we extend the Langevin equation approach to anisotropic rotational diffusion of an asymmetric top [66]. In the context of the Fokker-Planck equation, the theory of rotational Brownian motion of an asymmetric top was developed by Perrin [57] and others [9,22,58-65] in particular applications to the dielectric, NMR, and Kerr effect relaxation in liquids and in nematic liquid crystals.

7.6.7 Solution of the Euler-Langevin equation for an asymmetric top in the noninertial limit

We consider the three-dimensional rotational Brownian motion of a rigid asymmetric top molecule in an external potential V. The orientation of the molecule is described by the Euler angles Q. = a,p,y\ (here, the notations from Ref. 25 are adopted; see Fig. 7.6.1.1). The Euler angles completely determine the orientation of the molecular (body-fixed) coordinate system xyz with respect to the laboratory coordinate system XYZ. The dynamics of the molecule are described by the Euler-Langevin equation for the angular velocity ca(t) written in the body-fixed coordinate system xyz [22]:

—7(0(0 + <o(t)xI(o(t) + £m(t) = -VV[Q(t),t] + k(t) , (7.6.1.1) dt

where 7 is the inertia tensor of the molecule, g(0(t) is the damping torque due to Brownian movement, £is the rotational friction tensor, X(t) is the white noise driving torque, again due to Brownian movement, so that k(t) has the following properties:

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Chapter 7. Three-Dimensional Rotational Brownian Motion 379

Figure 7.6.1.1. The Euler angles.

4 ( 0 = 0 , Ai(tl)Aj(t2) = 2kTgiiSiiS(t1-t2). (7.6.1.2)

Here, the overbar means a statistical average over an ensemble of Brownian particles which all start at time t with the same angular velocity and orientation, and indices i,j = 1, 2, 3 correspond to the Cartesian axes x y z of the molecular coordinate system. The term -V V in Eq. (7.6.1.1) represents the torque acting on the molecule, V (Q, t) is the potential energy of the molecule, and

V = S/S(p

is the orientation space gradient operator (the properties of V are described in detail in [67]), Sep is an infinitesimal rotation vector so that (d(t) = &p/St. The torque - W in Eq. (7.6.1.1) can be expressed in terms of the angular momentum operator J [25]:

-VV = - i JV , (7.6.1.3)

where the components of J in the molecular coordinate system are [25]

Jr = 1 (J~l-J+1), / y = _ U r i + / + i ) , jz=j\ (7.6.1.4)

and

J±l = 7T

»Tf>

V2"

. a d . d 1 3 ' ±cot/?— + i — +

dy dp sin J3 da

J°=-dy

(7.6.1.5)

One can now determine the corresponding components of the orientation

space gradient operator V = z'J from Eqs. (7.6.1.3)-(7.6.1.5). The Euler-

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380 The Langevin Equation

Langevin Eq. (7.6.1.1) is a Stratonovich vector stochastic differential equation. In writing Eq. (7.6.1.1), it was assumed that the suspension of Brownian particles (molecules) is monodisperse, nonconducting, and sufficiently dilute to avoid interparticle correlation effects. Memory and quantum effects were also ignored (the last assumption fails in liquids and solutions only for the lightest polar molecules such as HF and HC1).

The Euler-Langevin Eq. (7.6.1.1) takes into account the inertial effects. The inclusion of these effects causes the theory of orientational relaxation of an asymmetric top to become very complicated even in the absence of an external potential (see, e.g., [22,68,69]. A radical simplification of the theory can be achieved in the noninertial limit (or in the Debye approximation), IfoKfix;^ «1, i.e., at low frequencies, when the inertia terms in Eq. (7.6.1.1) may be neglected. Here, the angular velocity <o may be immediately obtained from Eqs. (7.6.1.1) and (7.6.1.3) as

(0(0 = <Xp/St = Dk(t) -iJV[a(t),t]/(kT), (7.6.1.6)

where D - kTg~l is the diffusion tensor. The noninertial approximation is valid in the low-frequency region (< 10 GHz) for the majority of molecules in liquid solutions. In this approximation, one can readily obtain from Eqs. (7.6.1.3) and (7.6.1.6) the equation of motion of an arbitrary function f(a,jB, y):

±f = &±f+fi±f + tJLf = „.v)f dt da dp dy (7.6.1.7)

= ±[i(Dk)-Jf) + ((DJV)-Jf)].

On noting that for a symmetric tensor D and for any V(Q.,t) and f(Q.) the following mathematical identities hold

JDj(Vf) = j(fDJV+VDJf)

= (jf)-(DJv) + flDJV + (jv)-DJf) + VJDJf (7.6.1.8)

= 2(DJv)-(jf) + JJDiV + VJDJf,

one has 2(DiV)-if=VVlf + fV2

aV-VlVf), (7.6.1.9) 2 where the operator Va is defined as [58]

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Chapter 7. Three-Dimensional Rotational Brownian Motion 381

^km-'k-'m V2a=-JDJ = - I

k,m=x,y,z

Thus Eq. (7.6.1.7) becomes

(7.6.1.10)

(7.6.1.11) df ^ ~'J kTx " ' 2kTi

+f[0.(tW2aV[Q.(t),t]-V2clV[Q(t),t]f[Qit)])].

Equation (7.6.1.11) is a stochastic differential equation with a

multiplicative noise term iiDV) • J / l(kT). As usual, in order to average Eq. (7.6.1.11) over an ensemble of

particles which all have started with the same orientation £1, one must write Eq. (7.6.1.11) in integral form:

. r+T

/[«(*+*•>]=/(Q)+-5- J (DW)-mm')])dt' kl t

, t+T

M \vmat']vim(t')]+m(t')]vlv[n(t'),t'] (7.6.1.12) + -2kT

-v2av[w'),t']m(t')]]dt'

with £l(t) = Q., and then evaluate the limit

1 r->0 T kT r->0 X

\ Dk(t')-3f[a(t')]dt' t

+-2kT

V[Q(at']Vlf[n(t')] + f[Q(tWlV[£l(t'),t']

-^vmanma')] dt' . (7.6.1.13)

The right-hand side of Eq. (7.6.1.13) again comprises two terms, namely: the deterministic drift and noise-induced drift. The evaluation of the deterministic drift term [the last term in Eq. (7.6.1.13)] yields

1 t t+T

limif T * 2kTr-*0T

dt' vm\t']vim(t')]+fmwlvm\t')

- V ^ V W ) / ] / [ n ( f ' ) ] j (7.6.1.14)

=^\v(aj)vlf(Q)+f(a)v2£lv(a,t)-v2

a[V(Q,t)f(a)]] In order to evaluate the noise-induced drift term, we first note that a rotation through the infinitesimal angle Sip(t') transforms the function / (Q) into m(t')] as [25]

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382 The Langevin Equation

/ [ f l ( 0 ] = e'^'yjf(a)»(l + iW) • ! ) / ( « ) , (7.6.1.15)

where according to Eq. (7.6.1.6)

<XP(O = (P(O-<P(0=^J £MO-riv[a(o,f']K. (7.6.1.16) Thus, on noting Eqs. (7.6.1.15) and (7.6.1.16), we have

i t+T

r ^ l i m - f m ( 0 - J / [ ^ ' ) ] ^ '

, /-W

=—lim- f Z»X?)J kTr^Ot

1J Dx(0-av[n(0,f']^' J/(0) df'

=-lim- f $~lKt')-i T-)OT

J g-\t')dt" jm)dt'=-jDjm)=v2nm).

(7.6.1.17) Here, we have used Eqs. (7.6.1.2). Noting Eqs. (7.6.1.14) and (7.6.1.17), Eq. (7.6.1.13) yields [cf. Eq. (7.2.20)]

f^lf~[^iVf)-V^2af-fV

2av]. (7.6.1.18)

Equation (7.6.1.18) constitutes the averaged equation of motion of an arbitrary function f(a, fi, f). We reiterate that Q. in Eq. (7.6.1.18) and Q.(t) in Eq. (7.6.1.11) have different meanings, namely, the Cl(t) in Eq. (7.6.]|J1) are stochastic variables while the £1 in Eq. (7.6.1.18) are the sharp (definite) values Q. (t) = Q. at time t.

In rotational diffusion problems, the quantities of interest are averages involving Wigner's D functions defined as [25]

DJMJi<Q) = e -iMajJ

aMM ifi)' -M'y (7.6.1.19)

where dJMM'/3) is a real function whose various explicit forms are

given, for example, in Ref. [25]; a typical example is

(-1) J-M r

dMM'P)-

M'+M X(1 + C0S/?) 2

(J + M)\ -t\l 2

(J-M)\J + M')\J-M')\

M'-M (1-COS/?) 2

jJ-M 1

_[ (1 -C0S£) ' - " ' ( 1 -C0B# ' + " ' ] . (dcosfiy

The orthogonality and normalisation conditions for D functions are [25]

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Chapter 7. Three-Dimensional Rotational Brownian Motion 383

2/r In n %nL

J I j DtiWD^(Wn = —SjlhS¥A;i> (7.6.1.20) o o o lh + L

(dQ. = sin/3d/3dady, Dn*m = (-l)"~m Dln_m, and the asterisk denotes

the complex conjugate). In quantum mechanics, Wigner's D functions represent the wave functions y/JMK of a rigid symmetric top [25]

WJMK («. P, Y) = iJ \\-^DKM («• 0> Y)> (7.6.1.21)

— nJ = v2nj — dt n'm Q n'm 2kT

where J, K, and M are the corresponding quantum numbers. The properties of Wigner's D functions are described in detail, e.g., in [25].

Equation (7.6.1.18) written in terms of D functions yields

yi(VDim)-VVlDim-DimV2Qv].(1.6.l.22)

Here, it is convenient to use the molecular coordinate system in which the diffusion tensor D is diagonal so that the operator V^ , Eq. (7.6.1.10), is simplified to [58]

Vl=-(Dj2x+DyyPy + Dj2

z). (7.6.1.23)

We remark that the principal axis system of the diffusion tensor D for an asymmetric top molecule may not coincide, in general, with the principal

axis system that diagonalises the moment of inertia tensor / [60]. Thus, if the orientation of the diffusion tensor principal axis system is unknown, the nondiagonalised diffusion tensor form, Eq. (7.6.1.10), must be used.

Furthermore, the operator V^ defined by Eq. (7.6.1.23) can be represented as [58]

Vl=~(D^+Dyy)F+[2D„-(D^+Dyy)]Pz+(D„-Dyy)(Px-Py)

2TD[

where

J 2 + 2 A J 2 + E

D,

n'+in1

Dxx+Dyy Dx*+Dyy 2 ' " xx yy

the operator j 2 = J2 + J2 + J2 is given by [25]

J2 = a2 _ a

+ cot/?-^r + d/3 dj3 sin2 J3 da1

, « d2 d2 ^ -2cos/?_ _ + dady By

(7.6.1.24)

(7.6.1.25)

(7.6.1.26)

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384 The Langevin Equation

and we have used the fact that J2-J2 = (j+l) +(j~]) • Equation

(7.6.1.24) can be simplified by using the known properties of the angular momentum operators and D functions, viz., [25]

pDija) = JU+I)DU&) , J2zDiJW = m2Dim(a),

JVDUn) = -Jj(j + l)Cf™ZDL+An) (v=±l), and

Dh (Q.Dh (£1)= V r J " 2 rJ'mi+m2 r)J (ry\ Hh-h\

Here, Cj/, , , are the Clebsch-Gordan coefficients [251. Thus J\ .'l <J2,'2

ZTD (7.6.1.27)

+ ,Y;+n ,='rr-/'m+1r-/>+2 jy +r^m-x rj-m~2 jy 11 ^ y U + 1^|_ t-;,m,l,l ,-'j,m+l,l,l'^,m+2 ^ L-7,m,l,-lL-j,m-l,l,-l-Ln,m-2 J J •

Noting the above equations, one may show that for any potential V which may be expanded in D functions as

Vn,t)/(kT)= X vRXQ(t)DlQ(£l), (7.6.1.28) Q,S,R

the following equalities are valid

vMv)w=Z Z ^M^teUtQ Q,S,RJ=\J-R\

•"b e,s,K/=j./-^

A \ tV,m*<?,l,lL7,mt<2t-l,l,l urHSjr*Q& ^J ,nHQX-\^J ,nHQ-\X-lLJn+S,rrHQ-2 )\> K'-U-l •£?)

WlDimKkT) = --^- X v^GD sf l

G[; ( ; + D + 2Am2]D>m

1 I ' I r r -i

= - T - Z I vs,5,eC^[;(7 + l) + 2Am2]c^eG^-lD Q,S,RJ=\j-R\

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Chapter 7. Three-Dimensional Rotational Brownian Motion 385

-u JY y A- n~ (ri'm+x ri'm+2 rJ'm+1+® nJ

^JyJ ^ l)~ I *- ;>, l , l ' -y ,m+l, l , lL ' i ,m+2,R,Q l yn+S,m+2+Q (7.6.1.30)

i_r,J'm~l CJ'm~2 /^J,m-2+Q nJ II " r , -y ,m , l , - l < - ; ,m- l , l , - l , - ' y ,m-2 , f l , e Z y n+5 ,m-2+e j j -

D Q,S,R

+j?(i? + i-\'z[rR'Q+lrR'Q+2 nR J . ^ " 1 r*'G~2 nR 1\ +K(K -I- i ; - | ^ ' - f i ,Q , l i l « - f f ,Q + l J , l - l - ' s i (2 + 2 "+" l - R , Q , l , - l ^ R , e - l , l , - l l y S , j 2 - 2 J j

z +/?<•/?+ nWr / ? ' G + 1 r R ' 2 + 2 ry-m+G+2 n y

+ / H A + 1 J - \ ^ R , e , l , l L ' R , e + U , l L ' y > , f i , e + 2 - L V

Z Z ^Sjec;;;/4[/?(^+i)+2Ae2]c^ee^+5,m+Q

2 ^ D 2,S,R y=|j—/el

n+5,m+2+2

(7.6.1.31) ,rR,Q-\ rR,Q-2 rJ,m+Q-2 nJ \\

" t " L - R , e , l , - l L - R , 2 - l , l , - l c ' 7 > , f i , e - 2 - t y n + S , m + e - 2 y J -

On substituting Eqs. (7.6.1.27)-(7.6.1.31) into Eq. (7.6.1.22), we have

JyJ "l" *) • j f^J .m+l ,-../,m+2 n y .~y,m-l fj,m-2 p.j ~| 2 w L^y.m.l.l L;,m+l,l,l ^ ^ + 2 "*"u i,m,l,-l '-y,m-l,l,-l ^ . B r f J

1 j+R t

+7 Z Z v R A e C ^ [ y ( y + l) + 4Ame-7(7 + l ) - ^ + l)]C^eGD„y

+,m+Q 4R,S,Qy=|yW?|

, ^ f / / j , -i\(-iJ,m+Q+l f-,J,m+Q+2 rJ,m+Q _ y • 1 s.-.y',m+l ^j,m+2 rJ,m+2+Q -l-w^V (,J -r 1;>-y,m+Q,l,l>-'y,m+G+l,l,l,-y,m,R,Q VU "•" L^j,m,],\^j,m+l,hl'-j,m+2,R,Q

-R(BA-\ \rR&+1 rR'Q+1 W.m+G+2 1 nJ

l\\n. -I- lJ^Rgtil^R,Q+l,\,\'~jsn,R,Q+2jL'n+S,m+<2+2

j.^[ it r-t-nr,J'm+0~i rJ'm+Q"2 rJ'm+2 _ it; J.nr,-''m~l W."1-2 rJ,m-2+Q

•Tw ^ (,J -r ly«-y im+g,i _i W,m+e-l , l , - l *-j,m,R,Q A 7 "•" V<-y,m ,l-l '-y.m-U -1 *~j,m-2,R,Q

- ^ + l ) C , X . C f i T i V . O GG - 2 ] ^ , m + e - 2 ) - (7.6.1.32)

As we have already mentioned, all the quantities £>' in Eq. (7.6.1.32)

are, in general, functions of the sharp (definite) values Cl(t) = Q. at time t, which are themselves random variables with the probability density function W(£l, t) such that W(Q., t)dQ, is the probability of finding Q, in the interval (Q, £1 + dd). Therefore, in order to obtain equations for the moments governing the relaxation dynamics of the system, we must also average Eq. (7.6.1.32) over W. Thus we have from Eq. (7.6.1.32) a hierarchy of differential-recurrence equations

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386 The Langevin Equation

d/^i \_ fy(7+D

J(J + 1) ~rrj,m+l rJ,m+2 IpJ \ rj,m-l rj,m-2 I j-J \ 1 2 " [S ' . " i .U L ' J . "H-Ul \ ^ , m + 2 / + < - j ' .m, l - lS ' .« -U- l \ i n , m - 2 / J

1 ; + R r

+TE x ^,e^«M[y(y+1)+4z^-^+l)-^+i)K,^K+5,m+e) 4 R,S,Q J^j-F\ X '

4.W T It J 4- 1\W.'"+2+l W,m+£>+2 f-J,mU2 _ Y ; , is^ ' .m+l ^j.nH-2 ,-,J,m+2+Q "•"" I / ^ "•" ^W.m+flUW,m+2+l, l , lS ' , '" ,«.e J(J + ^ S ' ^ . u S ' . m + U l S ' . m + W . e

_p/p, n r R , 2 + l / - R . e + 2 rJ,mU2+2 1/nJ \ A(K r V-R,Q,l,i<-'R(!g+i,i>i»-_,->m,x>e+2 J\£7n+5,m-hQ+2 /

j ^ T T Y f 4- n r J ' m + 2 _ 1 W.m+fi-2 pljirtQ _ .v ; . i\/-y.m-l rJ<m~2 /-J.m-2-tfi + - [ J V -r ^W,m-H2,l,-lW,m45-l,l,-lS',m,«.e •' l-' + ^VwnJ,-l*-j^n-l,l -l^j,m-2,R,Q

- ^ + l ) ^ | , t i ^ u - , C f j ^ 2 ] ( £ ^ , I I t t C . 2 ) . (7.6.1.33)

If the system is in equilibrium, all averages in Eq. (7.6.1.33) are either constant or zero. Thus, in this case, one must construct from Eq. (7.6.1.32) a set of differential-recurrence relations for the appropriate equilibrium correlation functions (see, e.g., Chapter 8, Section 8.7).

Equation (7.6.1.33) contains three phenomenological constants, namely: the three-diagonal components of the diffusion tensor D„, Dyy, and D^. The values of Du may be estimated either in the context of the so-called hydrodynamic approach, where the components of the diffusion tensor depend only on the shape of the particle, or in terms of microscopic molecular parameters [60]:

D , = H 7 g , = T < * 7 7 / ; ,

where /, is the principal moment of inertia about the i-axis and x) is the angular velocity correlation time about that axis. The components of the rotational friction tensor g„ may be related to the intermolecular potential function of the asymmetric rotor [60]

where V is the potential energy of an assembly of N molecules with positions and orientations specified by the 6yV-dimensional vector RN and 0, is the angle of rotation about the /-axis. Unfortunately, it is very difficult to evaluate T) for a model system, however these times can be measured experimentally using nuclear magnetic resonance techniques [60].

Equation (7.6.1.33) is a general result, which may be applied to the anisotropic rotational diffusion problem. The advantage of the

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Chapter 7. Three-Dimensional Rotational Brownian Motion 387

approach we have developed is that it is valid for any potential V. Another advantage of the present treatment is that it is not based on the quantum theory of a free asymmetric rotor (as is that of Favro [58]) so that many results obtained in the context of the anisotropic rotational diffusion model may be rederived from the general Eq. (7.6.1.33) in a much simpler way than before (see, e.g., Section 7.6.2). In particular applications, Eq. (7.6.1.33) can considerably be simplified. For example, in isotropic rotational diffusion, where D^ = Dyy = Dzz, both A and S in Eq. (7.6.1.33) are equal to zero so that Eq. (7.6.1.33) reduces to

T £-/r>J \ _ 7'0' + l ) / n . / \ + I V ^V v CJ,n+S rJ,m+Q ZD AUn,m ~ n \Un,m + . ZJ Zu VR,S,Q^j,n,R,S^j,m,R,Q

dt 2 4R,S,QJ^j-R\

x[j (J +1) - j(j +1) - R(R +1)] (DJn+Sjn+Q ). (7.6.1.34)

Furthermore, if the Brownian particle may be approximated by a symmetric top and the rotation about the axis of symmetry does not influence the physical properties of interest, one may ignore the dependence on the angle / so that Eq. (7.6.1.34) yields

T ±LlDi \= ^ + 1VD j \ + I y J\ v rJ'n+s rJ-° LD iAunS) 0 \Un,0 r

A Zu ZJ vR,S,0L'j,n,R,S^ j.O.R.O dfX ' 2 \ ' "*R,SJ=\j-R\

x[j(J + l)-j(j + l)-R(R + l)](DJn+SS)). (7.6.1.35)

A radical simplification can also be achieved for axially symmetric problems, where one may ignore the dependence of the quantities of interest on the angles a and y. On using a known property of the Wigner D functions, viz., [25]

Di0(a,/3,Y)^Pj(cosp), (7.6.1.36)

where Pj(z) is the Legendre polynomial of the order;', and noting that

now the potential V may be expanded in Legendre polynomials as

V(P,t)/(kT) = 2Z vR(t)PR(cosP), R

with vR(t) = vR>0i0(0 , we have fromEq. (7.6.1.34)

^--^hhb t M<&»? (76137) ai z ^ R J=\j~f\ (7.6.1.37)

x[y( / + i ) - ; ( ; + i)-7?(/? + i)](p,). We remark that the recurrence Eq. (7.6.1.33) for the expectation

values of Wigner's D functions may also be obtained from the

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388 The Langevin Equation

corresponding Fokker-Planck (Smoluchowski) equation for the distribution function W(Q.,t) of the orientations of asymmetric top molecules, which is [8,58]

(7.6.1.38) Thus, the solution of Eq. (7.6.1.38) can be obtained by the same method as that of Eq. (7.6.1.18). The Langevin and Fokker-Planck equation treatments are equivalent and yield the same results. However, the Langevin equation approach has, in our opinion, the advantage that it allows one to derive Eq. (7.6.1.33) in a much simpler manner.

7.6.2 Linear response of an assembly of asymmetric tops

Here, we demonstrate how the linear response theory results for noninteracting dipolar asymmetric top molecules may be obtained from Eq. (7.6.1.32). Let us suppose that an external spatially uniform small dc electric field E (£= juE/ (kT) « 1 ) had been applied to the system of asymmetric top molecules at t = - °o in the direction of the Z-axis of the laboratory coordinate system and at time t = 0 the field has been switched off. The electric polarisation Pz (t) is defined as

Pz(t) = N0(juz)(t), (7.6.2.1)

where No is the concentration of dipolar molecules and juz is the projection of the dipole moment [i onto the Z-axis given by [25]

Mz=Ml, (-l)"u-p)Dl0p(a). (7.6.2.2)

P =- i

Here u0) =uz and w(±1) = + (ux ±iuy)lS

are the irreducible spherical tensor components of the first rank [25], and u„ uy, and uz are the components of a unit vector u in the direction of the dipole moment vector \i.

We are interested in the decay of the polarisation Pz (t) of the system of the molecules starting at t = 0 from the equilibrium state I with the Boltzmann distribution function

/ / Z (Q)£" wI(£i) = z-lef*ia)Em)»— 8;zr2

1 + ^ kT

(7.6.2.3)

to the equilibrium state II with the uniform distribution function

W n ( ^ ) = l/(8^r2), (7.6.2.4)

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Chapter 7. Three-Dimensional Rotational Brownian Motion 389

which is reached at t —> °°. According to Eqs. (7.6.2.1) and (7.6.2.2), the behaviour of the electric polarisation Pz (t) is completely determined by

( 4 o ) ( 0 , ( < _ 1 ) ( 0 , a n d ( D ' 1 ) ( 0 .

Equations of motion for these functions at t > 0 can be obtained from Eq. (7.6.1.32) by setting VRJS,Q = 0,j= 1, n = 0, and m = 0, ±1. We have

^ ( 4 o ) ( ' ) + (flo.o)(0 = 0, (7.6.2.5)

^ ^ ( 4 ± i ) ( 0 + d + A ) ( 4 ± 1 ) ( 0 = - ( S / 2 ) ( 4 T l ) ( 0 . (7.6.2.6)

In Eqs. (7.6.2.5) and (7.6.2.6), the initial values of (Dl0p\(t) at t - 0 are

determined from the following equation

( D ; P ) ( 0 ) = J DlP(a)Wi(a)da.

Thus, we can obtain from Eqs. (7.6.2.2) and (7.6.2.3) in the low field strength limit [juElikT)«1]

( 4 0 ) ( 0 ) ~ £ M z / 3 , (D^± 1)(0)-+^(M ; c±^)/(3V2). (7.6.2.7)

Here, we have also used Eqs. (7.6.1.19) and (7.6.1.20). The solutions of Eqs. (7.6.2.5) and (7.6.2.6) are

(Dl0fi)(t) = (£uz/3)e-(D™+D»]', (7.6.2.8)

-(Dyy+D^t -(D„+D„)t uxe

v w ' ±iuye v " zz; (7.6.2.9) (4±1(0 = +(£/3V2)

where the relations

TDl(l + A-Z/2) = Dzz+Dyy, TD

l(l + A + E/2) = D:z+Dxx (7.6.2.10)

have been taken into account. Now, one has from Eqs. (7.6.2.1), (7.6.2.8), and (7.6.2.9)

pz(t)=E^^uy(D-+D-)'+uyiD-+D^+uyiD-+D-)t .(7.6.2.11)

Having determined Pz(t), one may also evaluate from Eq. (7.6.2.11) other dielectric characteristics such as the complex susceptibility aoi):

,-2 M2N0

3kT

2 2 ? ui uv u1

-+ + y , "z l+ia>/(Dw+Da) l+ia>KD„+Da) l+;<y/(Z^+Dvv)

yy z z ' *""\"xx zz' """\"xx yy > which is the result of Perrin [57]. We remark that the quantum theory of a rigid asymmetric rotor (which was the basis of the previous theoretical approaches [58-63]) has not been used here in order to derive Eq. (7.6.2.11).

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390 The Langevin Equation

7.6.3 Response in superimposed ac and strong dc bias fields: perturbation solution

For the purpose of illustration, we also calculate the dielectric response for rigid rodlike molecules in superimposed external ac and strong dc bias electric fields. As far as symmetric top molecules with the dipole moment H directed along the axis of symmetry are concerned, this problem was solved by Coffey and Paranjape [70]. However, these results are not applicable to asymmetric top molecules.

In what follows, let us suppose, for simplicity, that the diffusion tensor D has only two distinct components D^ -Dyy =Dj_ and D^ -D\\. This approximation is reasonable for rodlike molecules, where D^ ~Dyy, so that Dp and D± are the rotational diffusion coefficients about the long and short axes of the molecule, respectively. Thus,

r D = ( 2 D ± ) - \ A = ( D n / D ± - l ) / 2 , and E = 0

Furthermore, let us suppose that the molecules are subjected to superimposed external electric ac Ei(f) and strong dc bias E0 fields (both directed along the Z axis) and consider an ensemble of rigid nonpolarisable polar molecules, where the dipole vector \i is oriented at an angle 0 to the direction of the long axis of the molecule. Due to the cylindrical symmetry about the Z-axis, only the moments DJ

nm)t) with n = 0 are required in the calculation of Pz(t), so that from a mathematical viewpoint, Eq. (7.6.1.32) becomes a recurrence equation, where only two indices vary. Here, the polarisability effects are ignored (equations, which take into account these effects, are given in [66]). Without loss of generality, the molecular coordinate system for a rodlike molecule can always be chosen so that uy = 0, whence ux = sin 0, uz = cos 0 in Eq. (7.6.2.2). The potential energy of the molecule is then

V/(kT)= X v 1 A Q < e , (7.6.3.1) e=-i

where

v i A o = - c o s 0 t # o + 4 ] . vi,o,±i=±sin0 Kfr)+&yJi > £0 =juE0/(kT), and £(?) =juE(t)/(kT) so that Eq. (7.6.1.32) yields

^f(^' , m ) = -(Am2+j(; + l)/2)(Di>m)

- T Z Z \o,QCiZuoC^$[jU + » + 2-J(J + l)-4AmQ](Dim+Q)

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Chapter 7. Three-Dimensional Rotational Brownian Motion

which for n = 0 may be written as

391

i±i±ll+m2A Dim) *4K>-2(27 + 1) L \ ' x ' .

+ ' ' (7 + 2AW)V(;' + m + l ) ( ; + m + 2) (Dj j ;1

+ 1 ) - (7-2Am)

xV(;-m + l)(7-m + 2)(D0^1_1)+(;' +1 - 2Am)V0"-m-l)0"-m)(D&, ,+,)

- (y +1 + 2Am)V(7 + /n-l)(y + m) ( D ^ 1 . , ) ] . (7.6.3.2)

Insofar as the values of the field parameters §Q and £are very small ( « 1 ) for the majority of polar molecules even at field strengths ~107 V/m, one may apply perturbation theory in order to calculate the nonlinear response. Here, we shall restrict ourselves to the ac response nonlinear in

the dc bias field £> (up to third order) and linear in the ac field £xeim

(higher-order terms may be calculated in like manner). Now, one can

obtain from Eq. (7.6.3.2) equations for (.Do-I) - O o I) and ( D Q 0 ) ,

which are necessary in order to evaluate the nonlinear response, viz.,

*D^(<o) + (4o) = tf(') + £0] dt

x COS0[ sin0 ['-K>]-fx[K.)-K

(7.6.3.3)

dt

= tf(0 + &)(^(1+frine+^^(fl8o)

cos©

2>/3 K.>-K); 3 3V2

(l + 2A)sinG

(7.6.3.4)

2^3 (DI_2)+(D12)

On seeking a solution in the form

(4 ,o ) " [*o(£o ) + 6 **%(£<> ,<y')]cos0/3, (7.6.3.5)

(D O _,) - (D01) » [2s0 (<f0) + £ eimxy (<f0, *>')]sin 0/(3V2), (7.6.3.6)

where s0 (£0) = £0 (1 - ^ /15) and «/ = <yr0 , one can evaluate

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392 The Langevin Equation

(D20,0), (Dll)-(D

20l) and (D0

2_2) + ( D 22 )

from Eq. (7.6.3.2) in the linear (in £0 approximation:

£ o ( l + 3cos20)f +^e'

60 3 + id

, 4xncos 0 - x , sin © 1+—- l-

l + 3cos20

1+-X,

(7.6.3.7)

(7.6.3.8)

(6 + 4A)x0 +3xj

4(3 +A)

(7.6.3.9) Having determined x0 and xi from Eqs. (7.6.3.3) - (7.6.3.9), one can calculate the polarisation Pzt) from Eq. (7.6.2.1) and the nonlinear dielectric increment Se - £nonlin(co,^0) - elin(co), i.e., the difference between the nonlinear dielectric permittivity £nonUn(co,^0) given by

£nonlin (® .5 ) ) ~ « - « COS2 0 X Q ( £ , , 0)') + s in2 0Xx (£„, fl/)/2

and the linear permittivity £lin(a>) = enonlin(co,0). The increment <?£is Se = Se'+iSe" = A(es,eJF(d,^0), (7.6.3.10)

where

FK) ~£I. c o s z 0

30 [ 1 + itf/

2 2(1 + A) + ico'

3 (3 + iV)(l + A + /<»')

s i n 2 0

3 + A + id 3 + A +

2(l + A)(3 + A) + (6 + 5A)ia>'

2(1 + I V ) ( 1 + A + I V )

'\

cos2 0

3 + id 3 +

3(1 + A) + (3 + A)IQ>"

(l + rV)(l + A + jft/)

, ' \

sin2 0

2(l + A + ?V)

( l -2A)(6 + I V) + c o s 2 0

3(3 + 1'©')

3(1 +A)

2(2A-1) ( 1 - 2 A ) ( 6 + J V )

(l + id)(3 + ifi/)

2(6 + 2A + jft/)N

(3 + «V)

(3 + A + I V ) ( 1 + A + J V )

3 + 2A

(l + ia/)(3 + A + »y')

(3 + A + ifl/)

/ + sin2 0

(l + A)(l-2A)

(l + A + id)(3 + id)

(1 + A)(1 + 2A)(3 + 4A) (l + 2A)(6 + 8A + zft/)

(1 + A + id)(3 + 4A + ico') (3 + 4A + ico') (7.6.3.11)

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Chapter 7. Three-Dimensional Rotational Brownian Motion 393

The function A(£„£u) takes into account the internal field effects; this function depends on the model of the local field used (appropriate equations for A(es,eJ) are given, e.g., in Ref. [71]). For 0 = 0 (symmetric top) or A = 0 (isotropic diffusion), Eq. (7.6.3.11) reduces to the corresponding result of Coffey and Paranjape [70]:

F(^)^-&,+(' + "°:f+"°'!3). (7.63.1?) v °> ° 45(1 + ia>)2(l + /o>73)

The principal difference between Eqs. (7.6.3.11) and (7.6.3.12) is that Eq. (7.6.3.11) contains the contribution of the rotation about the long molecular axis to Se.

Experimental and theoretical dielectric increments Se of dilute solutions of mesogenic 10-TPEB molecules (CioH2i-0-0-CH2-CH2-0-N=C=S, where 0 = C6H4) in benzene have been compared in Ref. [66]. The spectra of Se were measured in superimposed strong dc (1.1 x 107 V/m) and small (« 100 V/m) ac electric fields in Ref. [71]. For the 10-TPEB molecule, the angle 0 is markedly different from zero, viz., 0 = 42° ±2° [71]. In the fitting, the experimental value of the relaxation time xD- 8.57-10~10 s [71] has been used so that the only adjustable parameter was A. The least mean squares fitting procedure yields A ~ 3.85. The comparison of the real and imaginary parts of the experimental and theoretical Seand of the nonlinear Cole-Cole plot for a dilute solution of 10-TPEB molecules in benzene at 15 °C is shown in Fig. 7.6.3.1. It appears that the theory correctly describes the shape of the observed spectra; here, five modes with different characteristic frequencies (due to molecular rotation about the long and short molecular axes) contribute to the spectra. Moreover, the calculation demonstrates that the theory also explains the temperature dependence of the nonlinear dielectric decrement Se: it describes the nonlinear spectra measured at 6 and 25 °C in Ref. [71]

It is of interest to compare the value of A so obtained with that estimated in the hydrodynamic limit when the Brownian particle is much larger than the fluid molecules and exhibits no slip with the fluid as it rotates; here the diffusion tensor depends only on the particle shape [64]. For long rods (L » R, where L and R are the half-length and radius of a rod), the hydrodynamic theory yields [72]

x2 1 A = . (7.6.3.13)

41n(2x) 2 Here x = LI R and equations for Dn and D± from Ref. [72] have been used in order to obtain Eq. (7.6.3.13). The value A = 3.85 corresponds to a shape parameter x ~ 6.7 in Eq. (7.6.3.13) and reasonably characterises the geometrical structure of the 10-TPEB molecule [71].

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394 The Langevin Equation

k '©

3s cx 'o

0

-10

-20-

-30-

-40-

^ V % 2 ,'

1, 1' - &' ^\*J*# 2,2'- Se" \ / Y

V J "

„ i . „ o j j U

a)

0-

-5-

3s X

b-io-"*

-15-

f « •I

•\ • \ •\

^ • >*

b)

* / • • </.

• . *

••

1 2 3 4 log10[/(MHz)]

-30 -20 -10 0 W^xSe1

Figure 7.6.3.1. a) Nonlinear dielectric decrement Se(f) \f= (D/2TC\ of solution of 10-TPEB molecules in benzene at 15 °C. Circles are the experimental data [71]; solid lines are the best fit from Eq. (7.6.3.11); dashed lines are the Coffey-Paranjape Eq. (7.6.3.12). b) Nonlinear Cole-Cole plot for a solution of 10-TPEB molecules in benzene at 15 °C. Solid circles are the experimental data from Ref. 71; solid line is Eq. (7.6.3.11).

Our approach can also be used for the evaluation of the dynamic

Kerr effect, where the quantities of interest are (Dgm)(0 [61]. Moreover,

it can be applied (with small modifications) to the calculation of the nonlinear magnetic response of ferrofluids. Thus, the method we have developed provides a useful basis for future studies of the nonlinear response of various physical systems comprising Brownian asymmetric top particles. An example of the application of the anisotropic rotational diffusion model to dielectric relaxation in nematic liquid crystals will be given in Chapter 8, Section 8.7.

References

1. 2. 3. 4. 5. 6

9. 10. 11. 12. 13. 14. 15.

R. Ullman, J. Chem. Phys. 56, 1869 (1972). A. Morita, J. Phys. D: Appl. Phys. 11, 1357 (1978). A. Morita and S. Watanabe, Adv. Chem. Phys. 56, 255 (1984). W. T. Coffey, Yu. P. Kalmykov, and K. P. Quinn, J. Chem. Phys. 96, 5471 (1992). J. T. Waldron, Yu. P. Kalmykov, and W. T. Coffey, Phys. Rev. E 49, 3976 (1994). J. L. Dejardin, Yu. P. Kalmykov, and P. M. Dejardin, Adv. Chem. Phys. 117, 275 (2001). A. J. Martin, G. Meier, and A. Saupe, Symp. Faraday Soc. 5, 119 (1971). P. L. Nordio, G. Rigatti, and U. Segre, Mol. Phys. 25,129 (1973). R. Tarroni and C. Zannoni, J. Chem. Phys. 95, 4550 (1991). W. T. Coffey and Yu. P. Kalmykov, Adv. Chem. Phys. 113, 487 (1998). M. Bee, Mol. Phys. 47, 83 (1982). W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963). W. F. Brown, Jr., IEEE Trans. Mag. 15, 1196 (1979). W. T. Coffey, P. J. Cregg, and Yu. P. Kalmykov, Adv. Chem. Phys. 83, 263 (1993). Yu. L. Raikher and M. I. Shliomis, Adv. Chem. Phys. 87, 595 (1994).

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Chapter 7. Three-Dimensional Rotational Brownian Motion 395

16. J. L. Garcia-Palacios, Adv. Chem. Phys. Ill, 1 (2000). 17. C. Scherer and H. G. Matuttis, Phys. Rev. E 63, 011504 (2001). 18. H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1984; 2nd Edition, 1989. 19. W. T. Coffey, M. W. Evans and P. Grigolini, Molecular Diffusion and Spectra,

Wiley, New York, 1984; Russian edition, Mir, Moscow, 1987. 20. R. L. Stratonovich, Conditional Markov Processes and Their Application to the

Theory of Optimal Control, Elsevier, New York, 1968. 21. E. A. Milne, Vectorial Mechanics, Methuen, London, 1948. 22. J. McConnell, Rotational Brownian Motion and Dielectric Theory, Academic, New

York, 1980. 23. C. W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin, 1982. 24. G. A. Korn and T. M. Korn, Mathematical Handbook, McGraw Hill, New York,

1968. 25. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of

Angular Momentum, World Scientific, Singapore, 1998. 26. L. Neel, Ann. Geophys. 5, 99 (1949). 27. C. P. Bean and J. D. Livingston, Suppl. J. Appl. Phys. 30, 1205 (1959). 28. T. L. Gilbert, Phys. Rev., 100, 1243 (1955) [Abstract only; full report, Armour

Research Foundation Project No. A059, Supplementary Report, May 1, 1956]. 29. L. J. Geoghegan, W. T. Coffey, and B. Mulligan, Adv. Chem. Phys. 100, 475

(1997). 30. I. Klik and L. Gunther, J. Stat. Phys. 60, 473 (1990). 31. Yu. P. Kalmykov and W. T. Coffey, Phys. Rev. B. 56, 3325 (1997). 32. Yu. P. Kalmykov and S. V. Titov, Fiz. Tverd. Tela (St. Petersburg) 40, 1642 (1998)

[Phys. Solid State, 40, 1492 (1998)]. 33. Yu. P. Kalmykov, S. V. Titov, and W. T. Coffey, Phys. Rev. B 58, 3267 (1998). 34. Yu. P. Kalmykov and S. V. Titov, Zh. Exp. Teor. Fiz. 115, 101 (1999) [Sov. Phys. -

JETP 88, 58 (1999)] 35. Yu. P. Kalmykov and S. V. Titov, Phys. Rev. Lett. 82, 2967 (1999). 36. R. Kikuchi, J. Appl. Phys. 27, 1352 (1956). 37. L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion, 8, 153 (1935). 38. Yu. P. Kalmykov and S. V. Titov, J. Magn. Magnet. Mater. 210, 233 (2000). 39. L. Landau and E. Lifchitz, Mecanique Quantique, Mir, Moscow, 1967. 40. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and J. T. Waldron, Phys. Rev. B

51, 15947 (1995). 41. Yu. L. Raikher and M. I. Shliomis, Zh. Eksp. Teor. Fiz. 67, 1060 (1974) [Sov. Phys.

-JETP 40, 526 (1974)]. 42. M. I. Shliomis and V. I. Stepanov, Adv. Chem. Phys. 84, 1 (1994). 43. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, E. S. Massawe, and J. T.

Waldron, Phys. Rev. £49, 1869 (1994). 44. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and J. T. Waldron, Physica A

213,551(1995). 45. M. Abramowitz and I. Stegun, Eds., Handbook of Mathematical Functions, Dover,

New York, 1964. 46. H. S. Wall, Continued Fractions, Van Nostrand, Princeton, 1948.

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396 The Langevin Equation

47. A. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series, More Special Functions, vol. 3, Gordon and Breach, New York, 1990.

48. W. T. Coffey and D. S. F. Crothers, Phys. Rev. E 54, 4768 (1996). 49. W. T. Coffey, Yu. P. Kalmykov, and E. S. Massawe, Adv. Chem. Phys., 85, 667

(1993). 50. Yu. P. Kalmykov, Phys. Rev. E 61, 6320 (2000). 51. P. J. Cregg, D. S. F. Crothers, and A. W. Wickstead, J. Appl. Phys., 76, 6320

(1994). 52. W. T. Coffey, Yu. P. Kalmykov, and J. T. Waldron, Liq. Crystals 18, 677 (1995). 53. H. Bateman, A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher

Transcendental Functions, vol. 1, McGraw Hill, New York, 1953. 54. M. A. Martsenyuk, Yu. L. Raikher, and M. I. Shliomis, Zh. Eksp. Teor. Fiz. 65, 834

(1973) [Sov. Phys. - JETP 38, 413 (1974)]. 55. H. Block and E. F. Hayes, Trans. Faraday Soc. 66, 2512 (1970). 56. P. C. Fannin, B. K. P. Scaife, and S. W. Charles, /. Magn. Magn. Mater. 85, 54

(1990). 57. F. Perrin, J. Phys. Radium, 5, 497 (1934). 58. D. L. Favro, Phys. Rev. 119, 53 (1960); in Fluctuation Phenomena in Solids, Edited

by R. E. Burgess, Academic, New York, 1965, p. 79. 59. J. H. Freed, /. Chem. Phys. 41, 2077 (1964). 60. W. T. Huntress, Adv. Magn. Reson. 4, 1 (1970). 61. W. A. Wegener, R. M. Dowben, and V. J. Koester, J. Chem. Phys. 70, 622 (1979). 62. W. A. Wegener, J. Chem. Phys. 84, 5989 (1986). 63. W. A. Wegener, J. Chem. Phys. 84, 6005 (1986). 64. K. Hosokawa, T. Shimomura, H. Furusawa, Y. Kimura, K. Ito, and R. Hayakawa, J.

Chem. Phys. 110,4101 (1999). 65. E. Berggren, R. Tarroni, and C. Zannoni, J. Chem. Phys. 99, 6180 (1993). 66. Yu. P. Kalmykov, Phys. Rev. E 65, 021101 (2002). 67. H. Brenner and D. W. Condiff, J. Colloid Interface Sci. 41, 228 (1972). 68. A. Morita, /. Chem. Phys. 76, 3198 (1982). 69. D. H. Lee and R. E. D. McClung, Chem. Phys. 112, 23 (1987). 70. W. T. Coffey and B. V. Paranjape, Proc. R. Ir. Acad. Sec. A 78, 17 (1978). 71. J. Jadzyn, P. Ke.dziora, L. Hellemans, and K. De Smet, Chem. Phys. Lett. 302, 337

(1999). 72. W. A. Wegener, Biopolymers 20, 303 (1981).

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Chapter 8

Rotational Brow nian Motion in Axially Symmetric Potentials:

Matrix Continued Fraction Solutions

8.1 Introduction

The exact analytic solutions obtained in Chapters 4-7 are available because the Langevin equations may be reduced to the solution of a scalar three-term recurrence relation. In the majority of problems, the Langevin equation may not be reduced to a scalar three-term recurrence relation. Hence, the method based on conversion of the recurrence relation to an ordinary continued fraction no longer applies. Examples of this are problems, which involve diffusion in phase space and diffusion in configuration space where the form of the potential is such as to give rise to a four- or higher-order term recurrence relation. These difficulties may however be circumvented since, as we have shown in Chapter 2, there exists a method of converting a multi-term scalar recurrence relation to a three-term matrix one. We recall that such a matrix recurrence relation, in the notation of Risken [1], may be written down as

recp(o=Q;cp_1(o+Q /7c /,(o+Q;cp+1(o, CS.I.D

where the Cp(t) are column vectors and the Q~, Qp , and Q+p are time

independent non-commutative matrices. As shown in Section 2.7.3, the

solution of Eq. (8.1.1) for the Laplace transform Cp(s) is given by

Cp(s) = Sp (*)£„_! (s) + r£ [sreI -Qp- Qpp+[ (s)Jl C„(0)

+1 ffl Q^-iT^-Qp^-Q^s^^c^T'lc^co) 71 = 1 V* = l /

(8.1.2)

397

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398 The Langevin Equation

Equation (8.1.2) constitutes the exact solution of Eq. (8.1.1) rendered as a sum of products of matrix continued fractions in the s domain. In this way, quantities such as the complex polarisability, the correlation time, etc., may be calculated exactly in terms of sums of products of matrix continued fractions, so extending our previous solutions to vector valued functions.

First, we shall illustrate the method of solution by referring to the problem of dielectric relaxation of a single axis rotator subjected to an anisotropy potential [2]

V(9) = U0 sin2 6 (8.1.3) arising from the crystalline field and a strong uniform field E applied parallel to the anisotropy axis. In Eq. (8.1.3) UQ is the potential barrier between the sites and 8 is the angle describing the orientation of the dipole about its axis of rotation. Next, we consider the three dimensional analogue, viz., the rotational Brownian motion in a uniaxial potential subjected to a uniform field. This simple model is of interest in the context of the problem of dielectric relaxation of nematic liquid crystals and molecular crystals [3] in the presence of a bias field. The model may be applied with a small modification to the mathematically similar problem of magnetic relaxation of single domain ferromagnetic particles [4,5]. These models were treated in Chapters 4 and 7 in the absence of the field.

8.2 Application to the Single Axis Rotator

We suppose that the dielectric consisting of an assembly of dipolar molecules with dipole moment \i in the crystalline potential V, Eq. (8.1.3), has been influenced for a long time by the strong dc field E. We consider here, following [6], an assembly of the molecules with each molecule compelled to rotate about an axis normal to itself. We also suppose that the inertia of the molecules may be neglected and that the electrical interaction between each member of the assembly may be ignored so that on the average all molecules of the assembly behave in the same way. Thus, it suffices to consider the behaviour of one molecule only. Hence, the problem is reduced to considering the rotational Brownian movement in two dimensions of a dipole or rigid rotator in the presence of an external potential (Chapter 4).

8.2.1 Longitudinal response

In order to study the longitudinal relaxation, we write the potential as

V/(kT) = 2a sin2 0-(^ + )cos0, (8.2.1.1)

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Chapter 8. Rotational Brownian Motion in an External Potential 399

where

£ = //£/(*r),£=//£1/(*7')«l are the external field parameters, and 2cris, as usual, the barrier height parameter. We suppose that the small probe field Ei is switched off at time t = 0.

The appropriate set of differential-recurrence relations can be obtained from Eq. (4.2.23), Chapter 4, and is given by

TDfp(t) + p2fp(t) = ap[fp_2(t)-fp+2(t)]

+ ^ [ / P - i ( 0 - / P + i ( 0 ] / 2 ,

wherep = 0, ±1, ±2..., TD is the Debye relaxation time, and fp(t) = (cos P0)(t)-(cos Pe0

(the symbol ( )0 denotes the average in the absence of the probing field Ei) with f_p(t) = fp(t). The quantity of interest is the decay of the

polarisation of the system P(t) defined as P(t) = juN0 [(cos 6) (0 - (cos 6)0] = fiN0Mt), (8.2.1.4)

where No is the concentration of the dipoles, so we require an expression for fi(t). Here it has been assumed that the polarisation arises entirely from molecular orientation so that the induced dipole moment is ignored.

Equation (8.2.1.2) is a five-term recurrence relation. However, it may be cast into the form of Eq. (8.1.1). Indeed, Risken [1] has shown that generally the recurrence relation with L nearest neighbour coupling

(8.2.1.2)

(8.2.1.3)

*„(*)= ZAta+tCo (8.2.1.5) k=-L

may be cast in the form of Eq. (8.1.1) by creating the column vectors with L components

Cp(t) =

LL(p-\)+\

-L(p-l)+2

(0' (0

%(0

(8.2.1.6)

and the matrices Qt»QD with the matrix elements

mmn=Aii;^m, (Q,)m

where one must set Af = 0 for |A:| > L.

Here, it will be convenient to define

"^(p-lj+m '

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400 The Langevin Equation

Cp(0 = l % - i ( 0 N

flpit),

Hence, Eq. (8.2.1.2) becomes for/? > 1

*DCP (0 = Q^C,.! (0+QpCp (0+ Q;c p + , ( 0 ,

where Qp,Qp and Qp are 2x2 matrices which are given by

(a(2p-\) t(2p-l)/2^

\ 0 2(7/7 ,

(8.2.1.7)

(8.2.1.8)

%=\

QD = -2p-\)2-^p-\)l2\

p "4p2 (8.2.1.9)

Q:= -o2p-\) 0 ^

-#P -2crP) We note that for/? - 1, our set, Eq. (8.2.1.2), can be reduced to a two-term recurrence relation since it has the form

(*1 ' f \J2J

a <f/2N

.0 2<7, +

f-\ -£/2 f f.\ (.

J

fx +

-a u \(f3

-Z -^UA . (8.2.1.10)

I4J

(8.2.1.12)

On noting that f_(t) = /j(f) , Eq. (8.2.1.10) is evidently of the form

rDC1(0 = AllC1(0 + Q^C2(f), (8.2.1.11) where

Jo-I -/2\ U -4 I

On applying the Laplace transform to Eqs. (8.2.1.8) and (8.2.1.11), we have the recurrence equations in the s domain:

( ^ • 2 ) I - A I I ) C 1 ( J ) = T D C 1 (0 ) + Q + C 2 ( * ) (8.2.1.13) and

(srDl-Qp)Cp(s)-Q-Cp_l(s)-Q+pCp+1(s) = TDCp(0), (p>2). (8.2.1.14)

According to Eq. (8.1.2), the exact solution of Eqs. (8.2.1.13) and

(8.2.1.14) for C^s) in terms of matrix continued fractions is

C, (s) = TD [szDl - A11 - QtS2(s)Jl [C, (0) + f \ f l Q t i S , (s) (<£) ( n m

n=2\k=2

C„(0)

(8.2.1.15)

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Chapter 8. Rotational Brownian Motion in an External Potential 401

where the matrix continued fraction Sp(s) is given by (see Section 2.7.3)

Sp(s) = (sTDI-Qp-Q+

pSp+l(s)Yi Q~;. (8.2.1.16)

Having determined the Laplace transform f(s) of the decay function fx(t) from Eq. (8.2.1.15), we may calculate the longitudinal complex susceptibility ^n(<w), namely,

^ - - = l-icoCMco), (8.2.1.17) 4(0) "

where . 2 ,

4 ( 0 , = ^ M < 2 ) = ^ [ ( c o s ^ < 0 ) ) o - ( c o s S ( 0 ) ) ; (8.2.1.18)

is the static susceptibility and

Cn(ja» = Mm/M0) (8.2.1.19)

is the one-sided Fourier transform of the normalised longitudinal autocorrelation function C\\(t) defined as

C | | W Jcos g (0 ) cos 9 (0 ) „ - ( C Q S g (0 ) ) ; ( 8 2 ] 2 0 )

(cos20(0) -(cos0(0));; /o

The longitudinal correlation time, t\\, is as usual the area under the curve of the normalised longitudinal autocorrelation function. We have

Tj, = f CM)dt = limCn(s) = ^^-. (8.2.1.21)

" J " '-><> " / , ( 0 ) Having written down formal expressions for the correlation time and the complex polarisability, it is necessary for the purpose of computation to have expressions for the initial values ,(0).

The initial values ^,(0) can be calculated in terms of matrix continued fractions as shown in Section 2.7.3. The fp(0) in the linear approximation in £,\ are given by

lit

J Wo(0)[l + £cos0]cosp&/0 2x

fp(Q) = JL— J Wo(0)cosp0d0

J Wo(0)[l + £lcos0]d0 ° 0

(2x 2n IK 1 = 4i\\ Wo(0)cos0cosp0d0-jWo(0)cosp0d0JWo(0)cos0d0i

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402 The Langevin Equation

where

U(cos(p -l)0)Q + (cos(p +1)6>)0) -(cos6>)0 (cos p0)Q , (8.2.1.22)

2/r

Wo(0) = Z-le'JcosW+tcose and Z = J e°«»™+t"»°de. (8.2.1.23)

According Eq. (8.2.1.2), the equilibrium averages fp = (cosp#0 satisfy

the set of equations

p2f°p-ap(f;_2-f0p+2) + ^p(f0

p_1-f^)/2, (8.2.1.24)

which can be transformed into a matrix three-term recurrence relation as follows

J2p-3

J2p-2)

\J2 J +Q,+

(f0\ J3

J4 \U J ( fO ^

J2p-\

f° +Q:

K2a j

J2p+\

(8.2.1.25)

f° \J2p+2)

= 0,(p>l), (8.2.1.26)

where Q„,Q~, and A" are the same 2x2 matrices as in Eqs. (8.2.1.9) and (8.2.1.12). The solution of Eqs. (8.2.1.25) and (8.2.1.26) is given by

f f° >

J2p-\

f° V J2P )

In particular,

= Sp (OS,. , (0).. .S2 (0) [-A11 - Q+S2 (O)j 2a)

. (8.2.1.27)

^]=[-A'-«s.«»r(^ \J2 J V Thus, on using Eqs. (8.2.1.22) and (8.2.1.27), we can now evaluate the

initial conditions Cp(0) in terms of matrix continued fractions,

C„(0) = 6 (0 l/2^/2

Op_3),f-<cos0>o 1/2 ^

0 0 \hp-2)

(0 1/2 (~(cos0)o 1/2 ^

1/2 -(cos 6%

Sp(0) + 0 0 ) 1/2 -<cos6>>0J

xSp_1(0)...S2(0)[-All-Q^S2(0)]

/2p-l

KJ2P J

0 0' 1/2 0,

-if<f/2^ 2C7

f ,-0 ^ J2p+1

Jlp+2,

0 01 [l/2 0)

S,+1(0)Sp(0)

(8.2.1.28)

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Chapter 8. Rotational Brownian Motion in an External Potential 403

On the other hand, the initial values /p(0) can also be evaluated analytically in terms of the modified Bessel functions I£z) of the first kind [7] as described below. The integrals appearing in Eqs. (8.2.1.22) and (8.2.1.23) may be evaluated by noting the Fourier series

<rcos20+£cos0 oo oo

= Z E ImWntfV"0''2"" • (8-2.1.29) m=-°° n=-°°

By the orthogonality property of the circular functions and the equality

In=I-n [7], we have

Z = 2^^Im(.a)I2m^) = 2x /o«r)/0(£) + 2£/m(<T)/2m(#) m=\

27C cosp0)o=^- / o ^ ^ + I ^ W ^ ^ + V , ^ ) ] ,

m=l

(cosecosp0)o=^lo(c7)[lp+^) + Ip_l^)]

12m+p-l

so that, finally, we have

/p I " ) - Si 2[/0(cr)/0(^) + 2X:=1/m(cT)/2m(^)]

(/0(cr)/0^) + 2X:=1/m(tT)/2m^))

(8.2.1.30)

8.2.2 Transverse response

In the analysis of the transverse response, the perturbing field Ei is applied perpendicular to the easy axis of the potential Eq. (8.1.3) (see Chapter 4, Section 4.2). Thus, we have

*D8p<!) + P28p(t) = <rp[gp-i(t)-gp+2(t)]

+^-[gP-i(t)-8P+i(t)], (8.2.2.1)

where gp(t) = (sinpd)(t) and g_p - -g . On defining

*2p(0 J ' Cj(0 = (8.2.2.2)

we can rearrange Eq. (8.2.2.1) as

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404 The Langevin Equation

TDtt(t) = ALCi(t) + QtC^(t), (8.2.2.3)

TD&PV) = Q-pC±

p^(t)+QpC±

p(t)+ Q+pC

Lp+l(t), (p > 2), (8.2.2.4)

where Q~,Q„, and Qt are the matrices given by Eqs. (8.2.1.9) and

'-a-\ -^/i) (8.2.2.5)

The solution of Eqs. (8.2.2.3) and (8.2.2.4) is given by [cf. Eq. (8.2.1.15)]

CiL(5) = T D [5 r D I -A i -Q^( J )T 1 [ c^ (0 ) + | ; f n ( ^ _ A w ( « r 1 W ( 0 )

n=2 U=2 J

(8.2.2.6)

where the matrix continued fractions Sp(s) are defined by Eq. (8.2.1.16).

The initial conditions differ from the longitudinal case. We have

gp(O) = ^(Sm0smpd)o = | ( c o s ( p - l ) 0 ) o - ( c o s ( p + l)0)o] . (8.2.2.7)

Just as the longitudinal response, we can express gp(0) in terms of the

modified Bessel functions as follows

-w[w£>-Vi<£>] 8p(0) 2/0(cT)/0(^) + 2X:= 1 /m(c7)/2 m(#) '

2m+p+l (£)" 12m+p-l

( # ) - W l ( # ) + '2."-p-l(#)] (8.2.2.8)

allowing us to calculate Cp(0). On using Eqs. (8.2.1.27) and (8.2.2.7), we can also evaluate the initial conditions Cp(0) in terms of matrix continued fractions, viz.,

c£(0) = £ ' 0 1 / 2 ^ / 2 % ) ( 0 - 1 /2

0 0 \Jlp-2J + 1/2 0

(fo \ J2p-\ +

0 1/2

0 0)

0 -1 /2

1/2

f° V J2P J

0 (ft

0 0 -1/2 0

2p+l

0 /:

,S_(0) + , 0 J p 1-1/2 0

xSp_I(0)...S2(0)[-All-Q1+S2(0)J

\j2p+2

S,+,(0)S„(0)

if£/2)

2a J

The transverse complex susceptibility %±(GJ) and the transverse

correlation time T± are given by

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Chapter 8. Rotational Brownian Motion in an External Potential 405

Zl(0) S,(0) and

XM- = l-iaM*>. (8.2.2.9)

= J£i(0) (8.2.2.10) «i(0)

8.2.3 Relaxation times

In order to check our algorithm, we have compared numerically the correlation times ^ and Tx from Eqs. (8.2.1.21) and (8.2.2.10) with ^ and T± calculated by means of the definition

\ L = % \ k , (8.2.3.1)

where the A,k are the eigenvalues and the Ak are the corresponding amplitudes. Exact agreement was found between ^ and rL calculated by the both methods. However, the calculation based on Eqs. (8.2.1.15), (8.2.1.21) and (8.2.2.6), (8.2.2.10) is more efficient than that from Eq. (8.2.3.1). The correlation times ^ and T± are shown in Fig. 8.2.3.1.

The computation of the matrix continued fractions appearing in equations for the correlation times ^ and 7X was carried out as follows. In Eq. (8.2.1.20) with s = 0, we assume that

ro o i S„(0)= (8.2.3.2) f [o oj

for a particular large P. This allows one to evaluate numerically all the S/>_i(0), SP_2(0), ..., S^O). This procedure is implemented for successive large values of P until convergence is obtained. As far as practical computations of the graphs shown in Fig. 8.2.3.1 are concerned, P - 14 and the first 9 terms in the infinite sums were sufficient for convergence. The leading term of Eq. (8.2.1.15) for s = 0, namely,

[ W D I - A I I - Q + S 2 ( J ) ] " 1 C 1 ( 0 ) , (8.2.3.3)

leads, just as £=0 , to exponentially large behaviour of the longitudinal correlation time \. Our analysis demonstrates that the matrix continued fraction is an effective algorithm with the advantage that it is much faster than other numerical methods.

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406 The Langevin Equation

H 0 . 5 -

Figure 8.2.3.1. Correlation times T^ITD and TL/TD VS. cfor £ = 0 (curves 1), £= 1.5

(curves 2) and £ = 3 (curves 3); the effective eigenvalue Eq. (8.2.3.4) for 1/(TDA^) is

plotted by dashed lines.

An analytic formula for the transverse relaxation time may be easily written down using the effective eigenvalue method [7] (see Section 2.12) since exponential behaviour does not exist and agreement with the result predicted by the lowest eigenvalue is very close. All one has to do to determine the effective eigenvalue is to evaluate Eq. (8.2.2.1) at t = 0 thus

± gi(0) 1 a A e f - — + •

Si(0) <-D 1 + g3(Q)

*i(0). + £ g2(Q)

2TD gl(0) (8.2.3.4)

where the gp(0) are given by Eq. (8.2.2.8). The transverse relaxation time computed using this formula is also compared with the exact numerical solution in Fig. 8.2.3.1.

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Chapter 8. Rotational Brownian Motion in an External Potential 407

8.3 Rotation in Three Dimensions: Longitudinal Response

S.J. / Uniaxial particle in an external field

In Chapter 7, the relaxation behaviour in three dimensions has been examined for the simple uniaxial potential

V(*?) = £sin2tf. (8.3.1.1)

In applications to superparamagnetic particles (see Chapter 7, Section 7.3), this is an axially symmetric bistable potential with anisotropy constant K representing the free energy per unit volume of a particle. The second type of uniaxial potential which is of interest is when a constant field of arbitrary strength is superimposed on the anisotropy potential Eq. (8.3.1.1) [4,5,8-13]. In general, such a field can be applied at some angle to the crystalline field (Chapter 9). However, in order to preserve the axial symmetry of the problem and the consequent mathematical simplifications, we shall suppose as in Ref. [8] that the field is applied along the polar axis so that the potential V is of the form

^ ^ - = crsin2 tf-£costf = <r(sin2 tf-2hcos&), (8.3.1.2) kT

where a = Kvl(kT) is the barrier height parameter and £ = vHMs /(kT) is the external field parameter, with h = ^/(2a). Thus, we have a complete three-dimensional analogue of the two-dimensional problem considered in Section 8.2.

We have seen that this potential was originally introduced in the theory of superparamagnetism by Neel [8] who gave an expression for the time of reversal of the magnetisation using the discrete orientation approximation (Chapter 1, Section 1.17). It was further studied by Brown [14] who obtained approximate expressions for the lowest eigenvalue in the limit of large and small a using the Kramers escape rate method (Chapter 1, Section 1.18.1) and perturbation theory, respectively. Later, the smallest nonvanishing eigenvalue X\ was calculated numerically by Aharoni [4]. However, he did not calculate any other A,'s or the associated amplitudes including that of X\. Thus, it is impossible to ascertain the role they play in the relaxation process, neither was it possible to test the accuracy of the single mode approximation. On the other hand, the analysis presented by Garanin et al. [5] enabled them to derive an integral expression for the correlation time zj| from the Sturm-Liouville equation. However, rather than trying to calculate Ty exactly from their equation, they presented various asymptotic formulas for t\\ and the complex

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408 The Langevin Equation

susceptibility. The longitudinal dynamic complex susceptibility X\\(°^ a n d relaxation time t\\ have been calculated by using the matrix-continued fraction approach in Ref. [9]. Various aspects of the kinetics of uniaxial particles in a strong external field have also been studied, e.g., in [10-13]. It is the purpose of this section to study in detail (following the exposition of Coffey et al. [9]) the longitudinal relaxation behaviour of the system with V given by Eq. (8.3.1.2).

We have already obtained in Chapter 7 the differential-recurrence Eq. (7.3.2.13) for the equilibrium correlation functions//(f), namely:

2TK

1(1 + 1)

where

//(') + 1 — 2a

+

(2/-l)(2/ + 3)

2CT(/-1)

/i(0 = Tr - r [ /M(0- /w(0]

(2/-l)(2/ + l) / / -2<0-

2/ + 1

2<7(/ + 2)

(2/ + 3)(2/ + l )

(8.3.1.3)

//+2(0,

f,(t) = (cos^O)/ (cost?(0))0 -cas0)Q(P, (costf))o .

Now, we are interested in the behaviour of f\(t). This is more involved than the case £= 0 considered in Chapter 7 because Eq. (8.3.1.3) does not decouple into separate sets for even- and odd-index/; (t).

The set of equations Eq. (8.3.1.3) may be solved to yield the relaxation behaviour of fv(t) by either of two methods. The first is to arrange it in the form

X(r) = AX(0, (8.3.1.4) where the matrix A and the column vector X(0 are determined by Eq. (8.3.1.3) and are given by

A =

2a

5

-M 5

24<7

35

0

# 3

i-*I 7

_M 7

20a 21

2a

5

M 5

6 - ^ 15

9

— 0

24<r

35

7 20a

0 0

0 0

10-77

20<T

21

9

40(7'

33

(8.3.1.5)

and

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Chapter 8. Rotational Brownian Motion in an External Potential 409

X(0 =

rMt)^

flit)

flit)

(8.3.1.6)

V • J

The second is to arrange this five-term differential-recurrence Eq. (8.3.1.3) as a three-term matrix one, the Laplace transform of which may then be obtained in terms of matrix continued fractions as described for the corresponding two-dimensional problem earlier in this chapter.

The first method proceeds as follows (see Chapter 2, Section 2.7.1). In Eq. (8.3.1.4), / is taken large enough (equal to L say) to ensure a desired degree of accuracy. The lowest nonvanishing eigenvalue corresponds to the smallest root of the characteristic equation:

det( / l I-A) = o. (8.3.1.7) The relaxation modes of ft) may be found from Eq. (8.3.1.4) by assuming that the matrix A has a linearly independent set of L eigenvectors (R 1 . . . ,R / J , so that [15]

X(t) = b^'^% + b2e~^"R2 + ... + V ~ 4 ' R z , - (8-3-1 -8) where the bt are to be determined from the initial conditions. Equation (8.3.1.8) allows us to calculate the normalised correlation function Ci(0 = / i (0 / / i (0) defined as

1 CM): 2j*=i4fc

- \ t

The corresponding complex susceptibility Z\\(°^ is t n e n given by

Zn«o) = vj3M2sN0Yd-

A

(8.3.1.9)

(8.3.1.10) k^\ + iO)l\

Here we have made use of the linear response theory formula (Chapter 2)

X\\(o) = x'\\a>) ~ iX\\(a>) = ajj(0) l-io)j Cxt)e~imdt , (8.3.1.11)

where

4 ( 0 ) = vfiMJN0 [(cos2 tf)o - (cost?),

"2(^)o+i-3</>1>; VJ3MZ

SN0

3

(8.3.1.12)

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410 The Langevin Equation

(equations for the equilibrium averages (P1)0 and (P2)0 are given

below). We recall that the correlation time Tn may be evaluated more simply by noting that

Til = lim f Q (t)e~s'dt = C, (0) = / i (0) ,v->0

(8.3.1.13) / i (0)

with the tilde denoting the Laplace transform, where /j(0) is determined from the matrix equation

X(0) = -A_1X(0). (8.3.1.14) Here X(0) is the initial value vector, that is,

X(0) = A(0) /3(0)

(8.3.1.15)

(see Chapter 2, Section 2.9 for details). Equation (8.3.1.14) is obtained from the Laplace transform of Eq. (8.3.1.4) for s = 0. The correlation time -zji may also be calculated in terms of Ak and Xk as in Chapters 4 and 7.

The initial conditions for / ;(0) in Eq. (8.3.1.15) may be

determined as follows. We have, noting Eq. (7.4.2.11),

f!(0) = (PlPl)0-(Pl)0(Pl)0

l + \(^^Mo-WoWo-(8.3.1.16)

21 +

2a

The equilibrium quantities (Pz) satisfy the set of equations

2a(l + 2)

(2/-l)(2/ + 3)

2cr(/-l) (8.3.1.17)

te-2>0- : < ^ + 2 > 0 . (2/-l)(2/ + l ) w " ° (2/ + 3)(2/ + l)

which is Eq. (8.3.1.3) with /j = 0 . The (P;)0 for any / can be evaluated

by upward iteration from Eq. (8.3.1.17) by taking into account equations

for the equilibrium averages (P0)0 = 1, (Pi0, and (P2)0, viz.,

g^sinh^ £

aZ 2a ^ . ) o = : (8.3.1.18)

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Chapter 8. Rotational Brownian Motion in an External Potential 411

<*>>0 2aZ\ ( £ ^ | cosh£——sinh£ 2a +

3f 8<r2

_3_

4a 2' (8.3.1.19)

where

z = f ?fx+ax2dx

-A—e -Z2/4a erfr 4o~ + 2-Ja

+ erfr V ^ -2y[a.

(8.3.1.20)

is the the partition function and erf/' (x) is the error function of imaginary argument. We remark, however, that the upward iteration is unstable and one should use it with caution.

We have formulated above, the problem of determining the initial values //(0) from the recurrence Eq. (8.3.1.17). The (P/)0 may, however, be evaluated numerically by simply calculating A-1. First, we note that the set of Eq. (8.3.1.17) constitutes the inhomogeneous set

AF°=B (8.3.1.21) with

F° =

Wo' te>0

Wo

and B =

£/3 ' 20715

0

0

(8.3.1.22)

where we have noted that (P00 = 1. Thus, we can calculate numerically

all the equilibrium averages (Pz)0 from the equation

F° = A_1B. (8.3.1.23) This is the matrix solution of Eq. (8.3.1.17).

We now present the solution in terms of matrix continued fractions. The advantage of posing the problem in such a manner is that exact formulae in terms of matrix continued fractions may be written for the Laplace transform of the after-effect function, the correlation time and the complex susceptibility. The starting point of the calculation is the matrix differential recurrence relation Eq. (8.1.1) written as

^ ( 0 = Q r c w ( 0 + Q,C,(0 + QfC /+1(0. (8.3.1.24)

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412 The Langevin Equation

Equation (8.3.1.3) takes on the form of the matrix three-term differential recurrence relation Eq. (8.3.1.24) if we arrange it as follows

^ 2 ^

fv(t) = Q|-

/2/-3O'

V2/ -2W, /

1+

+ Q ! / 2 / - I ( 0 '

. /2/(0 . + Q»+ 72m(0 /2/+2(0.

, (8.3.1.25)

where the 2x2 matrices Q„,Q„ are

^4cr(/-l)(2/-l)/

Qi' =

Of"

1(21-1)

(21-1)1 (4/- l ) (4/-3) 4 / -1

2a(2l -1)(2/ + 1)Z 0

2(7

(4/ + l)(4/-l) J

-#(2Z-1)

(4Z-3)(4Z + 1)

#Z(2Z + 1) 4/ + 1

'-2cr(2Z + l)(2Z-l)Z

1(21 + 1)

(4/ + l)(4/-l)

2a

Qf+ =

(4/ + 3)(4/-l) \

0

--1

(4/-l)(4Z + l)

-1(21+ 1) -4a(l+ l)(2l+ 1)1 4/ + 1 (4Z + l)(4Z + 3)

(8.3.1.26)

Equation (8.3.1.25) may be solved for the Laplace transform

C,(S) =

rm^ using the method outlined already in this chapter to yield

C1(5) = rA,[5l-Qf-Qf+S|(5)J1

x[C, (0) + £ [ fl CfcsL (5)(QJ[-)"' 1 C„ (0)

where Sj[(.s) is the 2x2 matrix continued fraction defined as

SH(J) = (^I-^-Qrs"+,W)"1Qr-The initial value vectors

" ( ) 1/2,(0),

(8.3.1.27)

(8.3.1.28)

(8.3.1.29)

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Chapter 8. Rotational Brownian Motion in an External Potential 413

may also be determined from Eq. (8.3.1.17) in terms of matrix continued fractions. This is accomplished by transforming Eq. (8.3.1.17) into a matrix three-term recurrence relation as follows

or \^2/ i -3/0

.V 2n-2/0 . +Q1!

\ ^ 2 n - l / 0

V 2 n / 0 J +Q!+ V 2 n + l / 0

\\^2n+2/0j

= 0 , ( n > l ) , (8.3.1.30)

where Q[[,Q[r are the same 2x2 matrices as in Eq. (8.3.1.26). The solution of Eq. (8.3.1.30) is then given by

= S[l(0)SL1(0)...Sl1

l(0) ' ^ ( c o s t ? ) ^

In particular,

<P2n(costf)>0 ) vi;

V <P2(costf)>0

= S(0)

(8.3.1.31)

(8.3.1.32)

Thus, on using Eqs. (8.3.1.16) and (8.3.1.31), we can now evaluate the initial conditions C„(0), Eq. (8.3.1.29), in terms of matrix continued fractions, viz.,

In ^

C„(0) = 0

2 « - l \

4 n - l 0 0

< ^ - 3 > ,

.(^2/1-2/0/

-C?>0

2n

4 n - l

v0 0 2«

.4n + l

v4n + l

2n ^

4 n - l \^2n-l/0

<3»>0

0

2« + l

4n + l

V2n+1/0

0 . \^2/!+2/0,

4n- l s!(0)+ 0 d)

2« + l

V4n + 1

s!+1(0)sii(0) &Uo)-$(0) /r\\

\h

(8.3.1.33) Equations (8.3.1.27), (8.3.1.28), and (8.3.1.33) constitute the exact solution of our problem in terms of matrix continued fractions. Having

determined the Laplase transform f(s), one can calculate the susceptibi

lity J||(ft>) and the correlation time Zj| given by Eqs. (8.3.1.11) and

(8.3.1.13), respectively.

8.3.2 Characteristic times and magnetic susceptibility

In Fig. 8.3.2.1 (a), we show the behaviour of the lowest eigenvalue X\ calculated from the characteristic Eq. (8.3.1.7) as a function of a for various values of h showing that our calculation agrees with the

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414 The Langevin Equation

corresponding Fig. 1 of Aharoni [4]. A 45x45 matrix A was sufficient to obtain a value of k\ accurate to 3 significant digits for the range of parameters chosen. In Fig. 8.3.2.1 (b), we also show the behaviour of the inverse of the correlation time \ computed from Eq. (8.3.1.13) for different values of h. The most astonishing result of this calculation is that 1/TJI is markedly greatly from k\ above a certain critical value h (a discussion of this effect will be given below, see also Chapter 1, Section 1.20). This behaviour is further emphasised in Table 8.3.2.1.

More insight into the disparity between k[x and X\\ may be gained from the general equation

Lak\ (8.3.2.1)

101

10"

10"

a)

1 -A = 0.00 2-/1 = 0.10 3-/1 = 0.20 4 - h = 0.40

5-/1 = 0.50 6 - h = 0.70 7 - h = 0.80 8-/i=1.00

8

_— " 7 = = ?r~~~

^ ~ ^ ^ 6

^ \ \ \ ^ v ^ \ \ \ 3 \

l \ \ \

10" iou 10'

Figure 8.3.2.1. The smallest non-vanishing eigenvalue A] (a) and the inverse of the correlation time Tj (b) as a function of <rfor various values of the field parameter h.

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Chapter 8. Rotational Brownian Motion in an External Potential 415

Table 8.3.2.1. Product of the smallest eigenvalue Xx and the correlation time Zj| for

various values of the barrier height (7 and field h parameters.

h

0.01

0.1

0.2

0.4

0.5

0.7

1.0

TA (7=0.2 1.

1.

1.

1.

0.999

0.999

0.998

(7=0.5 0.999

0.999

0.999

0.997

0.996

0.992

0.984

(7=1 0.998

0.997

0.995

0.986

0.979

0.959

0.913

(7=2 0.994

0.99

0.979

0.926

0.878

0.746

0.544

(7=5 0.988

0.974

0.896

0.331

0.153

0.113

0.176

(7=10 0.996

0.964

0.455

0.00887

0.0171

0.0482

0.119

(7=20 0.999

0.725

0.00134

0.000315

0.00183

0.0176

0.084

Table 8.3.2.2. Amplitudes Ak and the corresponding eigenvalues Xk for various values of the field parameter h\ a- 10.

k

1

2

3

4

5

6

7

8

9

10

h = 0.01

\ 0.858

9.05 10"8

0.00283

4.9 10~5

8.07 10"4

2.18 10"6

3.61 10"5

8.78 10"8

9.45 10"7

1.92 10"9

KXH 0.00147

8.06

12.3

14.8

19.2

25.2

32.1

40.

49.

58.9

h = 0.2

4 0.0019

2.01 10-6

6.32 10"5

0.0014

6.68 10"

1.01 10"4

2.98 10~5

5.84 10"6

1.02 10"6

1.5 10"7

\TN

0.0192

7.32

12.7

17.6

20.3

26.6

33.4

41.2

50.1

60.0

h = 0A

\ 1.11 10"6

6.68 10"8

7.86 10"7

4.10 10"

1.06 10"3

2.90 10~5

2.27 10"5

6.14 10-6

1.18 10"6

2.06 10""7

\TN

0.192

7.12

14.4

22.6

23.6

31.2

37.7

44.8

53.6

63.5

The calculation of t\\ from Eq. (8.3.2.1) requires a knowledge of a sufficiently large set of eigenvalues Ak and their corresponding amplitudes Ak, as shown in Table 8.3.2.2. The reason for the disparity between Tn and X^ now becomes obvious. It is due to the fact that at short to intermediate times, the high frequency (intrawell) decay modes cannot be neglected as they contribute significantly to the correlation time. Indeed, it is apparent by using the values of Table 8.3.2.2, that the correlation time X\\ from Eq. (8.3.2.1) cannot be approximated by X[x in contrast to £=0 , where the Ak are negligible for all k>\ (see Table 7.4.5).

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416 The Langevin Equation

Figure 8.3.2.2. ZJ,, [Eq. (8.3.2.6); solid lines], *jf [Eq. (8.3.2.4); dashed lines], and \IXX

[Eq. (8.3.1.7); dotted lines] vs. cxfor various values of h. Filled circles: Eq. (8.3.2.2).

For <r>2, the lowest eigenvalue Xx can be approximated by Brown's formula [4,14] (see Chapter 1, Section 1.18)

\xN =x-U2<J3/2(l-h2)\(l + h)e-<7l+h)2 +\-h)e-°l-h)2\ (8.3.2.2)

For a< 2, this formula is inadequate (see Fig. 8.3.2.2); here, however, we can use an approximate equation for the effective relaxation time tef. (Chapter 2, Section 2.12). The effective relaxation time %is determined by evaluating Eq. (8.3.1.3) for I = 1 at t = 0, viz.,

-l

V / i ( 0 ) _

/ i (0) = TK 1-

2 g | g / 2 ( 0 ) 2<r/3(0)

5 3/ , (0) 5 ^(0) (8.3.2.3)

Thus, on using Eqs. (8.3.1.16)-(8.3.1.20), we can obtain after some algebra an analytic equation for % in terms of equilibrium averages or, in turn, in terms of known functions (see Ref. [16] for details), viz.,

rtf = 2T„ 2(P2)0 +1 - 3(P1)g = ^ (cos2 tf)0 - (cos tf>g

(8.3.2.4) l -(^2)o " l - ( c o s ^ ) 0

where (P^Q and (P2)0 are given by Eqs. (8.3.1.18) and (8.3.1.19), respectively. For low potential barriers, a< 2, Eq. (8.3.2.4) provides a good approximation to the relaxation time ^ (see Fig. 8.3.2.2).

For the uniaxial potential V given by Eq. (8.3.1.2), the dynamics of the system are described by a single variable Fokker-Planck equation [4,9] (see Section 1.17):

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Chapter 8. Rotational Brownian Motion in an External Potential 417

2 r „ — W = -1 d

dt sintfoW

d d \ sintfl—W + /?W— V

d$ dd ) (8.3.2.5)

so that the integral relaxation time zj| may be expressed in closed form as [17] (see Chapter 2, Section 2.10)

-\2

Tu = 2thl

Z(<cos2i^0-<cost?>g)i

oPViz) -dz, j (z ' -<cos t f ) 0 )<f^ ( z V

(8.3.2.6) where the partition function Z is given by Eq. (8.3.1.20). Equation (8.3.2.6) yields the same results as the matrix continued fraction solution and may serve as an independent check of numerical calculations. For either <x= 0 or £= 0, Eqs. (8.3.2.4) and (8.3.2.6) for rf and ZJ, can be considerably simplified. For £ = 0 , T\\ and rf are given by Eqs. (7.4.2.30) and (7.4.2.32), respectively. For a- 0, Z\\ is given by Eq. (7.5.2.22), while tf is given by Eq. (7.5.2.23).

For small a and £ % rf , and l/A\ have the following Taylor

series expansions: , 2 18 2 164 3

5 175 7875 9095625 33248 4

<7

/'l 442 794 2

- + cr + a 9 4725 23625

Y+—? 90 y

rf / i 2 16 2 11 N 5 175 2625

32 3

+ ...

192

2 206 1328 2 — + <T+ a 15 1575 23625

2 , 26 j-4 ^ +

1575

336875

and 32 a , 2 48 2

5 875 21875 <73+-

+ 1 1 686 — + a < 10 875 84375

15552 58953125'

7000 The above expansions were obtained from Eqs. (8.3.1.7), (8.3.1.13), (8.3.1.27), and (8.3.2.4) with the help of the MATHEMATICA program.

We compare the three time constants \IXU t\\, and rf in Fig. 8.3.2.2 for different values of the field parameters h. Here, the most interesting effect, as we have already mentioned, is the behaviour of the

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418 The Langevin Equation

integral relaxation time Tn as a function of a for sufficiently large bias parameter h. For values of h above a certain critical level hc = 1/6, the relaxation time zj| no longer has an activation character at large <r (solid curve 2 in Fig. 8.3.2.2). At this critical value of h, the relaxation switches from being dominated by the slowest overbarrier mode to being dominated by the fast intrawell modes. Thus, the relaxation time T\\ decreases as the height of the potential barrier increases. This effect was discovered numerically by Coffey et al. [9] and later explained quantitatively by Garanin [17] (see Chapter 1, Section 1.20 for more detail).

On the other hand, having determined l/Au % and r^ , we may also calculate the dynamic susceptibility from a simple analytic equation, as was described in Chapter 2, Section 2.13, where it was shown that the susceptibility Z\i°^ maY be approximated by a sum of two Lorentzians:

^ ) = ^ ^ + J - A L > (832J) Zn(0) \ + ial\ \ + icmw

where

T|,/T,f - 1 A>T» - 1 Aj = L J j - , % = — u

T . (8.3.2.8)

In the time domain, such behaviour of Zwi6^ IS equivalent to supposing that the longitudinal correlation function C\\(t) (which, in general, comprises an infinite number of decaying exponentials) may be approximated by two exponentials only:

C||(r)«A1e"^+(l-A1)e~ , /T"'. (8.3.2.9)

The correlation function Q(0 from Eq. (8.3.2.9) has the same biexponential form as that derived by Garanin [17] (see Section 1.20) in the low-temperature strong-bias limit 2a + £,» 1. For 2<r+ £ » 1, the asymptotic estimate for the Ai, 1-Ah and % may be evaluated from the simple equations [17]

4(0) ( l -A 1 )~(2(T + a " 2 ,

4(0)A, 2 ( 1 ~ / > ) j , (8.3.2.10) 11 (\ + h)eS +(\-h)e4

Tw~TN/(2a + ^).

The asymptotic estimate for X\ is given by Eq. (8.3.2.2). However, the biexponential approximation Eq. (8.3.2.9) unlike that of Ref. [17] may be used for any cand £if Ai and %are defined by Eqs. (8.3.2.8).

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Chapter 8. Rotational Brownian Motion in an External Potential 419

10"* 10"* 10"2 10° 102 104

Figure 8.3.2.3. %«(&>) and 2j'(<y) vs- COTN evaluated from the exact matrix continued

fraction solution (solid lines) for £ = 2 and various values of a and compared with those

calculated from the approximate Eq. (8.3.2.7) (filled circles) with numerical values of tj|,

T^, and \IA\ from Table 8.3.2.3 and with the low (dotted lines) and high (dashed lines)

frequency asymptotes Eqs. (8.3.2.11) and (8.3.2.12), respectively.

The results of the numerical calculation of the normalised complex susceptibility (V/JM^/VQ = 1) from Eqs. (8.3.2.7) and (8.3.2.8) and those obtained using the exact matrix continued fraction solution are shown in Figs. 8.3.2.3 and 8.3.2.4. The parameters used in the calculation are given in Table 8.3.2.3. One can see from these figures that the agreement between the exact matrix continued fraction calculations and the approximate Eq. (8.3.2.7) is very good. Similar (or even better) agreement exists for all values of £ and a. As has been shown in Chapter 2, Section 2.13, Eq. (8.3.2.7) predicts the correct behaviour of ,£j|(fi>) at low and high frequencies, viz.,

Z | | ( « ) « 4 ( 0 ) [ l - ^ | | ] + O(«2) (8.3.2.11)

as <u-» 0, and

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420 The Langevin Equation

10"

3

10"

10"

1

z

3

4

1 2 3 4

• 1 = 0

f=2 1=5 #=10

(7=10

10"' 10" 10 10"' 10" 10'

Figure 8.3.2.4. The same as in Fig. 8.3.2.3 for a= 10 and various values of £.

Table 8.3.2.3. Numerical values of -r„ [Eq. (8.3.2.6)], tf [Eq. (8.3.2.4)], lMj [Eq.

(8.3.1.7)], and %[Eq. (8.3.2.8)].

CT=10

i/(vW T||/Tw

T|| /TN

T\y/TN

Z=2

\ITM

TN/rw

T|, ltN

Ty/ltf)

£ = 0

693.92

691.02

16.644

0.07143

cr=l

0.97627

0.89117

0.75275

0.28660

£ = 2

232.54

224.15

1.6899

0.06146

a=5

6.3866

5.7200

1.0975

0.13831

<?=5

26.826

3.3982

0.05654

0.04949

cr=10

232.54

224.15

1.6899

0.06146

£=10

2.2650

0.03865

0.03839

0.03838

<7=15

16484

16221

2.3663

0.03774

Page 446: The Langevin Equation Coffey_Kalmykov_Waldron

Chapter 8. Rotational Brownian Motion in an External Potential All

X\\ m ~ - i 4 (0) /(flwjf) + O(O)-2) (8.3.2.12)

as CO—> 00. These asymptotes are also shown in Figs. 8.3.2.3 and 8.3.2.4. The figures demonstrate that the low- and high-frequency behaviours of the magnetic loss X^i®) are completely determined by T]| and rf , respectively.

8.3.3 Magnetic stochastic resonance

(In joint Authorship with Yu. L. Raikher and V. I. Stepanov)

Here, we discuss the stochastic resonance (SR) in uniaxial superparamagnetic particles. The basic aspects of the SR phenomenon [54] has been outlined in Chapter 1, Section 1.21. We recall that the manifestation of the SR is rather simple. An archetypal model for it is an overdamped oscillator with a bistable potential subjected to a noise, e.g. in contact with a thermal bath. Such an oscillator is excited by a weak ac force of a given frequency Q.. The alternations of the force favour the to-and-fro transitions between the two equivalent equilibrium positions of the oscillating "particle". The spectral density <£(<y) of the system is evaluated under these conditions whence the signal-to-noise ratio (SNR) at CO = Q. as a function of the noise intensity D is constructed. It turns out that the corresponding curve has a bell-like shape, i.e., passes through a maximum. Such a behaviour of the function SNR(Z)) is called stochastic resonance (SR). The maximum in the SNR is interpreted as being due to the surprising ability of noise to enhance the intensity of the interwell hoppings in the system.

The SR phenomenon is clearly manifested by uniaxial single-domain ferroparticles, as was discovered at the beginning of the last decade [55-59]. The magnetic energy of such a particle is given by Eq. (8.3.1.2). In the absence of the dc bias field, the magnetic moment of the particle has two equivalent stable orientations at $= 0 and z?= ;rso that it is a perfect example of a bistable system subjected to noise. The rate of transition between the potential wells is controlled by the parameter a, see Eq. (8.3.1.4), which relates the height Kv of the magnetic anisotropy barrier to the thermal energy. Assuming that the barrier height is fixed, one may regard the inverse of eras the dimensionless temperature, i.e., the noise intensity. The dc bias field H0 applied to the particle parallel to its anisotropy axis breaks the bidirectional symmetry of the potential. However, a two-minima profile of the potential v Vcosfy/kT survives as long as the bias field parameter h = MSH0/(2K) < 1 (see Fig. 1.18.1).

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422 The Langevin Equation

Proceeding, we recall first the basic results of the SR theory in the context of magnetic systems [57,59]. For weak ac magnetic fields, the theoretical approach is based on linear response theory [54,60] (see Section 2.8). This theory relates the Fourier components of the magnetisation Ma and the external ac magnetic field Ha, to the dynamic susceptibility %(a)) as

Ma)=z(O))H0. (8.3.3.1)

Thus, the spectral density &Mco) of the forced magnetisation oscillations in an ac field H(t) = Hcos(£lt) is given by [60]

<&M(a) = H2\x(Q.tf[S(a)+eL) + S(a)-Q.)l4. (8.3.3.2)

Here we have noted the parity condition z'i®) = X^-°^- By using the definition of the spectral density at the excitation frequency Q. as in [54]

^ " ^ = 2 i ™ o C > " (0))d(°' (8-3-33)

we have <S>^a)(Q) = H2\x(Q.f/2. (8.3.3.4)

On the other hand, according to the fluctuation-dissipation theorem (see Section 2.8), the noise-induced part of the magnetisation spectral density is given by [59]

&™™\a)) = — Z"m. (8.3.3.5) Ttatv

Here, the particle volume v occurs because we analyse magnetic relaxation processes in a particle using the magnetisation, not the magnetic moment. By noting that the total spectral density 4>M(Q) is

equal to <S>£ignal) (Q.) + 4><fse)(Q.), Eqs. (8.3.3.4) and (8.3.3.5) yield the signal-to-noise ratio as [59]

SNR= ,M. . = 7 Jl . (8.3.3.6) ^(noise) 2kTZ\n)

The linear response theory result Eq. (8.3.3.6) is very useful on account of its generality. It shows that the calculation of the SNR is reduced to the evaluation of the dynamic susceptibility, which is a fundamental dynamical characteristic of any relaxing system.

Since the longitudinal dynamic susceptibility ;jf(£2) of uniaxial particles has already been obtained for both H0 = 0 (Section 7.4.2) and H0 * 0 (Section 8.3.2), formally one may use the corresponding solutions for X(Q) in Eq. (8.3.3.6) [noting that here we consider the susceptibility

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Chapter 8. Rotational Brownian Motion in an External Potential 423

of a single particle and not that of an assembly of particles]. However, a deeper insight into the problem can be achieved by using the eigenvalue representation, Eq. (8.3.1.10), rewritten here as

*(Q>=;EiiZHi. . 5 . - • <8-3-3-7) 1 + /Q//L where

vMj

kT ^ l = ^ ( ( c o s 2 ^ ) o - ( c o s < ) ' (8.3.3.8)

is the static susceptibility and the weight coefficients ck are normalised by using the condition XI=ic* = * ^ tne ^ r s t t e n w e i § n t coefficients ck and the corresponding eigenvalues Ak can be obtained from Table 8.3.2.2). In order to emphasise the temperature dependence of the SNR, it is convenient to use the representation

S N R = — ( M SH / K)2 R(a,Q), 2r0

where the dimensionless function R(a,Q.) is given by [61]

(8.3.3.9)

R((T,Q.) = •4cosH-(cos<) y ° ° ck' \

^* = 1 l + ( Q / 4 ) 2 (8.3.3.10)

M J*=I i+(Q/Aky + v-i°° k k

^*= 11 + (Q//L)2

The time r0 = r^/ cr is a convenient reference time; its physical meaning is the damping time of the Larmor precession in a uniaxial magnetic particle with zero external field. Equation (8.3.3.9) shows that the quantity of interest in the present context is R(cr,£l). In general, R(a,Q.), besides the obvious dependence on the noise intensity (temperature) and on the constant (bias) field strength HQ, depends on the frequency of the exciting field Q. In the adiabatic limit, Q, -» 0, Eq. (8.3.3.10) yields

where

/?(<7,0) = - ^ ( ( c o s 2 t f ) o -(costf)^); (8.3.3.11)

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424 The Langevin Equation

is the longitudinal relaxation time [this time may also be calculated from the analytic Eq. (8.3.2.7)]. We remark that Eq. (8.3.3.11) may also be derived in the context of the single exponential approximation for the magnetisation relaxation, viz.,

Af(0 = Af(0)e~'/T'. Here, the low-frequency part of the dynamic susceptibility is given by

1 + liiTy

so that

Whence, the SNR from Eq. (8.3.3.6) is independent of Q. and reduces to

7tH2VY» SNR = ^ - , (8.3.3.13)

2Wzj| which corresponds to Eq. (8.3.3.11). Equation (8.3.3.13) justifies the previous treatment [57,59] of the magnetic SR in the limit Q. -» 0 as a process characterised by just one relaxation time \.

The characteristic function R vs. the temperature parameter a~l = kT/Kv is shown in Figs. 8.3.3.1 (for H0 = 0) and 8.3.3.2 (for H0 =* 0). For HQ - 0, the SR is most pronounced in the adiabatic limit Q —> 0, see curve 1 of Fig. 8.3.3.1. This case is relatively simple, since by setting £1 = 0, Eq. (8.3.3.10) reduces to Eq. (8.3.3.11). Here, the main maximum of R is attained at <T~ 1. In this limit, R —> 0 as a—> «>. As shown above, the time ^ is very close to 1 / Xx for any value of the parameter a so that either parameter may be used. Meanwhile, at any finite Q., i.e., outside the adiabatic limit, one must remember that 1 / Xx is exponentially large in crfor a» 1, i.e., it increases drastically as the system is cooled, while all the other 1 / Ak do not depend exponentially on a. Even for very low frequencies, at low temperatures, the ratio Q./Al

tends to infinity as the overbarrier transition is completely "frozen out". However, the magnetic moment, although it is confined to a particular potential well, is not completely immobilised: it may take part in intrawell motion, which is sensitive to the profile of the potential near the bottom of the well, and for the system in question is determined by the entire (infinite) eigenvalue spectrum. Thus, R —> 1/2 at <x-» oo. This finite limit is not expected on intuitive grounds, however, its validity is unambiguously confirmed by numerical calculations.

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Chapter 8. Rotational Brownian Motion in an External Potential 425

R

0.6 -i

0.5 -

0.4 -

0.3 -

0.2 -

0.1 -

0 -0 0.4 0.8 1.2

Figure 8.3.3.1. Signal-to-noise ratio as a function of dimensionless temperature in the absence of the dc bias field (£= 0). The adiabatic dependence (Q. = 0) follows the dashed curve at the bottom of the minimum on curve 1 and is then indistinguishable from curve 1. Solid curves: i i ^ = 0.01 (1), 0.1 (2), 0.5 (3), 1 (4), 2 (5), 10 (6).

R 0.5 -i

0 0.4 0.8 1.2 Figure 8.3.3.2. Signal-to-noise ratio at nonzero values of the applied constant field in the adiabatic limit, £2 = 0. The curves correspond to h = 0.05 (1), 0.1 (2), 0.25 (3), 0.5 (4), 1 (5), and 2.5 (6).

For HQ ^ 0, (see Fig. 8.3.3.2), the maximum of the SNR moves to higher temperatures with increasing the dc field strength. The explanation for this shift follows because the presence of the bias field radically alters the temperature behaviour of the static susceptibility of the system. At H0 = 0, when (cos t?)0 = 0, in the response to increasing temperature, (cos2^)0-(cos z?)0

2 decreases from 1 to 1/3. In a nonzero bias field, the effect of saturation of the longitudinal magnetisation is crucial as the saturation causes (cos2^)0-(cos &)0

2 to tend to zero at zero temperature. On heating the particle at Ho + 0, the function B grows from zero to the limiting value 1/3. Consequently, the combination

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426 The Langevin Equation

-()),[rad]

0 0.4 0.8 1.2 Figure 8.3.3.3. Phase shift as a function of dimensionless temperature for Q.T0 = 0.01 (1), 0.1 (2), 0.5 (3).

a 2 ( ( C O s 2 *V< C ° S ^o) , see Eq. (8.3.3.10), acquires an intrinsic temperature maximum. It is this specific (static) temperature-dependent factor that reverses the direction of the shift in the SNR maximum.

In like manner, one can study the phase shift (/><7,Q) of the magnetic SR [61]. This phase shift may be defined as

<p(o,£l) = -a rc tan(^r ' / / ) . (8.3.3.14) The phase here is understood in its conventional meaning, i.e., as the angular lag between the excitation force and the output signal [60]. The results of the calculation of the phase shift <j>(o,Ci) for a superparamagnetic particle are presented in Fig. 8.3.3.3 (for H0 = 0) and 8.3.3.4 (for Ho * 0). In a manner similar to the effect on the SNR curves, the bias field has a destructive effect on the noise dependence of the phase. This tendency is shown in Fig. 8.3.3.4. Note that a complete disappearance of the two-well potential taking place at h - 1, is reflected in this graph by an almost complete flattering and disappearance of the curve.

Here, we have considered the SR in uniaxial superparamagnetic particles in weak ac fields and in the simplest configuration (the direction of the ac and dc magnetic fields coincides with the easy axis of the magnetization). However, the method may readily be generalised to other interesting cases such as the nonlinear SR in strong ac fields, oblique configurations of applied fields [62], etc. Estimations of the effect in nanomagnetic systems are given in [63].

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Chapter 8. Rotational Brownian Motion in an External Potential All

0 0.1 0.2 0.3 0.4 0.5 Figure 8.3.3.4. Phase shift at the frequency Q-T0 = 0.01 for bias field strength parameters h = 0.05 (1), 0.1 (2), 0.25 (3), 0.5 (4), 1 (5).

8.4 Transverse Response of Uniaxial Particles

The primary goal of the present section is to evaluate the transverse complex magnetic susceptibility XA<^> of a system of noninteracting single domain ferromagnetic particles subjected to a uniform magnetic field. Just as the longitudinal response, we present (with the aid of linear response theory) the exact solution [18] for Xi_oi) in terms of matrix continued fractions and compare the results with various approximate solutions. The results can also be applied to dielectric relaxation problems by allowing the damping parameter a —> °°.

8.4.1 Matrix continued fraction solution

As shown in Section 7.3, the appropriate infinite hierarchy of differential-recurrence equations for the transverse equilibrium correlation functions gn (t) is

at n(n +1) i£

2 2a n(n + l ) - 3

(2n-l)(2n + 3)

= £ (n + l)

2«+ 1

n + l ia | . . n ' n ia ^

2 + c*f 8n+l(t) (8.4.1.1)

+a n(n + lY

(2/i-l)(2n + l)

where the gn (t) are defined as

S„-2(0-n2n + \)

(2n + l)(2n + 3) s«+20)

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428 The Langevin Equation

g„(t) = (cosp(0)P/[cosi?(0)]e,w/>B1 [cos^(0]) • (8.4.1.2)

The transverse complex magnetic susceptibility Z±«o) = Xl(a>)-iXlm (8.4.1.3)

is expressed in terms of the one-sided Fourier transform of g (?) as

id) Z±(a>) = Z±V —

2Re[^(0)]

where the asterisk denotes the complex conjugate,

[l!(iffl) + g*Hat)]\, (8.4.1.4)

Z± = v^MJN,

3kT *('-<i>„) (8.4.1.5)

is the static transverse susceptibility, (P2)o is g i v e n by Eqs. (8.3.1.19), and

gn(iO)) = ] gnt)e-i0dt (8.4.1.6)

For m = 1, Eq. (8.4.1.1) can be transformed into the matrix three-term differential-recurrence equation

rNCn (0 = <#-(;„_, (0 + Qn1C„ (0 + Q^+Cn+1 ( 0 ,

where

and

C„(0 = 8ln(t) ,

(8.4.1.7)

(8.4.1.8)

( 4n2(2n-l) , e . , , In N

a — (ng-ia/a)

Qi = (4/i- l)(4n-3) 4 n - l

2

<£ =

2n(2w-l)-3 % -n(2n-\) (4n-3)(4n+l) 2a

(2n+l)r

2n(2n + l)

(4n-l)(4n + l)

(2n-l)

(8.4.1.9)

(4n-l) [£(n-1/2)+i<7/a]

^ ( n + l /2 ) - to /a l <r 2n(-2n+l)~3 -n(2n + l ) - ^ (4n + l)L V ; J (4ra-l)(4n+3) 2a)

(8.4.1.10)

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Chapter 8. Rotational Brownian Motion in an External Potential 429

QT =

-<J-2n(2n-\f

(4n-l)(4n + l)

In -n^ + icrla) -a 4/T(2n + l)

(4n + l)(4n + 3)

(8.4.1.11)

(4n + l) By applying the general method of solution of matrix three-term

differential-recurrence relations (Chapter 2, Section 2.7.3), we have the exact solution for the one-sided Fourier transform Cx(ico) in terms of matrix continued fractions

g2ico) ••rN^(co) ^ ( 0 ) + ^

n=2 IIQ^A,1^) k=2

C„(0) (8.4.1.12)

where I is the 2x2 identity matrix, Q„,Q~are the 2x2 matrices, given in

Eqs. (8.4.1.9)-(8.4.1.11), and the matrix continued fraction A^(w) is

defined as

tf (co) = [rNicol - Q„x - Q„1+A^ (*»)(#" ] " ' . (8.4.1.13)

The initial condition vectors

C„(0) = 82n-^)]

82n(V

4 n _ ! ( ( P 2 "- 2 )o (P2n)0)

— (\^2 n - l )o~(^2 n +l)o)

(8.4.1.14)

V 4/i + ] may be evaluated in terms of matrix continued fractions just as the longitudinal response by taking into account Eq. (8.3.1.31).

8.4.2 Transverse complex magnetic susceptibility

Equations (8.4.1.12)-(8.4.1.14) allow us to evaluate i(ffi>) in terms of matrix continued fractions and to test the accuracy of various approximate solutions which already have been presented. Most of the analytical results have been obtained by the effective eigenvalue method. For example, on applying this method for cf=0 and cr^O, Raikher and Shliomis [19] derived an expression for ^i(<y), which in our notation is

1 + A + icor-,

where (1 + icor2 )(1 + icoT±) + A

2(l-Tx/TN).

(8.4.2.1)

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430 The Langevin Equation

Here t± and r2 are the effective relaxation times, which can be expressed in terms of the equilibrium averages (P2)0 of the Legendre polynomial of order 2, viz.,

*i = *-;v l~P2)o , (8.4.2.2) 1 W l + (P 2 ) 0 /2 '

r2 = tN ^ ^ (8.4.2.3) N2(j + (P2)0(a-6)

with 3ea 3 1

(P2)0 = ** _ L _ ± . (8.4.2.4) 2^/^:erf^•(^/5:) 4(7 2

For < r » 1, when ^ - T ^ O - " 1 , r2 ~ tNe~l, A~<j2a~2, Eq. (8.4.2.1) reduces [19] to the Landau-Lifshitz equation [20]

Z±(a>) _ (1 + a2)a>l + iaa>G)0

where Zx (1 + oc2)col -CD2 + 2zattKq,

a)o = 0aTNyl~yHa

(8.4.2.5)

is the precession frequency and Han = 2K/MS is the strength of the anisotropy field.

Furthermore, by using the effective eigenvalue method in the opposite limit <7= 0, %± 0, Garanin et al. [5] have obtained an equation for the circular magnetic susceptibility, which can be rearranged as the following expression for the complex transverse susceptibility

^ w = \rt+*nNx (8426) Z± \kr -arirN + 2ianNX

where k is the dimensionless complex effective eigenvalue given by

k = k' + irJ2 + ' - ^ + ^ - . (8.4.2.7) 2(£coth£-l) 2a

For high field parameters ( £ » 1), Eq. (8.4.2.6) again reduces to the Landau-Lifshitz equation (8.4.2.5) with Q)Q = %2azNTl. Equation (8.4.2.7) was derived for ^ 0, <7= 0 only. Since k is the effective eigenvalue for the equilibrium autocorrelation function g(t), Eqs. (8.4.2.6) and (8.4.2.7) can readily be generalised for <J* 0 and <J« £ as follows. According to Eq. (8.4.1.1), the first equation of the infinite hierarchy is

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Chapter 8. Rotational Brownian Motion in an External Potential 431

%Si(0 + 1 + — + Si(0 + ^ + | V 2 ( 0 + fg3(0 = 0. (8.4.2.8) 3or 6) 15 2a 5

The effective eigenvalue approach implies that the solution of Eq. (8.4.2.8) may be approximated by a single exponential, namely:

g\ (t) - g\ (0)e N with the effective eigenvalue X given by

^ g t ( 0 ) _ i , £ , ° , f^ , ^ ^ 2 ( 0 ) , 2<xg3(0) A = - 1 + -^- + —+ 1—+ -2or 5 U<* 6M(°) + -^(0) 2or 5 U « 6 7 ^ ( 0 ) 15 8l(0)

Equation (8.4.2.9) can be simplified after some algebra to yield

- + r-

(8.4.2.9)

(8.4.2.10) 2(1-<P2>0) 2a(\-(P2)0)

where (P^ and <P2)0 are given by Eqs. (8.3.1.18) and (8.3.1.19),

respectively. For a= 0, Eq. (8.4.2.10) reduces to Eq. (8.4.2.7) since

<P1)0=coth^-rI,

(P2>0 = l - 3 r 1 ( c o t h ^ - r 1 ) . The application of the effective eigenvalue method for ^ 0 and

£ « a requires two effective eigenvalues for the first two equations of the infinite hierarchy Eq. (8.4.1.1), since the resonance arises here due to the coupling between /u(f) and /2il(f) (this behaviour is termed 'entanglement' of the dipole and quadrupole branches of the response by Raikher and Shliomis [19].) Thus, we obtain the set of coupled ordinary differential equations:

xSi«Me /Si(0+Nr-+fWo = o, (8.4.2.H) dt

a a) where the effective eigenvalues are given by

- &

at M") 6/

2 gl(t) = 0, (8.4.2.12)

n 2a 5 *• + > 1 | 3 2<r-cf(P1)0

15^(0) 2a 4a 2 4a(\-(P2)0) (8.4.2.13)

and

$ - % | 3 a | 1 2 ( 7g4(°)

2a 7 35 g2(0)

ig t ff|^ , 3 2 tx(P1)0-£(P2)0

2« 2a ^((P2)0- l ) +3(^)0

(8.4.2.14)

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432 The Langevin Equation

Here, </>>„ and (P2)0 are given by Eqs. (8.3.1.18) and (8.3.1.19), respectively. We have noted in the derivation of Eq. (8.4.2.12) that

0 . . . 8i(t) (8.4.2.15) v 2a 5 J 2a

(in accordance with Eq. (8.4.2.13) the equality (8.4.2.15) is assumed to be valid at any time t). We note that in the limit £ cr—> 0, the effective eigenvalues have the following behaviour

K a ig , a So2 Aof ? =—+1+^- + +——+... 2a 5 175 175

A? « i l+3_ .o - . 16ff2 , ^ 7 + ' l 4 7 +-441

+....

On applying the one-sided Fourier transform to Eqs. (8.4.2.11) and (8.4.2.12), we can solve the set of linear equations so obtained for gx(ico) yielding

gx(ico) _ Xf +ia)TN)-S TN &\ (°) (tff + io)TN )(X? + icorN) + A

(8.4.2.16)

where

A = — + — 2 a ^HLA

6 3 a J / u ( 0 ) Ua 2a)

aj r

2

xn)o . l - < ^ 2 > 0 '

On substituting Eq. (8.4.2.16) into Eq. (8.4.1.4), we can evaluate Ji(ft>). For £= 0, this yields Eq. (8.4.2.1) of Raikher and Shliomis [19].

Furthermore, for arbitrary £and crand large damping (when one may ignore precessional motion) the transverse spectrum may effectively be described by the Debye relaxational equation [19]

XL l + ia)TN/A'' where A' is the effective eigenvalue which can also be expressed in terms of <P2)0 given by Eq. (8.3.1.29) [cf. Eq. (8.4.2.10)]:

l + <P2)0/2

(8.4.2.17)

X': (8.4.2.18) 1 - < ^ 2 > O

Typical spectra of the real and imaginary parts of Zii0^) a r e

shown in Figs. 8.4.2.1-3 (the calculations were carried out for

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Chapter 8. Rotational Brownian Motion in an External Potential 433

v2M^N0/kT = 1). The relaxational behaviour of the spectra Z±(°) is obtained for small anisotropy and field parameters (£ cr~ 0) or large damping (ct> 1). For small damping, as expected, the spectra have a pronounced resonant character and strongly depend on the damping parameter or (Fig. 8.4.2.1). The comparison of the effective eigenvalue solutions with the exact results allows us to estimate the accuracy of the former.

l-a=2 £=10,cr=10

Figure 8.4.2.1. Re[^x(ftJ)] and -lm[%Lco)] (solid lines) vs. anN for a= 10 and £ =

10 and various values of a. Filled circles: the overdamped effective eigenvalue solution, Eqs. (8.4.2.17) and (8.4.2.18).

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434 The Langevin Equation

10"' 101 103

Figure 8.4.2.2. Comparison of the exact (solid lines) and effective eigenvalue [filled

circles, Eqs. (8.4.1.4) and (8.4.2.16)] solutions for Re[z±(.a))] and -\m[xLco)] vs.

anN for a= 0.1 and £= 0.01 and various values of a.

In Fig. 8.4.2.1, the results of the exact calculations, Eqs. (8.4.1.4)-(8.4.1.14), are compared with the overdamped solution Eq. (8.4.2.17). In Fig. 8.4.2.2, the exact calculations are compared with the effective eigenvalue solution, Eq. (8.4.2.16). It is obvious by inspection of these figures that both solutions are in agreement for a~ 0 and for a» 1 only. However, for values of the barrier height parameter a from 1 to 5, the effective eigenvalue approach fails to describe the transverse response. The explanation appears to be as follows: at small to moderate barrier heights, there is essentially a spread of the precession frequencies

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Chapter 8. Rotational Brownian Motion in an External Potential 435

of the magnetisation in the anisotropy field. To a certain extent, this effect is analogous to inhomogeneous broadening; it considerably exceeds the true damping [5]. Therefore, it is impossible to describe asymmetric absorption and dispersion curves by the usual single resonance spectrum, which is predicted by the effective eigenvalue approach. However, in the high barrier limit (cr» 1), when the anisotropy potential may be approximated by a harmonic potential, the system may be effectively described by a single resonance with characteristic frequency C0Q given by Eq. (8.4.2.5).

0.8

0.4

0.0

-0.4

ar=0.1, <r=0.01

mmnm

10 an,,

Figure 8.4.2.3. Comparison of the exact (solid lines) and effective eigenvalue [filled

circles, Eqs. (8.4.2.6) and (8.4.2.10)] solutions for Re,[xLaJ) and -Im[^±(fl))] vs.

anN for a= 0.1 and a= 0.01 and various values of £

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436 The Langevin Equation

On the other hand, for a~ 0 and/or <x« <f, the effective eigenvalue solution provides (just as the longitudinal relaxation) perfect correspondence with the exact solution for all values of the field parameter £ (see Fig. 8.4.2.3) since we now have the natural resonance with the angular frequency co0 - g(2arN )_ 1 . Thus, when the external magnetic field considerably exceeds the anisotropy field or the anisotropy field is close to zero, the effective eigenvalue solution may accurately describe the transverse relaxation. However, when the influence of a constant magnetic field is negligible, the effective eigenvalue method requires careful investigation of its range of applicability before proceeding. Otherwise, it is possible to arrive at erroneous results as clearly demonstrated in Fig. 8.4.2.2.

Thus, the transverse response of an ensemble of noninteracting single domain particles can be evaluated from the exact Eq. (8.4.1.12). Furthermore, we have demonstrated that the effective eigenvalue, which yields simple analytical expressions, describes the main features of the transverse complex susceptibility with the exception of the intermediate barriers range a~ 1 to 5 and £< 1, where a spread of the precession frequencies of the magnetisation in the anisotropy potential field essentially exists. 8.5 Nonlinear Transient Responses in Dielectric and Kerr Effect

Relaxation

In the previous sections, we have demonstrated how matrix continued fractions can be used in the calculation of the linear response characteristics of a variety of dielectric and magnetic relaxation problems. The goal of this section is to present yet another application, namely, a transient nonlinear response theory for the dynamic birefringence and dielectric relaxation of polar and anisotropically polarisable particles (macromolecules) dissolved in nonpolar solvents when the magnitude of the dc field may suddenly be changed. The nonlinear response to a strong ac field may be treated in like manner [21] (see Section 8.6).

We consider a dielectric liquid consisting of an assembly of electrically noninteracting molecules (very dilute system) acted on by an electric field E. Hence, collective effects may be ignored, and the particles of the assembly can be considered as executing independent rotational motions. This allows us to reduce the problem to the orientational motion of an individual molecule. When a dielectric liquid comprised of polar and anisotropically polarisable molecules is acted on

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Chapter 8. Rotational Brownian Motion in an External Potential 437

by an external stimulus such as an electric field E, for example, it becomes birefringent, acquiring the same properties as a uniaxial crystal. If one assumes that the molecule under study carries a permanent dipole moment \i directed along its symmetry axis and is anisotropically polarisable (i.e., has an induced dipole moment) and if the magnitude of the electric field is not very great, then, the total dipole moment is given by

m = p + aE, (8.5.1)

where a is the molecular polarisability tensor of second rank. In Eq. (8.5.1), we have assumed that the induced dipole is linearly dependent on the electric field. When this condition is not fulfilled, it is necessary to include nonlinear terms characterised by the hyperpolarisability tensors ft (third order) and f (fourth order): this problem can be treated in a similar manner [22]. So, if Eq. (8.5.1) is valid and if the geometric axes of the molecule are chosen so that they coincide with those of the polarisability tensor, the orientational potential energy V will depend on z?only and is given by [23]:

V(i,t) = -juE(t)cos#--AaE2(t)cos2 &--a±E2(t). (8.5.2)

Here Aa= c\- a± is the difference between the principal electric polarisabilities parallel and perpendicular to the symmetry axis, the direction of which makes an angle z? with the electric field E at time t. The potential (8.5.2) has the same dependence on $as that considered in Sections 8.3 and 8.4.

The physical quantities, which are interesting from an experimental point of view and appropriate to dielectric and Kerr effect relaxation, are the electric polarisation P(f) and the electric birefringence function K(t) defined respectively by

P(t) = by+ iV0//(cos i?)(r) = bx + N0ju(Pl (cos i?))(f), (8.5.3)

K(t) = b2^^U-a°1)(P2(cos#))(t), (8.5.4)

where No denotes the number of molecules per unit volume, off and or°

are the components of the optical polarisability due to the electric field (optical frequency) of the light beam passing through the liquid medium, and n is the mean refractive index. The coefficients b\ and b2 depend on the particle depolarisation factors and the dielectric permeability of the solution. Here, we shall assume that these coefficients do not depend on the frequency of the electric field (in dielectric relaxation) and on the

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438 The Langevin Equation

light frequency (in Kerr effect relaxation). For simplicity, we shall also assume below that b\ = 0 and &2 = 1 • Here, the internal field effects are ignored. Thus, the effects of the long-range torques due to the connection between the average moments and the Maxwell fields are not taken into account. In the context of the dynamic nonlinear response, the internal field effects present a very difficult problem [24]. Nevertheless, these effects may be ignored for dilute systems in first approximation. Thus, the theory developed here is relevant to situations where the dipole-dipole interactions have been eliminated by means of suitable extrapolation of data to infinite dilution.

Here, we consider as a definite example a nonlinear step-on response, i.e., let us suppose that a strong uniform electric field Eo (directed along the Z axis of the laboratory coordinate system) is suddenly switched on at time t = 0 [21]. (Various results of nonlinear response theory as applied to dielectric and Kerr effect relaxation are presented in Ref. [22]). For t < 0, in the absence of the field, the system was at equilibrium with the uniform distribution function Wl(rS) = M2.

Having switched on the electric field Eo, the system will tend as t —> °° to a new equilibrium state with the Boltzmann distribution function

Wn(z) = Z-V v / ( * r ) =Z-V z + < J z 2 (z = cost?), (8.5.5)

where Z is the partition function, and

£ = juE0/(kT), a = AaE% l(2kT). (8.5.6) Our objective is to calculate the expectation values of the first

and second Legendre polynomials (Pi)(t) and (i^XO- The hierarchy of differential-recurrence relations, Eq. (7.3.2.12), for the expectation values of the spherical harmonics obtained in Chapter 7 can be applied directly to the present problem. Equation (7.3.2.12) [combined with Eq. (7.2.32)] for the potential (8.5.2) yields the infinite hierarchy of differential-recurrence relations for the moments (the averaged Legendre polynomials) at t>0 [cf. Eq. (7.3.2.13)]

2a 2TD —(p)(t) + 1-(2n-l)(2n+3)

W<0=^[fc)(0-fe)W] 2(7

+-2n+l

n_1-(^2)w-f^(c2)(o (8.5.7) _2n-V ' 2n+V

Here, it is convenient to introduce the relaxation functions f„(t) defined as /„(*) = (Pn (cos t?)> -(Pn (cos i>))(0 (8.5.8)

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Chapter 8. Rotational Brownian Motion in an External Potential 439

(in order to guarantee that fn(t) —> 0 at t —> °°). Thus, we have the five-term differential-recurrence relation [cf. Eq. (8.3.1.3)]:

2xD d

nn + \) dt /„(') + 1-

2<T

^ [/„-i(o-/B+i(o]+ 2a

(2n- l ) (2n + 3)

n-\

2n + l 2n + l •fn-iO)-

n + 2 (8.5.9)

fn+2(t) 2n-l"""' 2n + 3' In order to derive Eq. (8.5.9), we have noted that the equilibrium averages (Pn) satisfy the recurrence relation:

1 -2a

(2n- l ) (2n + 3) (P»)n=^T[(^)n-(P«+i)n]

+-2(7

2n + l

n-\

2n + \

n + 2 Vn-2/n 0 , , V n + 2 / l l

(8.5.10)

_ 2 n - T "" i / u 2n + 3 which follows from Eq. (8.5.7) in the stationary regime, t —> oo. The nonlinear step-on response to an electric field of arbitrary strength is an intrinsically nonlinear problem as small parameters are absent [21]. Hence, perturbation theory can no longer be used. However, on solving Eq. (8.5.9) in terms of matrix continued fractions, we can evaluate exactly the nonlinear transient responses of the electric polarisation P(t) and the birefringence function K(t).

Equation (8.5.9) can be transformed into a matrix three-term differential-recurrence equation, which is, from a mathematical viewpoint, identical to that considered in Section 8.3. Thus, we can apply the same method to its solution. First, let us arrange Eq. (8.5.9) as follows

^c„(0=QH-C„_1(0+^C„(0+Qli+cn+i(0, Cn(t)= Jln~iy>

fm(t) , (8.5.11)

where the matrix Q^.Q!!, Q"+ are defined by Eq. (8.3.1.26). On applying the general method of solution of the matrix three-

term recurrence equation, we have the solution for the Laplace transform Cj(s') in terms of matrix continued fractions

C1W=TD[-rD5l-^-^+S|(5)T1c1(0) + f ; n C f - i S ^ f Q ^ C ^ O ) , I n=2k=2 J

(8.5.12) where I is the 2x2 identity matrix and S\ (s) is the matrix continued fraction defined by the recurrence Eq. (8.3.1.28), viz.,

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440 The Langevin Equation

SI(S) = (STD1- .QII+S11 (^y1 QI The initial value vectors

C„(0) = /2„-i(0) ' ' ^ ( c o s i ? ) ) ^

</>2B(cost?)>n V

(8.5.13)

(8.5.14) /2„(0)

may be evaluated using matrix continued fractions as has been described in Section 8.3.1. We have according to Eq. (8.3.1.31)

C„(0) = Sll(0)St1(0). . .S lJ(0) |° Kh

(n>l). (8.5.15)

Having evaluated C(s) fromEqs. (8.5.12) and (8.5.15), one can

determine the transient nonlinear dielectric and Kerr effect response

spectra f^ico) and f2(iO)) as well as the corresponding relaxation times

*i = /i(0)//,(0) and T2 = / 2 (0) / / 2 (0) . (8.5.16) The solution in the form of Eq. (8.5.12) is mainly needed for the calculation of the relaxation times. On the other hand, the solution of Eq. (8.5.7) for the Laplace transforms of (Pj(cost?)>(0 and (P2(cost?))(0 has a much simpler representation, viz.,

| (Pl(cos&))(.t)e-stdt

J (P2(cos#))(t)e-s'dt

\o

•s-lShs) (8.5.17)

so that

= s-l[s\(0)-S\(s)] (8.5.18)

Thus, in order to calculate the nonlinear dielectric and Kerr effect step-on responses, we simply require the matrix continued fraction S ^ s ) .

The above results were given in terms of matrix continued fractions. However, the step-on response relaxation times X\ and Vi for the present problem can be calculated from the exact analytical equation:

l 2TD r 4>(z)¥„(z)e~g"z ~^ldz

(ra=l ,2) , (8.5.19)

where

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Chapter 8. Rotational Brownian Motion in an External Potential 441

<3>(z)=][Wu(z)-\/2]dz , (8.5.20) - l

%(z) = [P^-iP^X^^'dz , (8.5.21) - l

^ ( z ) = J [P2(z)-P2)U] ea*z'1+^'dz'. (8.5.22) - l

Here, the equilibrium ensemble averages (Pi)u and ( ^ n m a v be

calculated from Eqs. (8.3.1.18) and (8.3.1.19). Equation (8.5.19) is a direct consequence of the nonlinear transient response theory presented in Chapter 2, Section 2.14.2 for systems with dynamics governed by a single-variable Fokker-Planck equation. Here, the relaxational dynamics of Brownian particles at t > 0, is governed by a single-variable Smoluchowski equation for the orientation distribution function W [23] (see Section 1.15)

1 2Tn—W=-

dt sin 19 d& sini? -j-W + (#sini? + <7sin2i?)W (8.5.23)

Equation (8.5.19) allows us to evaluate the relaxation times T\ and Vi for various particular cases (see Section 2.14.2). Moreover, it provides us an independent check of the matrix continued fraction solution.

The behaviour of the real and imaginary parts of the one-sided Fourier transforms of the normalised relaxation functions defined as

Zn(a) = J n ^ : (8-5-24) W n ( ° )

is shown in Figs. 8.5.1-8.5.4. The spectra are evaluated from the exact solutions given by Eq. (8.5.18) and compared with the Debye spectrum

XDn(0) = 1 ^ - , (8.5.25) l + l(OTn

where Tn are the relaxation times calculated from Eq. (8.5.16). Equation (8.5.25) corresponds to the representation of the relaxation functions /„ (t) by the pure exponential

fn(t) = fn(0)e-"T" (8.5.26) It is apparent from Figs. 8.5.1-8.5.4 that Lorentzian behaviour is obtained for the spectra ;fr(<y) for arbitrary <f with a~ 0 and Xii®) f° r arbitrary a and £=0. Thus, in these cases alone, can the relaxation functions /i (/)

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442 The Langevin Equation

and f2(t) be approximated by a single exponential. Otherwise, the decay of the relaxation functions /„ (r) has a more complicated behaviour. This may be explained as follows. The relaxation dynamics in the potential given by Eq. (8.5.2) is determined by two relaxation processes. One relaxation (activation) process governs the crossing of the potential barrier between two positions of equilibrium. Another process describes relaxation inside the wells. For nonpolarisable molecules (<J= 0), where the potential (8.5.2) becomes a single well cosine potential, we observe one relaxation process only, namely, the reorientation of the molecule inside the well.

lcr4 io -2 io° io2

Figure 8.5.1. Real (curves 1,2, and 3) and imaginary (curves l',2', and 3') parts of #j(fiJ) at cr= 5. Curves 1,1'; 2,2'; and 3,3' correspond to £= 0.01,1, and 5, respectively.

l u -5 - 3 - 1 1 3

10 10 10 .„ 10 10 unD

Figure 8.5.2. Real (curves 1,2, and 3) and imaginary (curves l',2', and 3') parts of %2(o) at a- 10. Curves 1,1'; 2,2'; and 3,3' correspond to £= 0.01, 1, and 5, respectively. Filled circles and squares are the real and imaginary parts of ;trm(ft>) with relaxation time ^ given by Eq. (8.5.19).

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Chapter 8. Rotational Brownian Motion in an External Potential 443

10"* ltf4 1<T2 10° 102

onD

Figure 8.5.3. Real (curves 1,2, and 3) and imaginary (curves l',2', and 3') parts of Xii®)

at £= 2. Curves 1,1',2,2', and 3,3' correspond to <r= 0.01, 5, and 10, respectively. Filled

circles and squares are the real and imaginary parts of ZDI((°) w i t n relaxation time i\

given by Eq. (8.5.19).

Figure 8.5.4. Real (curves 1,2, and 3) and imaginary (curves l',2', and 3') parts of %2(a>)

at £= 2. Curves 1,1',2,2', and 3,3' correspond to a= 0.01, 5, and 10, respectively. Filled circles and squares are the real and imaginary parts of the spectrum XDI (<*>) w i t n

relaxation time ^ given by Eq. (8.5.19).

8.6 Nonlinear Dielectric Relaxation of Polar Molecules in a Strong AC Electric Field: Steady State Response

Previously, we have calculated the nonlinear step-on transient response for the dynamic birefringence and dielectric relaxation arising from sudden changes in magnitude of a strong external dc field. In order to obtain these results, we have used the matrix continued fraction technique. In the present section, we show how this technique can also be

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444 The Langevin Equation

applied to the calculation of the nonlinear ac stationary response of rigid polar molecules to an ac field of arbitrary strength. The approach is, in some respects, analogous to that used in Ref. [25] for the calculation of the harmonic mixing in a cosine potential and in Ref. [26] for the evaluation of the mean beat frequency of the dithered-ring-laser gyroscope.

Here, following [27,28], we shall consider the nonlinear dielectric relaxation of an assembly of rigid polar symmetric top particles (macromolecules) dissolved in a nonpolar solvent and acted on by strong external superimposed dc E0 and ac E t(0 = Et cos<2# electric fields. Each particle contains a rigid electric dipole ft directed along the axis of symmetry. Let us also suppose for simplicity that both E0 and Ex are directed along the Z axis of the laboratory coordinate system and that effects due to the anisotropy of the polarisability of the particles can be neglected. Thus the orientational potential energy of the molecule is

V(#,t) = -ju[E0 + El(t)]cos& (8.6.1) (notations are the same as in previous section). Just as in Secion 8.5, the problem we want to solve is intrinsically nonlinear because we assume that the magnitudes of both ac and dc fields are so large that the energy of the molecule in these fields may be comparable to or larger than kT. The recurrence equation for the expectation value of the Legendre polynomial of order n fn(t) = (Pn(cos*?))(0 is a particular case of Eq. (8.5.7) and is given by

• . . n(n + \) , . . W„(0+- iy- i/II(0

= [5,+^cosfl»]^±^[/l_1(0-/B+1(0]. 2(2n +1)

(8.6.2)

where £0 = juE0 /(kT) , $ = nEx /(kT) (8.6.3)

are the dimensionless field parameters. Our goal is to evaluate the ac stationary response of the electric

polarisation P (t), which is defined as P(t) = MN0(COS m) = MNoMt), (8.6.4)

where A^ is the concentration of polar molecules. Here the internal field effects are ignored. Since we are solely concerned with the ac response corresponding to the stationary state, which is independent of the initial conditions, we may seek the fn(t) in the form of a Fourier series:

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Chapter 8. Rotational Brownian Motion in an External Potential 445

/„(')= I Fkn(a»e ikat (8.6.5)

*=-«

As all the fnt) are real, the Fourier amplitudes Fkn must satisfy the

following condition

f-* =(*•*")*. (8-6.6)

where the asterisk denotes the complex conjugate. On substituting Eq. (8.6.5) into Eq. (8.6.2), we have the

following recurrence relations for the Fourier amplitudes Fk(co):

where

znJc(a))Fkn(a>)-2$0[Fr(.a»-Fr(a»]

- £ [ F - > ) + F£t«o) - Ftf(a» - F£(a»] = 0,

2icotDk zn,k(co) = 2(2n + l) 1 + -

(8.6.7)

(8.6.8) n(n + \)

The solution of Eq. (8.6.7) can also be obtained in terms of matrix continued fractions as follows. Let us introduce the column vectors

Cn(6» =

'

F!2m

vm F0

n(co)

Fxn((»)

F2n(co)

< '• J

and R =

( : "\

0

£ 2£> £ 0

I : / (By definition, the vector Cj contains all the Fourier amplitudes of ft), which are necessary for obtaining the ac nonlinear dielectric response). Next, the seven-term recurrence Eq. (8.6.7) can be transformed into the matrix three-term recurrence equations

Ql(0))Cl(0)) + qC2((o) = R, (8.6.9)

Qa((0)Cttm + qCa+lm = qCn_l((0), (n>2), (8.6.10)

where q and Q„(6>) are tridiagonal and diagonal infinite matrices, respectively, defined as

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446 The Langevin Equation

2<f0 <f 0 0 0 <f 2 £ 0 £ 0 0 0 £ 2#0 £ 0 0 0 £ 2<f0 £ 0 0 0 £ 2£0

(8.6.11)

• /

QniCO)-

\-2&) 0

0 *„,_,(©)

0 0

0 0

0 0

0

0

Z«fl(®)

0

0

0

0

0

z».i(®)

0

0

0

0

0

z»^(®)

(8.6.12)

where znk(0)) is given by Eq. (8.6.8).

The infinite system of Eqs. (8.6.9) and (8.6.10) can readily be solved:

Cl(0)) = S(co)R, where the infinite matrix continued fraction S is given by

S(6» = -I

Qi(©) + q-I

(8.6.13)

(8.6.14)

Q2(G)) + q—— q Q3(fi>) + ...

Having determined the column vector C^co) from Eq. (8.6.13), one can

calculate the stationary ac response function / , ( ( ) , which may be presented as follows:

flt) = F^O)) + lfj R e F , V y t o . (8.6.15) t=i

The term Fg(aJ) in the right-hand side of Eq. (8.6.15) is a time independent, but frequency dependent term. This frequency dependence is due to the coupling effect of the dc bias E0 and ac E](0 fields. In the absence of the dc bias field, i.e., for > = 0, the series (8.6.15) contains

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Chapter 8. Rotational Brownian Motion in an External Potential 447

only the odd components of F£ (all the even components are equal to zero) and reduces to

oo

/ i (0 = 2 £ R e f F ^ ^ y ^ - 1 ^ . (8.6.16)

The exact matrix continued fraction solution [Eqs. (8.6.13) and (8.6.14)] we have obtained is relatively easy to compute. As far as the calculation of the infinite matrix continued fraction S is concerned, we approximate it by some matrix continued fraction of finite order (by putting Q„ at some n = N). Simultaneously, we confine the dimensions of the matrices Q„ and q to some finite number M. Both N and M depend on the field parameters £ and £o and on the number of harmonics to be determined. They must be chosen taking account of the desired degree of accuracy of the calculation. For example, for the calculation of F£(GJ) up to k = 7 and for £ and ^ up to 20, the dimensions of Q„ and q need not exceed 50 and 15-20 iterations in calculating S are enough to arrive at an accuracy of not less than 6 significant digits in the majority of cases.

Here, we present the results of calculations for ^ = 0 only (many results for ^ ^ 0 are given in Ref. 22). The real and imaginary parts of the normalised nonlinear harmonic components of the electric polarisation varying in <yand 3ft>,

xG)) = 6Ft\a))lt and

X\(O) = 360Fl\(Q)l? are presented in Figs. 8.6.1-4. The normalisation was chosen in order to satisfy the condition

^ ( 0 ) | = l a t ^ 0 .

The spectra of R&x\co) (dispersion) and \mx\(co) (absorption) and the corresponding Cole-Cole diagrams of the first harmonic component are shown in Figs. 8.6.1 and 8.6.2. Here, it is clearly seen how the relaxation spectrum of Xii03) at £ « 1 (linear response) is transformed

to the nonlinear response spectrum in high fields: with increasing E, the absorption and dispersion curves are shifted to higher frequencies with corresponding decrease of the amplitude due to the saturation. A

saturation appears at £ - 5 , where all the Fourier components F\fiS) become comparable in order of magnitude. Moreover, the half-width of the spectra lmx\Q)) increases (Fig. 8.6.1) as cf increases.

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448 The Langevin Equation

1 i -£->o

0.01 0.1 1 10 100 COT

D

Figure 8.6.1. Rezl(co) (solid lines) and lmz\(G)) (filled circles) vs. onD for

various values of £ Curves 1 correspond to the linear response. Note that all the curves

merge in a single asymptote in the high frequency region (co —> °°).

ReU'l

Figure 8.6.2. Cole-Cole diagram for %\ (co) for various values of £ Curve 1 (semicircle)

corresponds to the linear response.

The frequency behaviour of the third harmonic component x\(a>) is shown in Fig. 8.6.3. For £ « 1, the real part of the Sty-component starts from - 1 at low frequencies, then reaches a positive maximum at cozD~Q.ll before decreasing monotonically to 0 when CO tends to <*>. The spectrum becomes more and more flattened as £ increases. The imaginary part of the 3 6>-component passes through a negative minimum at ft>zb«0.26 for £ « 1. This minimum is shifted to higher frequencies and its absolute value decreases with increasing £ All higher harmonics may be investigated in like manner.

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Chapter 8. Rotational Brownian Motion in an External Potential 449

fe.U ItelA-31]

•OBIOC^TD)

WxJ]

Figure 8.6.3. Re^j and Imjj as a function of log10(fiwD) and £

At small ac fields (£< 0.5), our results are in full agreement with the perturbation solution for the first and third harmonics previously obtained by Coffey and Paranjape [29]:

& 27 -13aJLr2D + i anD (42+ 2O)2TI)

60(1 + a?T1Df(9+4O/TI)

3-ncD2Tl + ia)TD(U-6a)2Tl)

(1 + a)2T2D )(9 + A(o2z2

D )(1 + 9 « 2 4 )

+ 0(<f4), (8.6.17)

+ 6>(£2). (8.6.18)

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450 The Langevin Equation

Thus, the steady-state ac nonlinear response of an ensemble of rigid polar molecules in strong superimposed ac and dc bias fields can be evaluated in terms of a matrix continued fraction. The method of the calculation of nonlinear ac responses based on matrix continued fractions, which we have presented, is quite general. For example, the method can also be applied to the calculation of the dynamic Kerr effect ac response of polar and anisotropically polarisable molecules [28] and to the evaluation of the nonlinear impedance of a microwave-driven Josephson junction [30,31] .

8.7 Dielectric Relaxation and Rotational Brownian Motion in Nematic Liquid Crystals

Here, we apply our method to the solution of a problem of dielectric relaxation in nematic liquid crystals. We recall, first, that dielectric relaxation processes in liquid crystalline materials are determined by structural properties, intermolecular interactions and reorientations of the molecules. The nematic liquid crystal mesophase has the simplest dielectric behaviour with the advantage that qualitative results obtained for nematics also provide, to some extent, a basis for understanding the molecular dynamics of other mesophases such as ferroelectric or antiferroelectric smectic. The complex dielectric permittivity of a nematic liquid crystal has very different dispersion regions and qualitative behaviour in the parallel (EII n) and perpendicular (E _L n) alignment (E is the measuring ac field and n is the director of the nematic). The transverse component of the dielectric susceptibility tensor always exhibits the Debye dispersion (due to reorientations of dipolar molecules) which usually occurs for liquid systems in the microwave region. The longitudinal component may also exhibit an additional dispersion at radio frequencies (the low frequency region). The origin of these dispersion regions is, in general, due to hindered reorientations of the molecules about the long and short molecular axes in the strong orientational forces of the nematic phase. If there are additional dipolar groups of the molecules, with their own internal degrees of rotation, the corresponding dispersion regions will again be similar to those in the isotropic liquid phase.

Dielectric relaxation in nematic liquid crystals is usually interpreted in the context of a model of noninertial rotational Brownian motion of a particle in a mean field potential field U (see, for example, [32,33]) although the mean field approximation has a restricted area of applicability, as it ignores local order effects. In spite of this drawback, the model nevertheless is easily visualised and, moreover, allows us to

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Chapter 8. Rotational Brownian Motion in an External Potential 451

carry out quantitative evaluations of the dielectric parameters of nematics. The usual approach to the problem is to solve the Fokker-Planck (7.6.1.38) of Chapter 7 for the distribution function W of the orientations of a unit vector u, fixed in the molecule. The basic model for the calculation of the components of the complex dielectric permittivity tensor and dielectric relaxation times of nematics is the rotational diffusion model of rodlike molecules in the self-consistent mean field uniaxial potential of Maier and Saupe [34]:

V/(kT) = -a cos2 j3 = -(2<7/3)P2 (cos /?) + const, (8.7.1)

where J3 is the angle between the axis of symmetry of the molecule and the Z axis of the laboratory coordinate system (here, we are using the notations of Section 7.6). The calculation of the complex dielectric susceptibility and associated relaxation times for the Maier and Saupe potential has already been considered (for instance, Martin et al. [35], Nordio et al. [36-38], Storonkin [39]) leading to a clear physical understanding of dielectric relaxation in nematics. Moreover, in Sections 7.4.2 and 7.4.4, we have presented both exact and simple approximate solutions for the rotational Brownian motion of a dipolar molecule in the uniaxial potential Eq. (8.7.1), when the dipole moment of the molecule \i is directed along the axis of symmetry of the molecule. Here, a more general case, where the vector \i is directed at an angle 0 to the molecular axis of symmetry, is considered so allowing us to substantially extend the scope of the model (the most general case of an asymmetric top molecule was treated by Tarroni and Zannoni [40] in the context of the Fokker-Planck equation). We shall show how simple approximate formulae for the longitudinal and transverse components of the complex dielectric susceptibility and corresponding relaxation times may be obtained. The results are presented in the form suitable for comparison with dielectric relaxation measurements.

The complex dielectric permittivity tensor $,(<2>) of a nematic is diagonal and has only two independent components one perpendicular, £±(a>) = exx(G)) = eYy(6)), and the other parallel, e^co) = e^ico), to the director vector n in the laboratory coordinate system XYZ, where the Z axis coincides with the director. The components of the complex dielectric permittivity tensor are determined by the relations [41]:

er(co)-£^= ^ i —

where

®r(0)-iG)j <S>r(t)e-ic*dt , (r=ll,l) , (8.7.2)

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452 The Langevin Equation

*I|C) = ( I ^ ' ( 0 W ( 0 ) and <S»±(f) = ( X // i(0)/ /(0) (8.7.3) W /o W /o

are the macroscopic autocorrelation functions of the parallel and perpendicular components of the dipole moment per unit volume M of N0

interacting dipolar molecules defined as No .

M(0 = Z n ' ( 0 , (8.7.4) ;=i

H' (?) is the dipole moment vector of the ith molecule, e^ is the high frequency limit of the components of the complex dielectric permittivity tensor, and Rr(a>) is the frequency dependent factor of the internal field. The internal field factor Rr(co) for an ellipsoidal cavity surrounded by an infinite dielectric continuum with the same complex dielectric permittivity tensor fy (co) is given by [41]

eJco) Ry(Q)) = r- , (8.7.5)

where <Tr(co) are the components of the depolarisation tensor defined as

a (co) = a * a ^ z 1 ds 7 lie^me^io^e^mf1 % [s + a2

r / er(co)JD(s)

with

D(s) = [s + a\ lew(o))~\ls + a*leYY(co)~\ls + a\Iezz(co)].

The a, are the semiaxes of the ellipsoidal cavity. Another equation for Rr(co) may be obtained in the context of the ellipsoidal cavity model [41], if we make the assumption that the permittivity tensor Sij(a>) of the material surrounding the cavity is frequency independent and equal to the static tensor £y (0). We shall then have

£J0)-aJ0)[eJ0)-eJco)] K((o).

According to the above equations, even if the macroscopic autocorrelation functions are known, it is still not possible to calculate the complex permittivity because the depolarisation tensor depends on the cavity dimensions, which are arbitrary.

This dependence on the geometry of the sample is an unsatisfactory feature of the analysis. However, as pointed out by Luckhurst and Zannoni [41], it does not present a major problem if the cavity contains many molecules since the cavity shape is then irrelevant.

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Chapter 8. Rotational Brownian Motion in an External Potential 453

It is therefore convenient to select a spherical sample since the depolarisation tensor is then (7r(0)) = \/3 and is independent of the sample size. Hence, the relationship between the permittivity tensor and autocorrelation function tensor for the net dipole moment of the spherical cavity is completely defined. Another essential feature of the problem consists in relating the macroscopic autocorrelation function tensor to that of a single particle:

C||(r) = <«z(0)Hz(f)>o, CL(t) = (ux(0)ux(t))0, (8.7.6)

where ux and uz are the projections of the unit vector u along the dipole vector n of a molecule onto the axes X and Z. The simplest one, by far, is to ignore the correlations between dipole moments of different molecules. The macroscopic dipole autocorrelation function then becomes that of a single molecule [41]

®r(t)~NM2Cr(t). (8.7.7)

We shall use this approximation below. As in Chapter 7, Section 7.6.3, we suppose that the dipole

moment vector i is oriented at an angle 0 to the direction of the long axis of the rod-like molecule. The orientation of a moving coordinate system xyz, fixed in the molecule, with respect to the laboratory system XYZ is defined by the Euler angles £1 = a,/3, f\ (see Fig. 7.6.1.1). We shall also take into account that the rotational diffusion coefficients about short and long molecular axes are different, i.e., Dn ?t D±. The autocorrelation functions from Eqs. (8.7.6) may be given as [38]

C„(f) = cos2 0 / ^ ( 0 + sin2 Qf^t) , (8.7.8)

Cj_(0 = cos2 &fio(t) + s i n 2 ®/A(0 - (8-7-9) where

/ ! ( 0 = << m (0)< m (0>o (8-7.10) is the equilibrium autocorrelation function (the definition of Wigner's D function is given in Chapter 7, Section 7.6).

Thus, our problem reduces to the evaluation of the four equilibrium autocorrelation functions defined by Eq. (8.7.10). We proceed as follows [42,43]. For the uniaxial potential Eq. (8.7.1), we have from Eq. (7.6.1.33) the set of moment equations for D]

nm at time t:

*D jKm = -[Am2 + j(j + l)/2]Dim

, j+2 (8-7.11) + T I v2,0,0C^2,0C;,m

m2,0[7(/ + l ) - ; ( ; + D-6]Dn

y,m, 4 H H

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454 The Langevin Equation

where v2 0 0 = -2cr/3, rD=(2D±) , and A = (Z|/£>x-i)/2, or explicitly

d T -~DJ

tDdtn'

j(j + 1) I m2\ aUU + -^m2][j(j + l)-3n2) 2 j(j + \)2j-\)(2j + 3) Km

a \(J + l)yl[f-n2][U-l)2-n2][j2-m2][(j-l)2^F] ,

2j + l 7(7-D(2j- l ) #

<7 + 3 ) V [ / - » 2 ] [ / - m 2 ] Z J j _ 1 wz(7 - 2)J[(j +1)2 - n2] [(j +1)2 - " ^ ] -+1

7(7 -1X7 + 1) n'm 7X7 +1X7+ 2)

7V(7 + 2 ) 2 - » 2 ] [ ( 7 + l ) 2 - » 2 ] [ ( 7 + 2 ) 2 -m 2 ] [ (7 + l ) 2 - m 2 ] f „ ,

(7 + 2X7 + 1X27 + 3) Dj;2 . (8.7.12)

Equation (8.7.12) is equivalent to that given by Nordio and Busolin [36]. We can now derive from Eq. (8.7.12) four systems of

differential-recurrence relations for the equilibrium correlation functions

/oi(0 = <£>o,o(0)Ao(0>o. foiit) = V;(7 + 1) /2(< 1 (0)D^(0) 0 ,

/iJ1(0 = <<i(0)A"!i(0>o. //o(0 = yJjU + D/2(D1

i;0(0)D10(0)0.

(where DJnmt) = DJ

nm [Cl(t)]). This can be accomplished by (i) obtaining

the differential-recurrence equations for D^0, D^, D0, and Dx (by

putting the indices m, n in Eq. (8.7.12) equal to 0,0, 0,1, 1,0, and 1,1, respectively), then by (ii) multiplying across these equations

by Z^o(0), £>oj(°). D*0(0),andDJ*(0), accordingly, and by (iii) averaging the equations so obtained over the equilibrium distribution function of the sharp initial conditions a, /J, and f- Thus, we obtain the three three-term recurrence relations for f^t), f0(t), and /jo(f):

2tr

JU + Vdt /o,(0 + 1 —

2a

(2 ; -l)(2y + 3) /diC)

2cr

27 + 1

2rD d

j(j + l)dt

2a

7-1

27-1

7 + 2

/ii(o+

/oT2(0-f^/of(0 27 + 3

2<7[l-3/y(7 + l)]

(27-1X27 + 3)

(8.7.14)

fii(t)

27 + I 7 + 1

.27-1 U~2(t)- 1 /ii+2(0

27 + 3

(8.7.15)

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Chapter 8. Rotational Brownian Motion in an External Potential 455

*Djtfoi(t) +

(Tjij + D

7U + D + A-

2j + l 7 + 1

27 -1 foi~2(t)-

g [ y g + i ) - 3 ]

( 2 j - l ) ( 2 ; + 3)

j fo\+2(t)

foW)

2j + 3

(8.7.16)

and the five-term recurrence relation for fx\ (t):

TDJtrt«)+ 7(7 + 1) + A — g[7(7 + l)~3f

;(7 + D(27-l)(27 + 3) /tf(0

27 + 1 °'-2)0'+1)-/Ir

a(o+^/Irw- ZZ2. f;+!(-,-, 7 (7 + 3) y-+2

; + l / n ( 0 27 + 3 / n ( ° 27-1 7 (8.7.17)

withy = 1,2,... and f^(t),f°m(t) = 0. We remark that Eqs. (8.7.14) and (8.7.15) are mathematically

identical to those describing dielectric relaxation when 0 = 0 [Eq. (7.4.2.1) and (7.4.4.1)]. Such equations have already been solved in Sects. 7.4.2 and 7.4.4 so we merely quote the main results here. The exact

solution of Eq. (8.7.14) for the Laplace transform f^(s) is [cf. Eq. (7.4.2.8)]

yo/«)(0) /<£>(*)=•

STD + 1 • 2a [i-4w] 15

x ~ (4n + 3)T(n + l/2)f^\0)n

n=\ 3V;zT(« + 2 ) / 0 0 ( 0 ) t=1

_, (8.7.18)

where S^is) is the continued fraction defined as

S"m(s) = 2a(n-l)

4n -1

2zns + 1 —

2a -+-

2cr(n + 2)

_n(n + l) (2n-l)(2n + 3) (2n + l)(2n +3) S£2(s)

- l

r2n+l The initial conditions f^ (0) are given by Eq. (7.4.2.23), viz.,

/ ^ + 1 ( 0 ) : anr(n + 3/2)M(n + 3/2,2n + 5/2,a)

2r(2n + 5/2)M(l /2 ,3 /2 , a)

Similarly, we have the exact solution for yY0(s) and /^(s) [cf. Eq. (7.4.4.7)]:

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456 The Langevin Equation

? (s)~ T° T Hr(n+3/4)r(n+l/2) f 2 ^ f m n ? 2 ^ i f r t

[01,

where

f + l)r(n+2) | o i | *£ , (8.7.19)

SUs) = on(n+l)

4n -1

and

TDS + rin+1) o[n(n+1) - 3] a n > + l ) +2

-r-l

(2n-l)(2rt+3) (2n+l)(2n+3) 10 (*)

S0B

1(J) = 5r0(* + A/TD).

The initial conditions /0o"+1(0) are given by Eq. (7.4.4.11), viz.,

f 2» + i , m _ r 2 t l + 1 / m _g B ( /» + l ) r (n + 3/2)M(n + l /2,2n + 5/2,g)

/io W - / 0 1 W - 2 r (2n + 5/2)M(l/2,3/2,<7)

In order to obtain /^(s) from Eq. (8.7.17), we, as usual, apply the general matrix method of solving multi-term differential-recurrence relations (see Section 2.7.3). Equation (8.7.17) may be transformed to the matrix equation

f/,2,'-1^ =Q;

with C0(0 = 0 and

v/ii7_2(0. + Q7-

/n'(0 ) +Q)

v/iV'+2(0. (8.7.20)

Q;= ( 4 ; - l ) ( 4 j - 3 )

0

Q,

(T[2j(2j- l ) -3] 2

2y(2 ; - l ) (4 ; + l ) (4 ; -3)

- ; ( 2 ; - l ) - A

<r(2j + 3)

27(47 + 1)

2o-(j + l)

(2y - l ) (47- l )

2cr(7 - l ) ( 2 ; + 02

(47-1X47 + D J

g(27-3)

27(47-1)

tj[27(27 + l ) -33 2

27X27 + 1)(47-1X47 + 3)

-7(27 + D - A

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Chapter 8. Rotational Brownian Motion in an External Potential 457

Q) =

f 2<7Q- + l)(2y-l)2

(4y- l ) (4 ; + l)

2o-(7-D 4af(2j + 3)

(2j + l)(4j + l) (4; + l)(4/ + 3), The exact solution of Eq. (8.7.20) in terms of matrix continued fraction is

rA\(^ = TD\\1(s)\C](0) + ZU QtAu(s)Cn(0)\, (8.7.21)

n=2 k=2 J

where the matrix continued fraction A", (s) and the initial value vectors

C„ (0) are given by

Ann(s) = [TDsl-Qn -QXiVXk* J* ,

C„(0) = ^ _ 1 ( 0 ) N

/,i"(0) V Ju

1

4 « - l [2n(P2n_2)0+(2n-l)(P2n)0]

Here

CP2„>o = a"r(n + \/2)M(n + l/2,2n + 3/2,a)

2F(2n + 3/2)M (1/2,3/2, <r)

[this equation follows from Eqs. (7.4.2.13) and (7.4.2.22) of Chapter 7] and we have used the equality

D11

1(Q)£»V1(Q) = ^ ;-+i 1±—PH(cos/3)+-±-Pj+1(cosj3) + Pj(cos/3) 2j + \ 2j + \

On using Eqs. (8.7.18), (8.7.19), and (8.7.21), we may calculate the spectra of the dipolar correlation functions:

C,(ifi>) = cos2 ®fm(i0J) + sin2 Qf^ (ico), (8.7.22)

C±(to) = cos2 ©//oCto) + sin2 ©fniico), (8.7.23) and so the components of the complex dielectric permittivity tensor. The exact solutions (8.7.18), (8.7.19), and (8.7.21) in terms of scalar and matrix continued fractions are applicable for any values of the parameters a, 0, and Dn/Dx. Furthermore, they allow us to determine the accuracy

of the various approximate solutions.

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458 The Langevin Equation

The equilibrium correlation function f^(t) = (D^(0)Z)Q 0 (0)O

can be approximated by a single exponential, as has been shown in Chapter 7, Section 7.4.5,

/oo(0 * /oo(0)e~'/%> = (1 + 2S)e-"T™ / 3 , (8.7.24) where

S = ! < c o s 2 / ? > 0 - i

3ea 3 1 (8.7.25)

AaM (1/2,3/2,0") 4<7 2 is the order parameter [32,33], %) is the longitudinal relaxation time given by the exact Eqs. (7.4.2.29) or (7.4.2.30). The approximate formula for the relaxation time % valid for all values of a is given by Eq. (7.4.2.38), viz., [44,45]

Too 2 a+- (8.7.26) TD(eff-l)[~ ' V^(1 + £T)_

The main contribution to the relaxation of f^(t) is due to the overbarrier relaxation mode, which has the smallest nonvanishing eigenvalue. However, Eq. (8.7.24) ignores high-frequency relaxation inside the wells, which is detected as a very weak peak in the dielectric loss spectrum (see

Section 7.4.5). Here, a better approximation for /^(t) is [42,43]

/oo(0 » e"'/T°° <cos P)2well + [<cos2 fi)Q - (cos P)2

well e-"T" ,(8.7.27)

where % ~TD/2cr is the time characterising relaxation inside the wells

[17],

(COS/?)we" = 2oM (1/2,3/2,0-)' •))weu means an average in a single potential well, for example, in the domain 0<f3<7tl2.

Now, as shown in Chapter 7, Section 7.4.5, the behaviour of the

(transverse) correlation function fiQ(t) = (Dj 0 (0 )^ (0)0 rnay accurately be described by a single exponential by means of the effective eigenvalue method:

/ > ) « /io(0)e"' /% = 0 - S)e-"T,° / 3 , (8.7.28) where

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Chapter 8. Rotational Brownian Motion in an External Potential 459

'10 z io /io(Q)

/io(0) = T r

1 + ^ +2 c J / o i ( 0 )

15/oi (0) (8.7.29)

= 2r r 1-5

2 + 5 '

Here, we expressed the effective relaxation time %f§ in terms of the order parameter S, Eq. (8.7.25). This representation is convenient for comparison with experimental data (the order parameter 5 can be

measured experimentally [32,33]). zfg is given explicitly by Eq. (7.4.4.18).

In like manner, the correlation functions f^ (t) and / j \ (t), which mainly characterise the rotation of the molecule about the long molecular axis, can also be described by the single exponentials

where

ti1(t)~(l-S)e-"T<»/3,

f1\(t)^(2 + S)e-"^/6,

''Ol *01 _ /oi(0)

= T r

= Tr,

1 + —+ A + — , 5 15/0^(0)

1-5

n- l

l + ( l -5 )A + 5/2

(8!7.30)

(8.7.31)

(8.7.32)

Ln -/ACQ)

= Tr SL A 10 2

A

KD± - 1 +

= Tr

4<T/13

1(0) af^O)

15/A(0) 6/^(0)

2 + 5

-|-1

(8.7.33)

2 + (2 + 5 ) A - 5 / 2

Here, TQ and iff are the effective relaxation times yielded by Eqs. (8.7.16) and (8.7.17), respectively, at t = 0.

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460 The Langevin Equation

a

Figure 8.7.1. \n(Tnm/rD) vs. a, calculated from Eq. (8.7.34) for £>(| /£>_,_ = 1 (solid

lines). Filled circles are the calculation from Eqs. (8.7.26), (8.7.29), (8.7.32), and (8.7.33).

The results of the calculation of the relaxation times

r„m=/„m(0)//n m(0) (8.7.34)

[which are evaluated from Eqs. (8.7.18), (8.7.19), and (8.7.21)] and the

effective relaxation times from Eqs. (8.7.29), (8.7.32), and (8.7.33) as

well as ?oo from Eqs. (8.7.26) are shown in Fig. 8.7.1 as functions of o.

Here Tmn and the effective relaxation times of / / 0 ( 0 , /o i (0 , and /^(t)

are clearly in complete agreement. The effective eigenvalue method is

successful in the evaluation of the decay of fi0(t), f^it), and /^(f)

because the overbarrier (activation) relaxation mode is not involved in

these relaxation processes so that the behaviour of the correlation times

% , % , and TH and TQ, TQ, and Tff is similar. This is not so for

/no(0 the effective relaxation time of which diverges exponentially from

%) on account of the activation process (see Fig. 7.4.2.1). On using the above results, we can now derive simple

approximate expressions for the normalised complex susceptibility,

XYG>)= V — ^ - = CJQ,)-icoCJico), (y=ll,± ) , (8.7.35) r 4^N0Rr(a))

which may be written as

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Chapter 8. Rotational Brownian Motion in an External Potential 461

z±(<y)=-

(l + 2S)cos20 (1-5) sin2 9

1 + icor^ 1 + ion^

( l -S)cos 2© (l + 5/2)sin2©'

(8.7.36)

(8.7.37) 1 + ian% 1 + ifinff

where T$, T<, rf0, and rf are given by Eqs. (8.7.26), (8.7.29), (8.7.32), and (8.7.33), respectively. The results of the calculation of the dielectric loss spectra Zyi®) from the exact and approximate formulas

are shown in Figs. 8.7.2 and 8.7.3. For a» 1, the longitudinal dielectric loss spectrum xli®) has two peaks. The low frequency peak, which

usually occurs in the microwave region, is due to the overbarrier mode of the parallel (to the long molecular axis) component of the dipole moment. On the other hand, both high frequency relaxation modes inside the wells and the rotation of the perpendicular component of the dipole moment around the long molecular axis manifest themselves in the high frequency (GHz) band. In contrast, in the transverse dielectric loss spectrum Z±(<&), the two dispersion regions are not widely separated as they are both located in the high frequency region. Nevertheless, if they can be distinguished (e.g., curve 1 in Fig. 8.7.2), one frequency dispersion band is associated with the rotation of the perpendicular component of the dipole moment about the long molecular axis, while the high frequency relaxation modes inside the wells contribute to another high frequency peak (or shoulder) of the spectrum zli10) • Despite the large number of high frequency modes involved in both high frequency mechanisms, essentially two near degenerate modes only (so that they have approximately the same characteristic frequencies) determine both xli®)

and %]_(co) in the high frequency relaxation process (see Section 7.4.5). Thus, both low and high frequency processes are still effectively governed by a single relaxation mode. As is apparent from Figs. 8.7.2 and 8.7.3, the results predicted by the approximate Eqs. (8.7.36) and (8.7.37) agree to good accuracy with the spectra calculated from the exact continued fraction formulas. This means that for 0 ^ 0 both longitudinal and transverse dielectric relaxation can be approximately described by two exponential decays.

We may formally introduce retardation factors for each effective mode (the retardation factor gnm is defined as the ratio of relaxation times

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462 The Langevin Equation

of the corresponding mode in the nematic and isotropic liquid phases). Thus, retardation factors for the longitudinal relaxation are given by

e°-\2oJcTln v ' £oo

and

Soi=-

l + <7

(1-S)(1 + A)

- + 2~a (8.7.38)

l + ( l -5 )A + 5/2 respectively. The retardation factors for the transverse relaxation are

(8.7.39)

NJ

(DT„

Figure 8.7.2. %\(a)) and j"(a>) calculated from the exact continued fraction equations at a= 10 and Dn IDL =1 (solid lines). Curves 1, 2, and 3 correspond to /?= nl 10, nlA and 2KI 5, respectively. Filled circles: the calculation from Eqs. (8.7.36) and (8.7.37). [for P=Jtl\Q, Eq. (8.7.27) was used].

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Chapter 8. Rotational Brownian Motion in an External Potential 463

10"'

lO"2

io-3

3--V.

'/ /

2 1

j •' ' • • *

10"" 10" 10'

corn

Figure 8.7.3. ^'(ca) and xli®) calculated from the exact continued fraction equations at

/?=;z74and £>||/DX=1 (solid lines). Curves 1, 2, and 3 correspond to <r=l,5,and 10,

respectively. Filled circles: the calculation from Eqs. (8.7.36) and (8.7.37).

1-S £io

1 + 5/2 and

Sn (2 + S)(l + A)

(8.7.40)

(8.7.41) 2 + (2 + 5 ) A - 5 / 2

In Eq. (8.7.38), the retardation factor goo is given as a function of the barrier height parameter a. One can also express goo as a function of the order parameter 5 by using the inverse function of Eq. (8.7.25) or by using the extrapolating equation [43]

Page 489: The Langevin Equation Coffey_Kalmykov_Waldron

464 The Langevin Equation

3S(5-xS) a~- (8.7.42)

TD ,, Dn,m ~ ' i<£tlW Di

2(1-5Z) which provides a close approximation to crS).

The approach we have given for the evaluation of the dielectric parameters of nematics can be generalised to a mean field potential of the form

^ = I v ^ ( c o S / 3 ) . (8.7.43) Kl R

Here, instead of Eq. (8.7.11), we have (see Section 7.6.1 of Chapter 7 for details)

d_

dt , j+H ' ' (8-7-44)

+ T Z I vRC];lRfiC]2Rfi[jJ +\)-jj + \)-R(R + \)]Dim. 4 R J=\j-R\

In like manner, it is possible to derive sets of equations for the equilibrium correlation functions foo(t),f0(t), f\i(t), and fx\(t), which can be solved by using matrix continued fractions.

Furthermore, the approximate relations (8.7.36) and (8.7.37) for the susceptibilities can also be applied with some modifications to a potential of the form of Eq. (8.7.43). Indeed, in this case, the longitudinal

relaxation time tm of the correlation function /0Q = (cos/?(0)cos/?(0)0

is given by the exact expression (see Chapter 2, Section 2.10):

Too _ 6zn

(25 + 1)Z I Vx)lkT)

f xe -V(x)KkT) dx, (8.7.45)

where x = cos/?,

S = -jP2(x)e-VMKkT)dx, Z = \ e-VMKkT)dx. (8.7.46)

For the uniaxial potential, Eq. (8.7.1), Eq. (8.7.45) reduces to Eq. (7.4.2.30). Equations (8.7.29), (8.7.32), and (8.7.33) for the effective

relaxation times (^oi'^iO'^n ) a^so r e m a m valid for a potential of the general form of Eq. (8.7.43), if we write them in terms of the order parameter S,

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Chapter 8. Rotational Brownian Motion in an External Potential 465

% = 4 = 2 ^ ^ | , (8.7.47)

T « zi = t \zl , (8.7.48) 01 01 Dl + (l-S)A + S/2

T « i f = TD — , (8.7.49) 11 n D 2 + (2 + 5 ) A - 5 / 2

where the order parameter 5 is given now by Eq. (8.7.46). Thus, on using Eqs. (8.7.45)-(8.7.49), we may calculate Xwi®) and Z±(°>) from Eqs. (8.7.36) and (8.7.37) for any potential of the form of Eq. (8.7.43). In particular, such an approach provides a simple analytical solution for the potential [46]

— = -A2P2(cosj3) - A4P4(cosytf) (8.7.50) kT

and agrees in all respects with the numerical calculations of Nordio et al. [38] for this potential. As has also been shown by Nordio et al. [38], it is possible to improve the agreement of the theory with experimental data by varying the value of A4.

The exact and approximate solutions for the retardation factors goo and g10 for the uniaxial potential Eq. (8.7.1) have been recently compared with experiment by Urban et al. [48-53]. They verified that the predictions of the theory are in qualitative agreement with the experiment. However, there are deviations from the experimental data in the temperature dependence of goo and gio predicted by Eq. (8.7.38) and (8.7.40). One would expect deviations of this kind as the mean field approximation provides only a qualitative description of relaxation processes in liquid crystals [32,33].

We have shown in this chapter how multi-term recurrence equations can be solved in terms of matrix continued fractions. We have illustrated the method by taking several examples from the theory of dielectric and magnetic relaxation pertaining to the problem of the rotational Brownian motion in an axially symmetric potential in the presence of an external field. We have also shown that both linear and nonlinear problems in that theory can be solved in similar manner. More complicated problems will be treated in the next chapter.

References

1. H. Risken, The Fokker-Planck Equation, Springer, Berlin, 1984; 2nd Edition, 1989. 2. J. I. Lauritzen, Jr., and R. Zwanzig, Jr., Adv. Mol. Rel. Interact. Proc. 5, 339 (1973).

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466 The Langevin Equation

3. A. J. Martin, G. Meier, and A. Saupe, Symp. Faraday Soc. 5, 119 (1971). 4. A. Aharoni, Phys. Rev. 177, 2 (1969). 5. D. A. Garanin, V. V. Ischenko, and L. V. Panina, Teor. Mat. Fiz. 82, 242 (1990)

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New York, 1964. 8. L. Neel, Ann. Geophys. 5, 99 (1949). 9. W. T. Coffey, D. S. F.Crothers, Yu. P. Kalmykov, and J. T. Waldron, Phys. Rev. B

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Engineers, Vol. 2, Mir, Moscow 1990. 16. W. T. Coffey, P. J. Cregg, and Yu. P. Kalmykov, On the Theory of the Debye and

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17. D. A. Garanin, Phys. Rev. E 54, 3250 (1996). 18. W. T. Coffey and Yu. P. Kalmykov, Phys. Rev. B 56, 3325 (1997). 19. Yu. L. Raikher and M. I. Shliomis, Zh. Eksp. Teor. Fiz. 67, 1060 (1974) [Sov. Phys.

JETP 40, 526 (1974)]. 20. L. D. Landau and E. M. Lifshitz, Phys. Z. Sowjetunion, 8, 153, (1935). 21. W. T. Coffey, J. L. Dejardin, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. E 54,

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29. W. T. Coffey and B. V. Paranjape, Proc. R. Ir. Acad. A 78, 17 (1978). 30. W. T. Coffey, J. L. Dejardin, and Yu. P. Kalmykov, Phys. Rev. E 61, 4599 (2000). 31. W. T. Coffey, J. L. Dejardin, and Yu. P. Kalmykov, Phys. Rev. B 62 3480 (2000). 32. L. M. Blinov, Electro-Optical and Magneto-Optical Properties of Liquid Crystals,

Wiley, Chichester, 1983. 33. W. H. de Jeu, Physical Properties of Liquid Crystalline Materials, Gordon and

Breach, New York, 1980.

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Chapter 8. Rotational Brownian Motion in an External Potential 467

34. W. Maier and A. Saupe, Z. Naturforsch. 13a, 564 (1958); Z. Naturforsch. 14a, 882 (1959); Z. Naturforsch. 15a, 287 (1960); G. Meier and A. Saupe, Mol. Cryst. 1, 515 (1966).

35. A. J. Martin, G. Meier, and A. Saupe, Symp. Faraday Soc. 5, 119 (1971). 36. P. L. Nordio and P. Busolin, J. Chem. Phys. 55, 5485 (1971). 37. P. L. Nordio, G. Rigatti, and U. Segre, J. Chem. Phys. 56, 2117 (1971). 38. P. L. Nordio, G. Rigatti, and U. Segre, Mol. Phys. 25, 129 (1973). 39. B. A. Storonkin, Kristallogr. 30, 841 (1985) [Sov. Phys. Crystallogr. 30, 489

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1031 (1972) 48. S. Urban, B. Gestblom,T. Bruckert, and A. Wiirflinger, Z. Naturforsch. 50a, 984

(1995). 49. B. Gestblom and S. Urban, Z Naturforsch. 50a, 595 (1995). 50. B. Gestblom and S. Urban, Z Naturforsch. 51a, 306 (1996). 51. S. Urban, D. Busing, A. Wiirflinger, and B. Gestblom, Liquid Cryst. 25, 253 (1998). 52. S. Urban, A. Wiirflinger, and B. Gestblom, Phys. Chem. Chem. Phys. 1, 2787

(1999). 53. J. Jadzyn, G. Czechowski, R. Douali, and C. Legrand, Liquid Cryst. 25, 253 (1998). 54. L. Gammaitoni, P. Hanggi, P. Jung, and F. Marchesoni, Rev. Mod. Phys. 70, 223

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1182 (1990) [JETPLett. 52, 593 (1990)]. 56. E. K. Sadykov, Fiz. Tverd. Tela (St.Petersburg) 33, 3302 (1991) [Sov. Phys. Solid

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Chapter 9

Rotational Brownian Motion in Non-Axially Symmetric Potentials

9.1 Introduction

In Chapter 7, we have demonstrated how differential-recurrence relations in two and three indices arise naturally in the solution of the noninertial Langevin equation pertaining to magnetic and dielectric relaxation. The most important results of this investigation are Eqs. (7.2.25), (7.3.1.32), and (7.6.1.33), which allow one to derive the differential-recurrence relations for any problem in which the potential may be expanded in spherical harmonics or in Wigner D functions. In this chapter, we shall illustrate the solution of differential-recurrence equations by considering the calculation of the dynamic magnetic susceptibilities, the relaxation times, and the smallest nonvanishing eigenvalues in the Neel-Brown model [1] for superparamagnetic particles when the assumption of an axially symmetric potential is abandoned. We shall consider two problems, namely, magnetic relaxation of (i) uniaxial superparamagnetic particles subjected to a strong external uniform magnetic field H which is directed at an arbitrary angle to the easy axis of the particle and of (ii) superparamagnetic particles with cubic anisotropy. The problem (i) was formulated by Stoner and Wohlfarth [2] and re-examined recently in the series of papers [3-12]. The study of the problem (ii) was initiated by Aharoni [13] and others [14-17] and continued in Refs. [18-21]. We have taken these two problems here only as particular examples since the methods developed are quite general and may also be applied, in principle, to an arbitrary anisotropy potential. These problems differ from those considered in detail in Chapters 7 and 8 insofar as they involve differences in two indices in recurrence relations. The numerical solution of such differential-recurrence relations poses special computational difficulties which, however, can be overcome as we shall see below. We remark that very similar problems also arise in other

468

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Chapter 9. Non-Axially Symmetric Problems 469

physical applications such as the transient nonlinear dielectric and Ken-effect response of polar and polarisable molecules in high electric fields (e.g., [22,23]) and the rotational Brownian motion and orientational relaxation of an impurity molecule in cubic crystals (e.g., [24-28]).

9.2 Uniaxial Superparamagnetic Particles in an Oblique Field

9.2.7 Recurrence equations

In Chapters 7 and 8, the effect of an external uniform magnetic field H on the dynamics of fine single domain ferromagnetic particles having simple uniaxial anisotropy was studied. For uniaxial anisotropy, the ratio of the potential energy v V to the thermal energy k T is

fiV=-a(u-n)2-^(u-h), (9.2.1.1) where u, n, and h are unit vectors in the direction of the magnetisation vector M, the internal anisotropy (or easy) axis, and the field H, respectively. In Chapter 8, the calculation assumes that the applied field and anisotropy axes are collinear. In this chapter, we study the dynamics of the magnetisation when the assumption that n || h is abandoned, as in practice the easy axis is usually in a random position [29]. The calculation proceeds by solving the infinite hierarchy of the moment equations for the averaged spherical harmonics, Eq. (7.3.1.32) obtained in Chapter 7, Section 7.3. That hierarchy can be presented equivalently

for the relaxation functions cl m(t) = (Yl m)(t) - \Yl m) :

TNCl,m ( 0 = 2-i.y,r el,m,l+r,m+sCl+r,m+s ( 0 ' (9-2.1.2)

where the elml>m> are given by Eq. (7.3.1.31), viz.,

/(2/ + l)(2/' + l) K/,m,/',m±.v

1 ,n i\x z < , v /0 /(2/ + l)(2 -l(l + l)SirSl0 + (-l) ^ ^

- f[z'(r + l ) -r(r + l ) - / (Z + l)] k '±s\ 2V27TT c/,o,r,oC/,m/,-mT,.

+ i_ j(2r + l)(r-S)\ t ! /(Z, + J - l ) ! ^ , 0

ai (r + 5)! L,hrJ(L-s + l)l'^° AL=2

x[m^(L + s)(L-s + l ) C ^ , _ m T ^ 1

(9.2.1.3)

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470 The Langevin Equation

(the definitions of all the quantities are given in Chapter 7, Section 7.3). In writing Eq. (9.2.1.2), we have noted that the equilibrium averages

Yt,m0 s a t i s fy

Z / / , m , / v ( 7 / v ) 0 = 0, (9.2.1.4) l',m

where the angular brackets ( ) designate the equilibrium average

defined as IKK

(A)0 = j f A(&,<p)W0(&,<p)sm#diM<p. (9.2.1.5) o o

Here, the equilibrium Boltzmann distribution function Wo is given by

W (& 0) = z~le<7cos2'+^7lCOS'psin'f+r2Sin'psin'f+r3COS^ (9 2 16)

where Z is the partition function and yl, y2>73 a r e t n e direction cosines

of the vector H. Equations (9.2.1.2) and (9.2.1.3) are valid for all those

anisotropy potentials that are characterised by the coefficients vr+.s. of the free energy density development as a series of spherical harmonics [see Eq. (7.3.1.25)]. The potential energy Eq. (9.2.1.1) can be expressed in terms of spherical harmonics as

pv=-<fV2^73[(K +ir2)*i,-i +yl2y3Ylfi -in - W u ] -(4a73)Vfl75Y20 + const.

Thus, we have equations for the coefficients /3vr±s in Eq. (9.2.1.3):

- . (9.2.1.7)

Then, on using Eqs. (9.2.1.7) and the explicit representation for the Clebsch-Gordan coefficients [30] in Eq. (9.2.1.3) (as mentioned in Chapter 7, the built-in function ClebschGordant^.m^.^.mzJ.ly'.m] of the MATHEMATICA program allows one readily to calculate these coefficients), we obtain the nonzero elements elml^m< [31]:

Z(/ + l)-3m2 tifan ll+l) (2Z-l)(2Z+3) 2a

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Chapter 9. Non-Axially Symmetric Problems 471

el,m,l-\,m ~ V V 4l2-l fy-i (l +1) iam

a t-7,0,/-l,0L7,m,/-l,-m

£r30+i) ic-m

ar

Z2-m2

4/ 2 - l

c/,m,;+i , m = ( - l ) m+1 1/(2/ + l)(2/ + 3) £y3l iam

2 a

2 „ 2

r-i.o ^i,o t-/,0,/+l,0 l-/,m,/+l,-m

^y3/ iam\ \ (/ + !) -m 2 <X , y(2/ + l)(2/ + 3)'

_2cr(l + l ) ( - l ) m I, ) ( > 2 , o 2,0 ei,m,i-2,m- , , v v z ' : 5M / ' + Ul-/,o,/-2,ot-/,m,/-2,-15

= g(/ + l) [ ( / - l ) 2 - m 2 ] [ / 2 - m 2 ]

2 / -1 ^ (2/ + l ) (2 / -3)

elMm = ( - I ) " 1 ^ f / V l ^ + l J l^ + S K ^ o C 2 ^ , -

\2 ...2 2 ...2-,

= - ( T -/ [(Z + l) - m 2 ] [ ( / + 2) - m 2 ]

e/,m,/-l,m±l

c;,m,(+l,m±l

2/ + 3^ (2/ + l)(2/ + 5)

-( iY"+1 ^7lTlYl\i \\\-J<u2 ir1'0 r w

4 l '\ 4 / 2 - l

el,m,l,m±\ - ( 1)

± ^( r .+^2)^ /(/±m + l)(/±m^2) 4 \ (2/ + l)(2Z + 3) '

m+\i4(7i+ir2) Aa

^7xTiy2) Aa

(21 + l)V(/ + m)(/±m + l)C/X,0C/0mV

A/(Z + m)(Z + m + l).

The elml'm' so obtained are in complete agreement with those derived

independently [7-10]. Equation (9.2.1.2) can now be written explicitly

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472 The Langevin Equation

at

n(n + l) iy-£m n(n + l)-3m2

2 2a (2n-l)(2n + 3) Cn,mO

[(n + 2f -m2][(n + l)2 -m2] <7(« + l) (n 2 -m 2 ) [ (n - l ) 2 -m 2 ] na . ., ... ,

2n-l i (2n-3)2n + l) C""2 'm U 2n + 3 ^ (2n + 5)(2« + l) C"+2 'mW

+ tf-a J\4n -1

I ^n imc\ I (n + 1) -m 2

2 1 C „ . 1 , m W - | ^ - + — J J ( 2 n + 3 ) ( 2 n + 1) Cn+l,mW

(n-m + l)(n-m + 2)

(2n + 3)(2n + l)

+! ^(n-m + l)(n + m)

c„,m-i(0 + (n + D (n + wi)(n + /n- l )

4 n 2 - l "-''m .,(0

(n-in) (n+m + \)(n+m + 2) n . I c,

(2n + 3)(2n + l) n+l,m+l (0

J(n + m + l)(n — m) -i-

a c«,m+i (') + (« + !) (n-m)(n-m-l)

4 „ 2 - l c"-1-m+1 (0

(9.2.1.8) According to linear response theory, the decay of the

magnetisation (M)(f) of a system of noninteracting single domain ferromagnetic particles when a small uniform external field H t parallel to H (v(MHi) / kT« 1) has been switched off at time t = 0, is given by

( M H X O - W O =ZiiHiq(t), (9.2.1.9)

where Cy(0 is the normalised relaxation function of the longitudinal

component of the magnetisation defined as

y:3^(0- ^ ( W - W Q ] |

^ c ^ W - V ^ R e K r , - / r 2 K i ( 0 ) ]

^i is the static magnetic susceptibility given by

vMsN0 \A7C Xt = ™jj>!sL ^(y3cuo(0)-^Re[(ri-ir2)cu(0)]),(9.2.l.n)

and TVo is the number of particles per unit volume. The longitudinal magnetic susceptibility Xwi®) 1S expressed in terms of CB(f) as follows:

X,ico) = xi(co)-ix;co) =zt[l-ia>Ct(ia>)], (9.2.1.12)

where

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Chapter 9. Non-Axially Symmetric Problems 473

C„ (id)) = j C„ (t)e~imdt o

V2x3q 0(i©) - (ft - 1ft ) c u (to) + (Yi + iy2 )q- i (iffl) (9.2.1.13)

^ 3 c i , o ( ° ) - ( y i - ^2 )c u (0 )+ ( r i+»y 2 K- i (0 ) Another quantity of interest is the relaxation time zj| which, as we have seen, is the area under the curve of the longitudinal autocorrelation function C«(t) so that zj| is

= Jq|(0* = ^3gi,o(Q)-(ri -^2)gu(Q)+<ri +»y2)gi,-i(Q) ^ 3 c i ,o ( ° ) - ( r i - ^ 2 Ki(0 )+ ( r i +»y2)

ci.-i(0) (9.2.1.14)

Thus, in order to study the longitudinal magnetic relaxation, we must evaluate the one-sided Fourier transforms of cx 0(t) and c, ±l(t). As was

shown by Kalmykov and Titov [10,12], this may be done using matrix continued fractions. We follow their solution below.

9.2.2 Matrix continued fraction solution

The eleven-term recurrence Eq. (9.2.1.8) can be transformed into a matrix three-term differential-recurrence relation

^ C n ( 0 = Q;C„_1(0 + QnCn(0 + Q„+Cn+1(0, ( n > l ) . (9.2.2.1)

The column vector Cn(t) and matrices Q~, Qn, Q^, included in this equation, may be set up as follows

0(0 = 0, G(f) =

-2,-2

'2,-1

(0^ (0

C2,o(0 c2l(t)

(0 (0

'2,2

-1,-1 ci,o(0 c i , i(0

C„(0 =

-In-In (0 C2n,-2n+l(0

-2n,2n (0 C2n-l , -2n+l( f)

c 2n- l , -2n+2(0

V, c2n-l,2n-l (0

(n > 1), (9.2.2.2)

and

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474

Qn =

with

A2n w2n X Q;=

The Langevin Equation

^ . Y,„ ^ __ ( V,

2n-\J

J2n

0 On -"2M-\ )

In

¥2n-l /

Q7 = ' v 2 ^ w,

(9.2.2.3)

(9.2.2.4) \)

The dimensions of the matrices Q„,Q^, and Q~ are accordingly equal

to 8nx8n, 8nx8(rc+l), and 8nx8(«- l ) . In turn, the matrices

Qn'Qn»Q« consist of submatrices. There are five distinct types of

submatrices: (

X/ =

-i-i

Xl,-M

0

0

0

n,-i

V,-l+l

0

*/,-/+!

Xl,-l+2 Xl-l+2

0

0

0

0

(..-

Y,=

?/,-/

0

0

0

0

yi.-i

yl-M

0

0

0

yi,-i

yi,-M

y 1,-1+2

0

0

w,=

w, i,-i

w, i,-i+\ w, 7-/+1

0

0

w, w. '1-1+2 wl-l+2 wl~l+2 W,

0

0

0

0

0

0

0

0

0

0

0

0

KlJ-\

xl,l

0

0

0

xu-l

(9.2.2.5)

0

0

0

0

0

0

'•' A2/+1M2/+1)

0 N

0

0

yu-\ yu-i

yjj yu

0

0

0

yh

0 N

0

0

,(9.2.2.6)

/(2/+l)x(2/+3)

W, '1,1-2 wl,l-2 W,

W, '1,1-1 W, 1,1-1

W, 1,1

(9.2.2.7)

/(2/+l)x(2/-l)

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Chapter 9. Non-Axially Symmetric Problems 475

(0 0 zZH

0 0 0

0 0

0 0

v,=

0

0

0

0

"1-1+2

0

0

0

0

0

0

Zl,-M

0

0

0

0

0

0

0

0

0

0

0

zl,l-l

0

0

0

0

0

0 0 0^

0 0 0

vU-\

0 0 0

zu 0 0

0 ^

0

0

0

0

(9.2.2.8)

/(2/+l)x(2/+5)

0

0

0

V/,/-2

0

0

(9.2.2.9)

'(2/+l)x(2/-3)

(The dimensions of the submatrices are indicated by subscripts). The elements of the submatrices X,,Y,,Zl,\l,Wl are composed from the ei,m,i',m a n d ^ g i v e n bY

_a(nn + l)-3m2) n(n + \) m ^ --i-

2a n'm ( 2n - l ) (2n + 3) :

<m = -fc-n, )* = -i^7\'Yl) 4(n + m + \)(n-m), 4a

, - n .am

a )

n + \f-m2

|(2n + l)(2n + 3)

+ / - \* ^^(n-iri) kn + m + \)(n + m + 2) v ' 4 y (2n + l)(2n + 3)

w, l,m - n+\ .amN

-m l(2n + l)(2/i-l)

w„ - ( < - m ) = ^(w + l ) (y i - iy 2 ) Un-m)n-m^r)

4 \l (2n + l ) (2n- l )

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476 The Langevin Equation

an [(n + 2)-m2][(n + l) -m2]

2« + 3V (2n + l)(2n + 5)

v = on + \) l[n2-m2][(n-l) -m2]

2 n - l M (2n + l)(2n-3)

On applying the general method of solution of the matrix three term differential-recurrence Eq. (9.2.2.1), suggested in Chapter 2, Section 2.7.3, we again obtain the exact solution for the Laplace transform C^s) in terms of matrix continued fractions:

Cyis) = TN [TNSI-Q1 -QtS^s)]1

\ °° T n r - i -x Q(0)+Z noTL^^-Ofc+i-^iS^^)]

(9.2.2.10) C„(0) ,

where I is the identity matrix, and the matrix continued fraction Sn(s) is defined as

Sn(5) = [ V I - Q „ - Q f l+ S n + 1 ( . ) J 1 Q ; . (9.2.2.11)

The initial condition vectors C„(0) appearing in Eq. (9.2.2.10) may also be evaluated in terms of matrix continued fractions. The initial values c„,m(0) in C„(0) [see Eq. (9.2.2.2)] are

27t7t

\ j y n i m (0 ,p )W o ( t f , 0 ef i t a r a *™^™'^* c ™ , ' ) sinMM<p

c„,ra(0) = - 1 1 In it \\w0i,(p)eiAr^9^+r^n^na+y^ahmiMTM<p

\ «.«/o

Yi + iYi

I 2 (n-m + 2)(n-m + l) \n + m)(n + m-\) /v

(2n + 3)(2n + l) In - l)(2n +1)

(2«-l)(2n + l)

( n - m ) ( n - m - l ) _ \(n + m + 2)(n + m + l) /v \ rn-l,m+l/0 \\ ,„__ , 1 u n . . , 1N v»+ l ,» t | / 0 (2« + 3)(2n + l)

+ 73 (n-m + \)n + m + \) \ n2-m2

lv \1n+l,m/0 + \l~ ,v/„__ . ^Vn-l .m/O

(2n + 3)(2n + l) (2n-l)(2n + l)

MYl0)0-yf2Re(ri-i72)(YM , (9.2.2.12)

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Chapter 9. Non-Axially Symmetric Problems All

where £, = vMsHl l(kT). In order to evaluate the equilibrium averages

(Ynm) , we shall transform Eq. (9.2.1.4) to the matrix recurrence

relation:

Q;Rn_, +QnR„ + Q X + 1 = 0 . in>\), (9.2.2.13)

where the matrices Q„, Q*, and Q~ are defined by Eq. (9.2.2.3) and

(Y2n,-2n0

V2n,-2n+l/0

R = with R0 = - = V4;r

(9.2.2.14) \Y2n,2n0

y2n-l,-2n+l ) 0

\Y2n-\,-2n+2/0

\Y2n-l,2n-l0 ,

The solution of Eq. (9.2.2.13) may be expressed in terms of matrix continued fractions as

R„ =S„(0)R„_1 =Sn(0)Sn_1(0).. .S2(0)S1(0)^L, (9.2.2.15) An

where S„(0) is defined by Eq. (9.2.2.11). Noting Eq. (9.2.2.12), the

initial condition vectors C„ (0) are given by:

C„(0) = ^KnR„_1+KnRn+K f lw

+1Rn+I, ( „ > l ) , (9.2.2.16)

where ' 0 (^

Kn = f*2n D2n ^

H \J*2n K2n-\)

, K_ VD2n-l 0 ,

with

* > = D

(9.2.2.17)

(9.2.2.18) \j

(the superscript H denotes the Hermitian conjugate, i.e., the transposition and the complex conjugate). The matrices K n ,K n consist of two types of submatrices:

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478 The Langevin Equation

D,=

dt-i dl,->

[An

T 0

Fi=-J^-RiMYuo)0->f2(n-ir2)Yu)0]i,

l+\ dl,-l+\

0

0

dl,-l+2 dl,-l+2 dl,-l+2

0

0

0

0

0

0

*l,-l+3

0

0

0

"1,-1+3 " / , - 1+3

0

0

0

0

0

0

0

0

\

0

0

0

0

0

0 0

"1,1-2

dl,l-\

d; ''•' /(2/+l)x(2/-l)

where the matrix elements of D; are given by

,*_ (fl+'ft) \(n+m-l)(n + mj dn,m - Yz\

•m dn,-m)

4n-l Noting Eq. (9.2.2.15), Eq. (9.2.2.16) yields

6

4 n 2 - l

Cn(0) = -^[Kn+[Kn+K„w+]Sn+1(0)]Sn(0)]s„_1(0)...S1(0) (9.2.2.19)

with

,-A C,(0) = - p = K , +[K, +K£s2(0)]Si(0)) . (9.2.2.20)

The exact matrix continued fraction solution [Eq. (9.2.2.10)] we have obtained is useful for computation. All the matrix continued fractions and series involved converge very rapidly, thus 10-20 downward iterations in calculating the continued fractions and 10-15 terms in the series (9.2.2.10) are enough to arrive at an accuracy of not less than 6 significant digits in the majority of cases of interest (the number of the iterations and the number of the terms must be increased with increasing a and/or with decreasing a). The greatest dimension of all the matrices involved is of the order 102 allowing one to perform the calculations on a personal computer.

9.2.3 Smallest nonvanishing eigenvalue, the relaxation time, and the complex susceptibility

Having obtained the matrix continued fraction solution for (^(s), Eq. (9.2.2.10), we can calculate the relaxation time zj. Eq. (9.2.1.14) and the

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Chapter 9. Non-Axially Symmetric Problems 479

longitudinal dynamic susceptibility ZH(CO) , Eq. (9.2.1.12). We first study the relaxation time zj| and the associated time constant: the inverse of the smallest nonvanishing eigenvalue. We may suppose without significant loss of generality that the spherical polar coordinates of u, n, and h are ( # <p), (0,0), and iff,0), respectively, that is, the field is in the x-z plane or confined to a single longitude. Thus, the potential of Eq. (9.2.1.1) is of the form

/ ^ = -<7cos2t?-£cos^cosz?-£sin^'sinz?cos^ + const, (9.2.3.1)

and the direction cosines % are given by

7i = sin yr, y2 = 0, y3 = cos yr. (9.2.3.2)

The results of the calculation of the relaxation time rn from the matrix continued fraction solution are shown in Fig. 9.2.3.1. It is evident from Fig. 9.2.3.1 that 7j| has a deep minimum at y/-nll (the curve is symmetrical about this line), because the alternating field Hi is perpendicular to the easy axis of the particle, corresponding to the condition for the observation of ferromagnetic resonance, which is determined by the transverse component of the dynamic susceptibility of the particle. Accordingly, the low-frequency (activation) mode associated with the reorientation of the magnetisation over the potential barrier does not contribute to the relaxation time. As the strength of the static magnetic field H is increased, the activation process becomes increasingly suppressed. Moreover, this process also tends to diminish the contribution of the low-frequency relaxation mode to \.

0 1 2 3

% rad

Fig. 9.2.3.1. log10 (f|| ITN ) vs. y/for a= 100 (high damping), <r= 10, and various values

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480 The Langevin Equation

We can also calculate the smallest nonvanishing eigenvalue k\ (as has been described in Refs. [8,9]) if we arrange the hierarchy of differential-recurrence Eq. (9.2.1.8) in the matrix form

X(f) = AX(f), (9.2.3.3) where

X =

'1,0

' i , i

'2,0

'2,1

'2,2

(9.2.3.4)

and A is the system matrix with elements determined by Eq. (9.2.1.8). The column vector X does not include qm for negative m because they may be calculated using the symmetry Eq. (7.2.19) for m < 0 . The eigenvalue X\ is as usual the smallest nonvanishing root of the characteristic equation

de t (Al -A) = 0. (9.2.3.5) On the other hand, the smallest nonvanishing eigenvalue k\ may also be evaluated using matrix continued fractions. An equation for X\ was derived in Chapter 2, Section 2.11.2, Eq. (2.11.2.13), which for the present problem reads [32]

det[S]

Sp[D] where the matrix S is defined as

S = -[Q!+Qrs2(0)]

.(0)]

(9.2.3.6)

(9.2.3.7)

-k+\ ~Qn-k+l^n-k+2y' n=2m=\ k=\

and the matrix elements of D are the minors of the matrix S. In the calculation of k\ given below, we shall use Eq. (9.2.3.6).

The calculation of asymptotic estimates of k\ for the oblique field problem is well documented [7,33]. This calculation requires the analysis of the Fokker-Planck equation for the density of orientations of M which is [17] (see the detailed derivation in Chapter 1, Section 1.17)

2rN-W = f, a 'u a a — V x — W du du

+ -du f a ^ w—v

3u + AW. (9.2.3.8)

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Chapter 9. Non-Axially Symmetric Problems 481

The term in a'1 in Eq. (9.2.3.8) is the precessional or gyromagnetic term which gives rise to ferromagnetic resonance (FMR) at high frequencies. If the potential is axially symmetric (HII n ) , then this term does not contribute to the longitudinal relaxation time. On the other hand, if H is not parallel to n as in the present problem, these terms may be neglected only if the phenomenological damping coefficient a »1. In other cases, the range of values of the damping factor a, for which a particular escape rate formula is valid, must be taken into account (see Chapter 1) just as in the conventional Kramers theory [34,35] of the escape of particles over potential barriers.

We consider first the detailed calculation of the behaviour of the potential energy as a function of the applied field H. We rewrite Eq. (9.2.3.1) in the form

^ M f ) = f / = s i n 2 ^ _ 2 / ? ( c o s ^ c o s t ? + s i n ^ s i n ^ c o s ^ ) , (9.2.3.9)

where h = %l(2a). The stationary points occur for <p=0 and q)-n (0 < •&< K, 0< (p< 27t). The stationary point for cp- # corresponds to a maximum of Eq. (9.2.3.9) and so is of no interest for us. The stationary points for (p - 0 correspond to a saddle point of Eq. (9.2.3.9) at t\ and minima at T3\ and ^ for h< hc, where hc is some critical value of h at which the potential (9.2.3.9) loses its bistable character (to be determined below). The saddle point is generally in the equatorial region, while ^ and t% lie in the north and south polar regions, respectively. The two equilibrium directions of the magnetisation and their associated polar angles ^ and ^ lie in the x-z plane (#>= 0) and are determined by the conditions for a minimum of U($,0), namely [7]

^ - = 0, Pi>0. (9.2.3.10)

The position of the saddle point follows from the conditions for a maximum of £/(#0), namely

W0, pJ<0. (9.2.3.11)

The critical value hc follows from the condition for a double root (point of inflexion) of f/(t?,0) namely

^LL = ^L = 0. (9.2.3.12) d# 8z?2

Equation (9.2.3.12) now yields sin2z? = -2/*csin(z?-^) and cos2z? = -ft ccos(tf-^), (9.2.3.13)

that is,

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482 The Langevin Equation

tan2tf = 2 tan( t f -^) . (9.2.3.14)

The only real root of this trigonometric equation in the range t?(0, 7t) is

tantf = -(tan^)1 / 3 (9.2.3.15) so that [using Eq. (9.2.3.15)]

. sintfcos*? / 2/3 • 2/3 r 3 / 2 tc\^i\c\ h=—:— = cos ' V + s n r ' V . (9.2.3.16)

s in^cos^-cos^s iny v ' Equation (9.2.3.16) may also be written in terms of tan yas

/ic = (l + t a n V ) 1 / 2 [ l + ( t an^ ) 2 / 3 ]~ 3 / \ (9.2.3.17) It should be noted that in the calculations of Stoner and Wohlfarth [2] and that of Pfeiffer [36], the external field axis is taken as the polar axis. Thus in their calculations the quantities &- \jfwsA e?in Eq. (9.2.3.14) are interchanged. In their coordinate system, Eq. (9.2.3.14) would read

tan2(t?-^) = 2tantf. (9.2.3.18) Our choice of angles has the merit that it is much easier to derive the set of differential-recurrence relations for the Qm using such a coordinate system.

For the purpose of the discussion of X\, we are interested in the behaviour of the potential V&, <p) in the x-z plane, <p= 0, where V$, (p) = V( &, 0) and in the range 0 < $ < nsince (p= 0 contains a saddle point of V(# (p), Fig. 9.2.3.2. In general, the potential (9.2.3.9) retains its asymmetric bistable form for 0 < h < hc and y/± 7tl 2, i.e., the potential V( $, 0) has two minima separated by a maximum [which corresponds to a saddle point of V(&,<p)] and two potential barriers B\ and B2. If h > hc, the two minima structure of the potential disappears and V( # 0) has a simple maximum. For h = hc, the second minimum becomes a point of inflexion. The two minima have in general different energies so that the energy barriers are not equal and only part of the magnetic moment relaxes. For \f/± 0, it is not simple to evaluate the barrier heights

S 1 =/ / [V( i i | ) 1 0) -V(^0) ] and B2 = £[v(i%,0)-V(t»j,0)]

as a function of h and ^from the Stoner-Wohlfarth analysis [2] given above. However, for particular values of y/, e.g., y/= 0, we easily find that (see Fig. 1.18.1 in Chapter 1)

J31=<T(1 + hf, B2=a(\-hf (9.2.3.19)

and the potential has an asymmetric bistable form. For \j/- nl 2, the barriers B\ and B2 coincide so that

Bl=B1=CJ(\-h)2 (9.2.3.20)

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Chapter 9. Non-Axially Symmetric Problems 483

«4.

0.0-

-0.5-

-1.0-

-1.5-

• S " ^ > v

• / "3 • \ • / - - " " " " • " - - • ^ v

• / - ' • * - „ • \ ^

• / ' " - .• / ,-; % ^— . * / . - > • ' 4 - • • , • . • - - -

• / . ' " ' • • / • ' ' •

" • • " * ••-'"// 1 - ^ = 0 , fc = 0 '"•••.... '*•.•.••"

/ - ' 2-|y=0, fc = 0.25 . . _ . . / 3-y=x/4, A = 0.25 J ^ 4 - |/= nl 2, /i = 0.25

i / 2 1

Figure 9.2.3.2. Variation with polar angle z? of the reduced potential function U (•&, 0) = p V (•&, 0) / (T in the JC-Z plane for various values of y/ the angle between the external field and the easy axis: filled circles (1) are h = 0, solid line (2) y/= 0, dashed line (3) y/= K/4, and dotted line (4) i//= n/2 (the last three curves are drawn for h = 0.25).

and the potential assumes a symmetric bistable form reminiscent of that for h = 0. For other values of y/, Pfeiffer [36] has shown that the lower barrier height Z?2 (which in certain cases determines almost entirely the relaxation process) is given by the approximate expression

B2 = a[\-hlhc(y,)f%MMh<(¥\ (9.2.3.21) where hc is to be calculated from Eq. (9.2.3.17).

In order to evaluate X\ in the intermediate to high damping (IHD) limit, i.e., for a> 1, it is supposed [7] (see Chapter 1, Section 1.8.2) that the free energy per unit volume V(M) has a bistable structure with minima at ni and n2 separated by a potential barrier that contains a saddle point at no (see Fig. 9.2.3.3). If ( a f 0 , ^ 0 , ^ 0 ) denote the direction cosines of M and M is close to the stationary point n, of the potential, then V(M) can be approximated to second order in or' as

V=V:+-1 2

'V'NW') (9.2.3.22)

Upon substituting Eq. (9.2.3.22) into the Fokker-Planck equation (9.2.3.8), the latter may be solved in the vicinity of the saddle point to yield [7,33]

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484 The Langevin Equation

Fig. 9.2.3.3. 3D plot of the oblique field potential, Eq. (9.2.3.9), for h = 0.25.

IHD

where

< f

K

JUJX)

a IXCOQ

(9.2.3.23)

<*€ y2

-(2).(2) «o=- ,(ou<>) qwc2

u; (9.2.3.24) , s i „ 5 MS

are the squares of the well and saddle angular frequencies, respectively, and the (over) damped saddle angular frequency Q0 is

a 0 = fi 4T„

-cj0) - 40) + ^ - c ^ W ^ c f (9.2.3.25)

Equation (9.2.3.23) is Kramers' IHD formula for the escape rate [34]

[see Chapter, Section 1.13, Eq. (1.13.4)]. Equations for c[°, 4 ° , and Vt

were derived in Refs. [7,33] and are given by

Vi = K [s in2$ - 2h cos (&t - y/)\,

W=2K[ cos &+hcos A~¥)\

4 ° = IK [cos2 i5>. - sin2 ^ + Acos(i5>. - yr)],

where the $ are the solutions of the trigonometric equation sin2*? = 2/isin(^-??).

The latter equation may be written as the quartic equation [7]: (x + hcosys)2(l-x2) = (xhsmi/f)2 (x = costf).

A very low damping (VLD) asymptotic formula for Xx in the energy diffusion controlled [34] limit, i.e., for a« 1, was derived by Klik and Gunther [37] (see Chapter 1, Section 1.13.2). Their formula applied to the present problem yields

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Chapter 9. Non-Axially Symmetric Problems 485

Jv=vn (l-cos2z?)- -Vd(p-- ^—Vdcostf (9.2.3.27)

3">>^£g-/»(Vo-v.) (9.2.3.26) KkT

where JL d_

dco&t"'T \-cos2dd(p is the energy loss per cycle of the almost periodic motion at the saddle point energy v V0. Here, instead of the numerical evaluation of the integral in Eq. (9.2.3.27) (which is of the order of the barrier height [37]), we commonly use the approximation AE ~ ccv\V0\.

The results of the calculation of the smallest nonvanishing X\ from Eqs. (9.2.3.6) and (9.2.3.7) are presented in Figs. 9.2.3.4 and 9.2.3.5, where they are compared with T^1 from Eq. (9.2.1.14) and with the IHD and VLD asymptotic estimates for X\. In the IHD limit, the lowest eigenvalue X\ from Eq. (9.2.3.6) is in good agreement with the asymptotic solution X[HD [Eq. (9.2.3.23)] at high cr(Fig. 9.2.3.4). Just as in the uniaxial problem, for h < hc, X\ and Tjj"1 are very close to each other for all barrier heights. However, for h > hc, the depletion effect appears, and X\ and r^1 diverge exponentially. In Fig. 9.2.3.4, X^° calculated from the asymptotic Eq. (9.2.3.26) is also presented. As one can see in Fig. 9.2.3.5, in contrast to the biased uniaxial potential (Chapter 8, Section 8.3) X\ for the oblique field problem strongly depends on the damping parameter a. Evidently, X\ is in good agreement with the asymptotic estimates for both the IHD and VLD limits. Here, the VLD limit corresponds to values of a< 0.001 that are in agreement with the independent calculation of Ref. [38]. However, for crossover values of or (about a ~ 0.05) neither the IHD formula (9.2.3.23) nor the VLD Eq. (9.2.3.26) yield reliable quantitative estimates. Hence, a more detailed analysis is necessary [38] in order to obtain asymptotic formulas.

We have calculated the complex magnetic susceptibility Z\\(°% by using a matrix continued fraction solution, viz., Eqs. (9.2.1.12) and (9.2.2.10). Re[^|(6J)] and -Im[j||(<i))] vs. corN are shown in Figs.

9.2.3.6 and 9.2.3.7 for a wide range of frequencies, bias field strength, and

damping (the calculations were carried out for vj3MJNQ = 1). The results

indicate that a marked dependence of Z\\(a% on a exists and that three distinct dispersion bands appear in the spectrum.

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486 The Langevin Equation

0 5 10 15 20

Fig. 9.2.3.4. X\ [Eq. (9.2.3.6) - solid lines] for the oblique field potential as a function of the barrier height a compared with the asymptotic IHD solution /l/™ [Eq. (9.2.3.23)] (stars and crosses) and the solution rendered by the inverse of the correlation time fjj"1

[Eq. (9.2.1.14)] (diamonds and filled circles).

Fig. 9.2.3.5. A, [Eq. (9.2.3.6) - curves 1-4] for the oblique field potential at h = 0.25 as a function of the barrier height cfor various values of the damping parameter or compared with the asymptotic IHD solution A("D [Eq. (9.2.3.23)] (stars) and the asymptotic VLD solution %° [Eq. (9.2.3.26)] (filled circles).

The characteristic frequency and the half-width of the low-frequency relaxation band (LRB) are determined by the characteristic frequency coob~\ I rn of the overbarrier relaxation mode. As a decreases, this peak shifts to higher frequencies and reaches its limiting value A^°. In addition, a far weaker second relaxation peak appears at high frequencies. This high frequency relaxation band (HRB) is due to the intrawell modes (for y/= 0 and c r » 1, the characteristic frequency of this band is 2a(\ + h)/TN [40]). The third FMR peak due to excitation of transverse modes having frequencies close to the precession frequency cq,r

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Chapter 9. Non-Axially Symmetric Problems 487

of the magnetisation appears only at low damping and strongly manifests itself in the high frequency region. As a decreases, the FMR peak shifts to higher frequencies since (Opr~ 1 / (ccrN). Moreover, at y/"=0 or 1//= n, the FMR peak disappears because the transverse modes no longer partake in the relaxation process. The dependence of the linear response on the bias field strength is illustrated in Fig. 9.2.3.7. Here, the effect of the depletion [41,40] (Chapter 1, Section 1.20, Chapter 8, Section 8.3.2) of the shallower of the two potential wells of a bistable potential (9.2.3.9) by a bias field is apparent: at fields above the critical field hc at which the depletion occurs, it is possible to make the low frequency peak disappear altogether (curves 3 and 3').

o.io-

£ °0 5-

o.oo-

\ \ i

\ ' ' * •

\\\ 1 - a = 1 >j\ '*> 2 - a =0.1 V \ \

4 - a = 0.001

a = 10 h =0.1 v =n/4

1 * 2/j j \

K r i 1 V

J L

10" 1 0 ' 10" 10 ioJ

10

3, 10"

10

LRB

f/f 1 -a = 1 * /,'/ 2-a =0.1 •Y 3-a=0.01

Y 4 - a = 0.001

, a = 10 h =0.1 V

\"-"5

= 7114

FMR

/ / \ i • ! • HRB. ' \ \ : ;

W 10 10 10 10 10

Fig. 9.2.3.6. Re[£(<»)] and -lm[x(co)] vs. OTN from the IHD (« = 1) to VLD (a = 0.001) limits for a= 10, h = 0.1, and i//= nl 4. Curves 1-4: numerical calculations based on the exact matrix continued fraction solution Eq. (9.2.2.10). Stars and filled circles: Eq. (9.2.3.28) with iSxhf =0.023 and z = 1/A,™ and r = \lX^°, respectively.

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488 The Langevin Equation

10 T 1,1' -h =0.1 LRB 2,2' -h =0.17 3, 3' -h =0.40

1,2,3 - a = 1.00 l',2', 3 ' - a =0.01

Fig. 9.2.3.7. Relxico) and -lm[z(eo)] vs. COTN for <7= 10, ^ = ^ / 4 , a= 1.0 (IHD: solid lines 1, 2, and 3) and a= 0.01 (low damping: dashed-dotted lines 1', 2', and 3'). Lines 1, 1' (/i = 0.01); 2, 2' (/i = 0.17); and 3, 3' (/i = 0.4) are numerical calculations based on the exact matrix continued fraction solution Eq. (9.2.2.10). Stars and filled circles: Eq. (9.2.3.28) with T = 1/AJHD and T = l/Al

LD, respectively.

Such behaviour of Zui®) implies that if one is interested solely in the low frequency (OVN < 1) part of X\i°^ (where the effect of the high frequency modes may be completely ignored so that the relaxation of the magnetisation at long times may be approximated by a single exponential with the characteristic time x ~ A[x), then the Debye-like formula, viz.,

Z,(o» = *f^ + AZhf (9.2.3.28)

yields an accurate description of the low frequency part of the spectra (see Figs. 9.2.3.6 and 9.2.3.7). Here T is given by r = \l^HD and

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Chapter 9. Non-Axially Symmetric Problems 489

7 = 1/4 i n t h e MD a n d V L D limits, respectively, X\\=Z\\(®)is the static susceptibility, and AXy is the contribution of the high-frequency transverse and longitudinal modes. The values of Xw a nd A/fo/ depend on £ y/, and cr and can be measured experimentally, calculated numerically, and/or estimated theoretically (an example of such theoretical estimations of Xw a nd &Zhf f° r ¥= 0 n a s been given by Garanin [40]). Our calculations indicate that Eq. (9.2.3.28) yields an adequate description of the low-frequency part of the spectra for a> 3.

It follows that the nonaxial symmetry causes the various damping regimes (IHD and VLD) of the Kramers problem to appear unlike in an axially symmetric potential, where the formula for t\\ is valid for all a. We remark that the intrinsic or dependence of Z\\(°f) for the oblique field configuration serves as a signature of the coupling between the longitudinal and precessional modes of the magnetisation. Hence, it should be possible to determine the evasive damping coefficient from measurements of linear and nonlinear response characteristics [39,42,43], e.g., by fitting the theory to the experimental dependence of X\\i°^ o n the angle ^and the bias field strength H, so that the sole fitting parameter is a, which can be determined at different T, yielding its temperature dependence. This is of importance because of its implications in the search for other mechanisms of magnetisation reversal of M (e.g., macroscopic quantum tunneling [46,47]) as a knowledge of a and its T dependence allows the separation of the various relaxation mechanisms.

Here, we have treated the "oblique field problem" in the context of its application to the magnetic relaxation of superparamagnetic particles. Very similar problems arise in the evaluation of the nonlinear transient dielectric and Kerr effect responses of polar and polarisable particles (macro-molecules) diluted in a nonpolar solvent to a sudden change both in magnitude and in direction of a strong external dc electric field [22,23]. By averaging the underlying Langevin equation, the infinite hierarchy of differential-recurrence equations for ensemble averages of the spherical harmonics can be derived [22,23] for an assembly of polar and anisotropically polarisable molecules. The hierarchy now describes the noninertial rotational Brownian motion in high electric fields. From a mathematical viewpoint, the hierarchy is completely equivalent to that considered here in the high damping limit. Thus, the calculations can be accomplished [22,23] using the matrix continued fraction method, which allows us to give the exact solution for the first- and second-order transient responses of the ensemble averages of the spherical harmonics (relaxation functions).

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490 The Langevin Equation

9.3 Cubic Anisotropy

9.3.1 Recurrence equations

Here, we shall consider the magnetic relaxation of single domain ferromagnetic particles with cubic magnetic anisotropy. As demonstrated above, the problem may be reduced to the solution of an infinite system of linear differential-recurrence relations for the averaged spherical harmonics (moments), which may be arranged in matrix form:

X(?) = AX(0 (9.3.1.1) where X(t) is the column vector consisting of the moments and A is the infinite system matrix. The numerical solution of Eq. (9.3.1.1) (eigenvalues and eigenvectors of the system matrix) may be obtained by consecutive increase of the number of the equations N until convergence is attained. Such an approach to cubic anisotropy has been used, for example, by Aharoni and Eisenstein in the context of the Fokker-Planck equation approach in Refs. [15,16], where several low order eigenvalues were evaluated numerically in the intermediate-to-high damping limit. Unfortunately, the application of this direct matrix approach to the present problem is inconvenient especially for low damping as it is necessary to carry out calculations for very large dimensions of the system matrix ~ 104-105, and convergence of the solution is consequently difficult to achieve. On account of the difficulties encountered in the previous numerical analysis which we have mentioned, the cubic anisotropy is treated here using the matrix continued fraction method for the solution of infinite systems of recurrence relations. Such a method was applied by us to uniaxial particles in a strong magnetic field in Section 9.2. Here, the method is extended to obtain the longitudinal magnetic susceptibility and the relaxation time tu of a system of noninteracting single domain particles having cubic anisotropy.

The free energy of unit volume of a particle possessing cubic magnetic anisotropy is [14,17]

/3V = a (sin4 tfsin2 2(p + sin2 2*?)

8a r v ACT fitter v V - I t (9.3.1.2) = ~~[5 4'° ~ T 5 \ ~ L 7 4 - 4 +F4,-4j + const,

where cr = fiK/4 is the dimensionless anisotropy parameter, K is the anisotropy constant, which may have either positive or negative values. For K > 0, the potential (9.3.1.2) has 6 minima, 8 maxima and 12 saddle points [17]. If K < 0, the maxima and minima are interchanged, see Fig. 9.3.1.1.

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Chapter 9. Non-Axially Symmetric Problems 491

Fig. 9.3.1.1. Cubic anisotropy potential for K >0 (a) and K < 0 (b).

For cubic anisotropy potentials, the moment system for the equilibrium correlation functions clm(t) = (cos&(0)Ylm(t)) governing the kinetics of the magnetisation is

1 4

V / , m W = Z Z < W , ^ / + r , m + 4 . v ( 0 . ( 9 . 3 . 1 . 3 ) ,s=-l r=-4

where the coefficients et m r m< are given by

9(/2-l)[(Z + l)2-l]-15m2[6/(Z + l ) - 5 - 7 m 2 ] /(/ + !)

(4/2_9)[4(; + l ) 2 _9] 2

e;,m,/,m±4 = ( - l ) 4J—"—-(2/ + l)C;o;0C/lm,/i-mT4

_15a>/(ZTm)(;±/n + 4)[ ;2-(m±l)2][ /2-(TO±2)2][Z2-(m±3)2]

2(4/2-9)[4(/ + l) - 9 ]

cl,m,l-2,m = (-If ^ V ( 2 / + 1)(2/-3)(2/+9)Cft»_2i0caJ>l_2i_

o-(2/ + 9 ) ( / 2 - / - 2 - 7 m 2 ) ( / 2 -m 2 ) [ ( / - l ) - m 2 ]

(2/-5)(2Z-l)(2/ + 3) (2/-3)(2/ + l)

8(7 ^,,-4,m = ("If ^ V(2/ +1) (2/ -7) (/ +1) C ^ - 4 , o C 4,-m

7<T(Z + 1)

(2Z-5)(2 / -3) (2 / - l ) H

l[(l-3) -m2][(l-2) -m2][(l-l) -m2][l2-m2]

(2 / -7)(2 / + l)

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492 The Langevin Equation

5 a e,,m,i-2,m±4 = ( - I f 2 ^ ^ ( 2 l + l)(2l-3)(2l + 9)CtZ_2fiCf:^_

a(2l + 9)

m+4

2 (21-5) (21-l) (21+ 3)

(l + m-5)(l + m-4)(l + m-3)(l + m)[l2-(m±2) ][l2-(m±l) ]

(2 / -3 ) (2 i + l)

e,,m,,-4,m±4 = (-1)"1 ^ ^ V ( 2 / + l ) ( 2 / -7 ) ( / +1) C* 5° ;-4,o«. •4,-m+4

<J(/ + I )

2 ( 2 / - 5 ) ( 2 / - 3 ) ( 2 / - l )

2iom

(l + m-1)1 + m-6)...(l + m-1)(/ + m) (2 / -7) (2 / + l)

c/,m,/-l,m 5«r

/•_i m J 4 / 2 _ 1 Tr1-0 r1-0 + r 3 ' 0 r3-0 1

3iam(3l2 - 5 - 7 m 2 ) / 2 - m 2

a(4l2-9) ^4l2-l

W l , » ± 4 = ± ^ ( " 1 ) m V ( 4 ' 2 - 1 ) ( ; ± ' " + 3 ) ( ; + ' » - 4 ) C /X-1 ,0«- 1 , - m T3

2 n r ; 2

= +-3iff (l + m)(l + m-4)[l2-(m±lY][l2-(m±2Y][l2-(m±3Y]

2a(4l2-9) 4 / 2 - l

e/,m,/-3,m Horn

5a (-irV4(/-l)2-9C3

bV3,o^m0

;_ 3,-m

7icrm ( / 2 - m 2 ) [ ( / - l ) 2 - m 2 ] [ ( / - 2 ) 2 - m 2 ]

a [ 4 ( / - l ) 2 - l ]

2i<7

4(Z- l ) 2 -9

e7,m,/-3,m±4

= +" Zc |(Z + m-6)(Z + m-5)...(/ + m-l)(Z+m)(Z±m + l)

2a[4(Z-l) - 1 ] ^

2 / - 7 el,m,l+2,m ~ 9 / , 1 T el+2,m,l,m ' e/,m,/+2,m±4 97 i 1 -J e'+2,m±4,/,m '

4 ( Z - l ) 2 - 9

21-1

I I cl,m,l+4,m ~ , c el+4,m,l,m ' el,m,l+A,m , c el+4,m±4,l,m ' el,m,l+l,m el+l,m,I,m '

c/,m,/+l,m: ±4 ~~ el+\,m±4,l,m ' el,m,l+3,m ~ el+3,m,l,m ' e/,m,/+3,m±4 el+3,m±4,l

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Chapter 9. Non-Axially Symmetric Problems 493

In order to obtain the coefficients elml-m>, we have used Eq. (9.2.1.3),

where the nonzero coefficients vr, are "r,s •

2K r K 10* K 10* -4,0 = " ^ V * . V4,4 = - _ ^ _ , V „ = — ^ — ,

Eq. (9.3.1.2), and the explicit representation for the Clebsch-Gordan coefficients [30]. The elml>m- so obtained are in agreement with those derived independently in Refs. [7,18,19].

9.3.2 Matrix continued fraction solution

Before proceeding, we first summarise the principal results of linear response theory in the context of its application to superparamagnetic particles with cubic anisotropy. According to this theory, the decay of the longitudinal component of the magnetisation (M^)(t) of a system of noninteracting single domain ferromagnetic particles with cubic anisotropy, when a small uniform external field Hi (v(MH 1 ) /&r« 1) applied along the z axis (which is at the easy axis of the particle for K > 0) has been switched off at time t = 0, is

<Af||>(0 = ^ i C | | ( 0 , where the longitudinal static magnetic susceptibility X\\ is

Os&LfcClo0) = ^ U o s ^ ) = ^ ^ l (9.3.2.1) 11 kT V 3 l'° kT \ /o 3kT

and the normalised autocorrelation function C\\(t) of the longitudinal component of the magnetisation of the particle is

^ ( 0 = ^ ( 0 / ^ 0 ( 0 ) . (9.3.2.2)

The longitudinal complex magnetic susceptibility %\\o^ is given by

Za((0) = X[\(0) - ixG>) = X\\ [l - toC||(i©)]. (9.3.2.3)

Another quantity of interest is again the integral relaxation (or correlation) time % which is defined as the area under Cn(t):

T„ = Cu(0). (9.3.2.4)

We shall calculate T\\ and Ziii0*) m terms of matrix continued fractions. Following [18,19], we introduce a column vector Cn(t):

C„(0 =

f c4B(0 N

C 4„- l (0 C 4 „ - 2 ( 0

VC4n-3(0y

(9.3.2.5)

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494 The Langevin Equation

This vector consists of 4 column subvectors c4nH(r) which are given by:

c4n-(,-4(n-l+<S;0)

L4n-i (0 = "4n-i',-4(n-2+£l0) (0

, (i = 0,l,2,3). (9.3.2.6)

CAn-i,A(n-l+Si0) (0

The vector Cn(t) has 8n-2 elements. Thus, Eq. (9.3.1.3) can be transformed into the matrix three-term differential-recurrence relation

^ C n ( 0 = Q;Cn_,(0 + QnC„(0 + Q;Cn + 1(0, (n> 1), (9.3.2.7) with

C0(0 = 0and C,(0 =

'4,-4 (0 c4,o(0

(0 (0

c2Q(t)

"4,4

C3,0

v Cl,0(0 )

The matrices Q„, Q^, Q~ in Eq. (9.3.2.7) are given by

' J

Q; =

'An 0 D 4n-l

M n - 2

V B 4n-3

•4n-l

Q„ = B

An T An

JAn™An

D

D4„-2

M n - 3

B 4 «

A

B 4 n - 1

0 0

Un-2 D 4n-3

0 0 0

J4n -3 /

An-l

T

Q+n =

&An+A* 4«+4

0

0

0

An JAn-l An-l

T uAn+3

#4n+3 J Ai

0

0

lAn

An-l

An-2

T An-2

T

D4„ A

B

/4«+2Mn+2

L4n-1

4n-2

An-3J

T B

D4

&An+2 J 4n+2

0

4n+3

J 4n+3 ^4(1+2

T

/4n+lMn+l

D An+l

§An+l^' An+lJ

where the index T means the transposition,

(9.3.2.8)

(9.3.2.9)

(9.3.2.10)

(9.3.2.11)

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Chapter 9. Non-Axially Symmetric Problems 495

2 w - l l n - 4 (9.3.2.12) 2n + 9 n + 1

The dimensions of Q„,Q^,Q; are (8n-2)x(8n-2), (8ra-2)x(8n+6) and

(8n-2)x(8n-10), respectively. The exception is J 4

Qr = 0

0 (9.3.2.13)

which degenerates to a column vector of dimension 6. The matrices Q„,Q^,Q^ comprises the three-diagonal submatrices A, B, D, J, and P. There are two kinds of submatrices in Eqs. (9.3.2.9)-(9.3.2.11). The submatrices A4„, A4„_!, A4„_2, A4„_3, B4„_i, B4„_2, B4„_3, D4n_i, P4„_i, P4„_2 have the form:

^ 4 n - i —

x4n-i,^(n-USi0) x4n-i,-4(n-\+5m) "

^ n - i . ^ n ^ + r f j o ) *4n-i,^Kn-2+.5j0) x4n-i,-4(n-2+Si0)

0 ^n-i.^Cn-S+^-o) x4n-i,-4(n-3+<5j0)

0 0 0

0

0

0

xin-iA(n-\+Si0) J

(9.3.2.14) (i = 0, 1, 2, 3) and have dimension [2(n +<?„,•) - l]x[2(n + S0i) - 1]. The

submatrices B4„, D4m J4„, P4n, D4„_2, D4„_3, J4„_i, J4„-2, J4n-3, P4«-3 are defined as

X 4 « - j

Hn-i,^(n-\+Si0) 0

•X4n-i,-4(n-2+<5j0) ^ n - i . ^ C n ^ + ^ - o )

0

0

x4n-i,-4(n-$¥Si0) X4n-i,^Kn-3+<5;0) - ^ n - i . - ^ n - S + ^ o )

0 0 0

0

0

0

X4n-iA(n-l+Si0) J

(9.3.2.15) (i = 0, 1, 2, 3) and have dimension [2(n +SQi) - l]x[2(n + SQi - 3]. The

its are formed from the et

and are given by

submatrix elements are formed from the elml>m> listed in Section 9.3.1

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496 The Langevin Equation

9(n-l)n(« + l)(« + 2)-15m2[6n(n + l ) - 5 - 7 w 2 ] „(„ + i)

(In - 3)(2n - l)(2n + 3)(2n + 5) 2 '

15<TJ (n + m)(n - m + 4) n 2 - (m-3 ) 2 n 2 - ( m - 2 ) 2 n 2 - ( m - l ) 2

2(2n - 3)(2n - l)(2n + 3)(2n + 5)

3iaml3n2—5—7m2) L 2 _ m2

tf(4n - 9 ) 4n2-l

3ia l(n + m-4)(n + m)

la(4n1 -9)

n 2 - ( w - 3 ) « 2 - ( m - 2 ) n2 —(m—1)

4n 2 - l

fl... c7(2n + 9 ) ( n 2 - n - 2 - 7 m 2 ) l [ « 2 - m 2 ] ( « - l )

\2 2

-m (2n-5)(2n-l)(2n + 3) V (2n + l)(2«-3)

cr(2« + 9) "n,m Pn,~m 2(2n-5)(2n-l)(2n + 3)

j(« + m - 5)(/z + m - 4)(« + m - 3)(« + m) « 2 - ( m - 2 ) « 2 - ( m - l )

(2n + l)(2n-3)

7tom (n2-m2) ( n - l f - m 2 (n-2) N2 2

-m (2n-5)(2n+l) "'m a(2n-3)(2n-l) 1

iO" l(n+m-6)(n+m-5).. .(n+m—\)(n+m)(n-m+\) un,m un,-m / - i ^ o ow-» i \

2a(2n-3)(2n-l) (2n-5)(2/i+l)

Jn,m 7o(n+l) (n-3)2-m2 (n-2)2-m2 (n-lf-m2 [n2-m2~^

(2n-5)(2n-3)(2n-\y

o(n+l)

(2n-7)(2n+l)

-/n,m .//i,-(n+m-7)(«+m-6).. .(n+m-l)(n+m)

2(2n-5)(2n-3)(2n-l)^ (2n-7)(2n+l)

On applying the general method of solution of the matrix three term differential-recurrence Eq. (9.3.2.7), we obtain the exact solution for the Laplace transform C(s) in terms of matrix continued fractions as before:

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Chapter 9. Non-Axially Symmetric Problems

Ci(s) = tN[TNsl-Q1 -<$S2(s)Jl

497

x ]C,(0) + X n=l

flQ;[%^i-Qt+,-Q,++iS,+2w]

,k=l

(9.3.2.16) Cn(0) ,

where I is the identity matrix and the matrix continued fraction S„(s) is defined as

S„(.) = [ r i v 5 I -Q„-Q n+ S n + I ( , ) ]" 1 Q-. (9.3.2.17)

The initial condition vectors C„(0) may also be evaluated in terms of matrix continued fractions. The initial values c„>m(0) are given by:

(n + Y) -m .,, . In -m /T, . , o . . , n n „ , ^< y n+l .«>0 + \ | „ 2 , <Fn-l,m>0-(2n + l)(2/i + 3) 4 « 2 - l

(9.3.2.18)

The recurrence relation for the equilibrium averages \Ynm\ [19], viz.,

1 4

2a 2-i en,m,n+r,m+4s \*n+r,m+4s/Q ~ ^ ' ( y . O . Z . i y ) ,s=-l r=-4

may be written as the three-term matrix recurrence relation:

Q ; R „ _ 1 + Q „ R „ + Q X + I = 0 . (n>l), (9.3.2.20) where the matrices Qn ,Q+, and Q~ are defined by Eqs. (9.3.2.9)-(9.3.2.11) and

R„

and the column vector r An-i

Hn

'4/1-1

Mn-2

V r4n-3 J

is

, Rr AK

(9.3.2.21)

r4n-i _ \Y4n-i,-4(n-2+Sl0))0 (i = 0,1,2,3). (9.3.2.22)

The solution of Eq. (9.3.2.20) is

R„ =S„(0)RB_1 =S„(0)Sn_1(0)...S2(0)S1(0)/V4^, (9.3.2.23)

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498 The Langevin Equation

where the matrix continued fraction S„(0) is given by Eq. (9.3.2.17) for s = 0. Thus, the C„(0) are given by:

c» ( 0 ) =^[^»+[K»+^ s»^H s»H s»- i ( 0 )-"S i ( 0 ) ' ( 9-3-2-2 4 )

where

K =

f 0 T An

u An

u 0 U

0 U

0

4 n - l

0 U

o o ^

4„-l 0

0 U J4n-2

4n-2 0

' K « -

' 0 0 0

V^4n-3

0 0 0^ 0 0 0

0 0 0 0 0 0,

The matrices K„ and Kn are comprised of submatrices U. The submatrices U4n-i,U4n_2,U4n_3 are

U4„ =

0

*4n,-4n+4

u4n_(- =

u, •An-i,-A(n-\)

0

0

0

0

0

0

u.

u

0 0

4n,-4n+&

0 0

0

0

0

u4n,4n-4

0

4n-i,-4n-2)

0

0

uL 4n-i , -4(n-3)

0

/ ( 2 n + l ) x ( 2 n - l )

0 ^

0

0

^-^n-V )(2n-l)X2n-\)

(i= 1, 2, 3). The elements of the submatrices U are given by

-1 -m2)/(4n2-l) un,m=yl(n2

The exact matrix continued fraction solution [Eq. (9.3.2.16)] we have obtained is again very easily computed. All the matrix continued fractions and series involved converge very rapidly, thus 10-20 downward iterations in calculating the continued fractions and 11-15 terms in the series (9.3.2.16) are enough to arrive at an accuracy of not less than 6 significant digits in the majority of cases. The greatest dimension of all the matrices involved is of the order 102 again allowing one to carry out the calculations on a personal computer.

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Chapter 9. Non-Axially Symmetric Problems 499

9.3.3 Complex susceptibility and relaxation times

The behaviour of the relaxation time q\ as a function of the barrier height parameter a for various values of the damping parameter a, is shown in Fig. 9.3.3.1. Here, the results of the calculation from the exact formulas are compared with those evaluated from the asymptotic formulas for rn

in the low temperature or high barrier limit (lot » 1) for various ranges of a. As far as physical interpretation is concerned, the relaxation time rn

is determined by the slowest low-frequency relaxation mode which governs transitions of the magnetisation vector over the barriers separating one potential well from another. The characteristic frequency of this overbarrier relaxation mode is determined by the inverse of the smallest nonvanishing eigenvalue k\ of the Fokker-Planck equation (9.2.3.8). As one can see in Fig. 9.3.3.1, just as the oblique field problem, % ltN for particles with cubic anisotropy strongly depends on the damping parameter a.

The asymptotic formulas for the cubic anisotropy potential in the intermediate to high damping (IHD) limit (a> 1) were obtained in [14,17]. We recall (see Section 9.2.3), that the escape rate T from a minimum i of the potential V to the minimum j separated by the barrier V0-Vi (VQ is the saddle energy) is given by

P 8;nv

.O'UO c) c „«V0) -C

(0)_„(0) - c ; ' + r-cf)2-4C[ (QUO)

a -Wo-V,)

(9.3.3.1)

10'

io2 i

10"

1 - a -»oo

, 2 - a = 1 . 0

3 - a = 0 . 1

, 4-or=0.01

a , , , ,

xz/

-10 0 a

10

Fig. 9.3.3.1. T||/rw vs. crfor various values of the damping parameter a. Solid lines: Eqs. (9.3.2.4) and (9.3.2.16) at a - > «, (curve 1); a= 1 (2); or= 0.1(3), and a= 0.01 (4). Filled circles and crosses: (9.3.3.4) and (9.3.3.5) for a—> °° and a= 1, respectively; stars: Eqs. (9.3.3.6)-(9.3.3.9).

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500 The Langevin Equation

where c'] are the coefficients of the expansions of V , Eq. (9.2.3.22),

near the stationary point n,. For two minima separated by two or more barriers, one must set T = 0. For a cubic crystal with K > 0 (see Fig. 9.3.1.1), one finds [17]

Vi = 0,V0 = K/4, c[i] = 4 ° = 2K, c[0) = K, 4° = -2K . (9.3.3.2) For a cubic crystal with K < 0, one has [17]

Vi = K/3, V0 = K/4, q(0 = 4 ° = -4K/3, c0) = -2K, c f = K . (9.3.3.3) As may be shown in the context of the discrete-orientation model (see Chapter 1, Section 1.16), the mean magnetisation decays with time constants l/(4r) for K>0 and 1/(2D for K<0 [7,14,17]. Thus, we have fromEqs. (9.3.3.1)-(9.3.3.3):

tNTtea

V « T | ^ , .N - Y , O - > 0 , (9.3.3.4) 2V2<7lV9 + 8 / a 2 + l l

3r ^l f fl /3

V--Z-H , / ,-,<7<0. (9.3.3.5) 2V2|cr|(V9 + 8/or2 - 1 ) '

In the opposite very low damping limit (a« 1), the appropriate asymptotic solution (9.2.3.26) was derived by Klik and Gunther [37,44] and reviewed by Coffey [45]. Their formulas applied to the present problem yield

2coAAE 8(7

\~X ^ - ^ — « ^ ^ - , (a < 0), (9.3.3.7) coAAE 8a2

where we note that the frequency of oscillation in the potential well is [ SoykT/(vMs), (cr>0),

l6\a\ykT/(3vMs), (cr<0), and the energy loss per circle of the almost periodic motion at the saddle point energy E0 is

AE~a(E0) = av\K\/4. (9.3.3.9)

The dependence on the damping parameter appears also in the spectra of the imaginary part of the complex susceptibility xli®) > shown in Fig. 9.3.3.2. In Fig. 9.3.3.3 the spectrum of ^a) is plotted for a = 0.1 and various values of a. (The calculations were carried out with

0A=\V, _,..,„.„,. , ,_ .„ (9-3-3.8)

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Chapter 9. Non-Axially Symmetric Problems 501

v2M*N0/kT = 1). In these figures, two peaks in the loss spectrum are visible. The first (low-frequency) peak is located at frequencies of the order of the average frequency of reorientation of the magnetisation vector. The characteristic frequency and the half-width of this low-frequency Lorenzian band are given by X\. The second peak is caused by the contribution of the high-frequency intrawell and transverse relaxation modes. Just as the oblique field configuration, there is an inherent geometric dependence of Xwi.®) a nd 3[/% for particles with cubic magnetic anisotropy on the value of the damping parameter a arising from coupling of the longitudinal and transverse relaxation modes.

10"* i<f 10"2 10° 102

corN

Fig. 9.3.3.2. x"\ vs- °^N f o r <7= 10 and or-> o° (curve 1), a= 1 (2), a = 0.1 (3), and a

= 0.01 (4).

10"5 10"3 10"' 10' 103

Fig. 9.3.3.3. X\\ vs. cotN for <x= 0.1 and a= 0 (1), a= 1 (2), a= 5 (3), and <r= 10 (4).

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502 The Langevin Equation

A simple description of the spectra can be presented in the high damping limit, a —> oo, in the context of the approach described in Chapter 2, Section 2.13. Hence, the contribution of the transverse modes becomes negligible, and the susceptibility X\iM maY be approximated as a sum of two Lorenzians:

X\\W) A, 1-A, ^ = i—-+ x—. (9.3.3.10)

X\\ \ + ia>l\ \ + ionw

The parameters % and Ai are expressed in terms of the smallest nonvanishing eigenvalue A.u the correlation time Tn and the effective relaxation time tf and are given by Eqs. (2.13.11) and (2.13.12) of Chapter 2, Section 2.13.

The effective relaxation time rff may be evaluated from Gilbert's equation for the averaged magnetisation written in the spherical polar coordinates (see Chapter 7, Section 7.3):

2TN— costf = -2costf + y#sin??— — £ — . (9.3.3.11) N dt H d& ad<p It follows from Eq. (9.3.3.11) that

c, n(0) = (cos*?—cost?) = ( cos i?-— +—cos?? ) \ dt /0 TN\ 4 3*? 2a 3 ^ / 0

^ - ( l - ( c o s ^ ) o ) . (9.3.3.12)

Thus, we have

tf =_fu>(0) <co,'tf)0 =

" q,0(0) "l-<cos2t?>o

(here, we have noted that in cubic anisotropy (cos2t?)0 = 1/3), i.e., the effective relaxation time is independent of a. Hence, the high frequency asymptotic behaviour of Im[;£j|(fi>)] also does not depend on <x, that is

[see Eq. (2.13.8), Section 2.13]

- lim flwwlm[^i|(©)/^||] = < / ^ = l . (9.3.3.14)

The numerical evaluation of X\ and t\\ requires the application of matrix continued fractions which is not useful in practice. Fortunately, for cubic anisotropy, these constants may be evaluated in a very simple manner because one can use simple extrapolating formulas for the correlation time zj| and A[l [28]:

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Chapter 9. Non-Axially Symmetric Problems 503

K '• T I I ~ T> iea-\) n + 1

7t

8V2 I 8V2

for <T> 0 and

K •• ?«« - 3 r . (e

-(7/3 •1) #

+ 1 -It ,<T/3

(9.3.3.15)

(9.3.3.16) 4V2 X 4V2 J

for <7<0. Equations (9.3.3.15) and (9.3.3.16) provide a good approximation to the exact matrix continued fraction solution for all a [28]. The high-frequency amplitude A! and characteristic time % can be evaluated from the following approximate equations (for a» 1) [28]

At «1 - 1/(4<T) and Tw » Tw /(4cr) (9.3.3.17)

for positive anisotropy constant and Ai «l-3/(8|<r|) and % « ^ / ( 8 | t r | ) (9.3.3.18)

for negative anisotropy constant. The difference between Eqs. (9.3.3.17) and (9.3.3.18) is because the frequencies of oscillations at the minima of the cubic potentials with positive and negative anisotropy differ by a factor of two-thirds.

The spectra of the imaginary part of the complex susceptibility Z\\(<d) vs. GKN and crin the high damping limit are shown in Fig. 9.3.3.4. It is clearly seen here that the spectra can be effectively approximated by two relaxation bands. The first (low-frequency) band is located in the vicinity of the average frequency of reorientation of the dipole moment vector. The characteristic frequency and the half-width of this low-frequency band are determined by X\. The second, high-frequency, process is caused by the contribution of the intrawell relaxation modes. In spite of the large number of high-frequency modes involved, this peak may be described by a single Lorenzian. Thus, according to Eq. (9.3.3.10), which is the superposition of two relaxation modes, both low-and high-frequency relaxation processes are effectively governed by a single relaxation mode in the high barrier approximation (Id » 1). Moreover, this two mode approximation provides sufficient accuracy for n(ft») even for moderate barriers (1 < lot < 3).

Equation (9.3.3.10) corresponds to the representation of the dipole autocorrelation function C\\(t) in the time domain by a sum of two exponentials

C | |(0 = A 1 e " ^ + ( l - A 1 ) e " ' / % . (9.3.3.19)

The simple representation in the high barrier limit which we have given is possible because the diffusion within the potential wells is much faster than the transition rate between the wells ( tw « A[l).

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504 The Langevin Equation

10°i

^ 10"2i

I ' T -4

10 ]

m-S

A. \

l - < 7 = 5

2- a=10

3 - <r=15

4 - cr= 20

10" 10" 10 10 10 10" 10' 10'

Fig. 9.3.3.4. Comparison of the exact and approximate calculations of - I m ^ ] vs. anN for positive (a> 0) and negative (a< 0) anisotropy constants. Solid lines: matrix continued fraction solution Eqs. (9.3.2.3) and (9.3.2.16). Stars: Eqs. (9.3.3.10) and (9.3.3.15)-(9.3.3.18).

The above examples demonstrate how the Langevin equation method may be successfully applied to the problem of the rotational Brownian motion in a cubic potential. Other examples of the application of the method to Brownian motion in a cubic potential are given in Refs. [20,21,28]-

To conclude this chapter, we have demonstrated the power of the matrix continued fraction approach in the solution of very complicated problems in the theory of superparamagnetism. With small modifications, the approach developed may be applied to the solution of similar problems in the theory of dielectric relaxation and Kerr effect in liquids and nematic liquid crystals. These are all based on the noninertial rotational diffusion

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Chapter 9. Non-Axially Symmetric Problems 505

model (see, e.g., Refs. [22,23]). In the next chapter, we apply the Langevin equation in order to analyse the inertial rotational diffusion model.

References

1. L. Ned, Ann. Geophys. 5, 99 (1949). 2. E. C. Stoner and E. P. Wohlfarth, Phil. Trans. R. Soc, Lond. A 240, 599 (1948). 3. W. T. Coffey, D. S. F. Crothers, J. L. Dormann, L. J. Geoghegan, Yu. P. Kalmykov,

J. T. Waldron, and A. W. Wickstead, J. Mag. Mag. Mater. 145, L263, (1995). 4. W. T. Coffey, D. S. F. Crothers, J. L. Dormann, L. J. Geoghegan, Yu. P. Kalmykov,

J. T. Waldron, and A. W. Wickstead, Phys. Rev. B 52, 15951 (1995). 5. L. J. Geoghegan, Ph. D. Thesis, University of Dublin, 1995. 6. W. T. Coffey and L. J. Geoghegan, J. Mol. Liquids, 59, 53 (1996). 7. L. J. Geoghegan, W. T. Coffey, and B. Mulligan, Adv. Chem. Phys. 100, 475

(1997). 8. W. T. Coffey, D. S. F. Crothers, J. L. Dormann, L. J. Geoghegan, and E. C.

Kennedy, J. Magn. Magn. Mater. 173, L219 (1997), 9. W. T. Coffey, D. S. F. Crothers, J. L. Dormann, L. J. Geoghegan, and E. C.

Kennedy, Phys. Rev. B 58, 3249 (1998). 10. Yu. P. Kalmykov and S. V. Titov, Fiz. Tverd. Tela (St. Petersburg) 40, 1642 (1998)

[Phys. Solid State, 40, 1492 (1998)]. 11. H. Kachkachi, W. T. Coffey, D. S. F. Crothers, A. Ezzir, E. C. Kennedy, M.

Nogues, and E. Tronc, J. Phys : Condens. Matter, 12, 3077 (2000), 12. Yu. P. Kalmykov and S. V. Titov, Fiz. Tverd. Tela (St. Petersburg) 42, 893 (2000)

[Phys. Solid State, 42, 918 (2000)]. 13. A. Aharoni, Phys. Rev. B, 7, 1103 (1973). 14. D. A. Smith and F. A. de Rozario, J. Magn. Magn. Mater. 3, 219 (1976). 15. I. Eisenstein and A. Aharoni, Phys. Rev. B 16, 1278 (1977). 16. I. Eisenstein and A. Aharoni, Phys. Rev. B 16, 1285 (1977). 17. W. F. Brown, Jr., IEEE Trans. Magnetics, Mag.lS, 1197 (1979). 18. Yu. P. Kalmykov, S. V. Titov, and W. T. Coffey, Phys. Rev. B. 58, 3267 (1998). 19. Yu. P. Kalmykov and S. V. Titov, Zh. Exp. Teor. Fiz. 115, 101 (1999) [JETP 88,

58 (1999)] 20. Yu. P. Kalmykov, Phys. Rev. B. 61, 6205 (2000). 21. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov, J. Magn. Magn. Mater. 241, 400

(2002). 22. J. L. Dejardin, P. M. Dejardin, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. E 60,

1475 (1999). 23. J. L. Dejardin, Yu. P. Kalmykov, and P. M. Dejardin, Adv. Chem. Phys. 117, 275

(2001). 24. C. Brot and I. Darmon, Mol. Phys. 21, 785 (1971). 25. B. De Raedt and K. H. Michel, Phys. Rev. B 19, 767 (1979). 26. R. W. Gerling and B. De Raedt, /. Chem. Phys. 77, 6263 (1982). 27. G. Moro and P. L. Nordio, Z. Phys. B 64, 217 (1986). 28. J. L. Dejardin and Yu. P. Kalmykov, J. Chem. Phys. I l l , 3644 (1999).

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506 The Langevin Equation

29. J. L. Dormann, in Magnetic Properties of Fine Particles, p.l 15, Eds. J. L. Dormann and D. Fiorani, North Holland Delta Series, North Holland, Amsterdam, 1991.

30. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of Angular Momentum, World Scientific, Singapore, 1998.

31. Yu. P. Kalmykov and S. V. Titov, J. Magn. Magn. Mater. 210, 233 (2000). 32. Yu. P. Kalmykov, Phys. Rev. E 61, 6320 (2000). 33. W. T. Coffey, D. A. Garanin, and D. J. McCarthy, Adv. Chem. Phys. Ill, 483

(2001). 34. H. A. Kramers, Physica 7, 284 (1940). 35. P. Hanggi, P. Talkner, and M. Borkovec, Rev. Mod. Phys. 62, 251, (1990). 36. H. Pfeiffer, Phys. Stat. Sol. 118, 295(1990); ibid. 122, 377(1990). 37. I. Klik and L. Gunther, J. Stat. Phys. 60, 473 (1990). 38. D. A. Garanin, E. C. Kennedy, D. S. F. Crothers, and W. T. Coffey, Phys. Rev. E

60, 6499 (1999). 39. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. B

64,012411(2001). 40. D. A. Garanin, Phys. Rev. E 54, 3250 (1996). 41. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and J. T. Waldron, Phys. Rev. B

51, 15947 (1995). 42. W. T. Coffey, D. S. F. Crothers, J. L. Dormann, Yu. P. Kalmykov, E. C. Kennedy,

and W. Wernsdorfer, Phys. Rev. Lett. 80, 5655 (1998). 43. J. L. Garcia-Palacios and P. Svedlindh, Phys. Rev. Lett. 85, 3724 (2000). 44. I. Klik and L. Gunther, J. Appl. Phys. 67, 4505 (1990). 45. W. T. Coffey, Adv. Chem. Phys. 103, 259 (1998). 46. B. Barbara, L. C. Sampaio, J. E. Wegrowe, B. A. Ratnam, A. Marchand, C.

Paulsen, M. A. Novak, J. L. Tholence, M. Uehara, and D. Fruchart, J. Appl. Phys. 73, 6703 (1993).

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Chapter 10

Inertial Langevin Equations: Application to Orientational Relaxation in

Liquids

10.1 Introduction

As we have seen in Chapters 3-9, an accurate method of solution of the Langevin equations for particles undergoing Brownian movement (rotational or translational) is essential in a variety of physical problems such as the dynamics of Josephson junctions, dielectric relaxation and dynamic Kerr effect of molecular liquids and nematic liquid crystals, magnetic relaxation of ferrofluids, etc. In the theory of dielectric and Kerr-effect relaxation, the inertia of molecules may be neglected at low frequencies. However, the neglect of the molecular inertia in dielectric absorption leads to incorrect predictions of the behaviour of dielectric spectra at high frequencies (e.g., to infinite integral absorption, Chapter 1, Section 1.15.1). Inertial effects may also appear in the magnetic susceptibility spectra of ferrofluids due to the physical rotation of the ferrofluid particles carrying the single domain magnetic particles. In Josephson junctions (Chapter 5), at high frequencies, one must also take into account the effect of the capacitance, which here plays the role of the inertia.

The theoretical treatment of inertial Brownian motion has hitherto been mainly based on the Fokker-Planck equation. Just as the noninertial case, the solution of the Fokker-Planck equation may be obtained by separating the variables yielding an equation of Sturm-Liouville type which possesses solutions in the form of known functions only in a few specialised cases (just as the analogous problem of the solution of the Schrodinger equation in wave mechanics) or by expanding the distribution function as an appropriate Fourier series in the phase space variables which yields an infinite hierarchy of linear differential-recurrence equations for the time-dependent moments (the expectation

507

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508 The Langevin Equation

values of the Fourier coefficients). An alternative approach to the problem is to derive this hierarchy directly by averaging the Langevin equation without recourse to the Fokker-Planck equation. The key step in applying the Langevin method is first to convert by appropriate transformation the Langevin equation into an equation for the quantity the statistical average of which is desired and then to average that equation over its realisations in phase space. The transformed Langevin equation contains not only the average of the desired quantity but also the next higher order average and so on. It is thus the generating equation of a hierarchy of averages. It is the purpose of this chapter to show how the hierarchies of differential-recurrence equations for inertial Brownian motion arise naturally from the Langevin equations thus bypassing the problem of constructing and solving the Fokker-Planck equation entirely. The advantage in computational labour that the averaging method has over the solution by the Fokker-Planck method is considerable as neither the derivation of that equation nor a knowledge of the intricate transformations used to effect separation of the variables in it, and to solve the resulting simultaneous recurrence relations is required.

In this chapter, we shall treat the inertial effects in orientational relaxation of Brownian particles. We commence our discussion by considering rotation about a fixed axis as this is the simplest example.

10.2 Step-On Solution for Noninertial Rotation about a Fixed Axis

In Chapter 1, Section 1.15, we saw that the original treatment of dielectric relaxation of assemblies of rotators in two and three dimensions due to Debye [1,2] proceeds directly from the noninertial Smoluchowski [3] equation as in Einstein's theory [4] of the translational Brownian movement, without making explicit reference to the Langevin equation [5]. In particular, Debye [1,2] found two solutions of the Smoluchowski equation for the Brownian motion of the planar rotator and the sphere. The first is the after-effect solution. This is the response following the removal of a steady (dc) field which had been applied in the infinite past. The second is the response to a continuously applied ac field. Furthermore, since the response is only considered in the linear approximation, the response to a dc field suddenly applied will be the mirror image of the after-effect response. The use of the linear approximation also means that the ac response may be found from the after-effect response using the well-known formulas of linear response theory (Chapter 2, Section 2.8).

Here we shall demonstrate how the three solutions (step-on, step-off, and ac field) may be obtained from the Langevin equation without

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Chapter 10. Inertial Langevin Equations 509

recourse to the underlying Smoluchowski equation. Our argument will almost exactly parallel to that given by Langevin [6] in his derivation of the mean square displacement of a Brownian particle. The key step in our calculation will be a transformation of the Langevin equation given by Frood and Lai [7] which allows one to see how that equation may be linearised in the presence of a driving field. The approach will be fully generalised in the next section to include both inertial and nonlinear effects.

The Langevin equation for a typical member of an assembly of non-interacting rigid dipoles free to rotate about a fixed axis is (Chapter 2, Section 2.6)

I0(t) + £0(t) + juF(t) sin 0(t) = X(t), (10.2.1)

where / is the moment of inertia of the rotator about the axis of rotation, 6 is the angle the rotator makes with the direction of the driving field F(t), 0 and X(t) are the frictional and random torques due to the Brownian motion of the surroundings. A(t) has the white noise properties

1t) = 0, A(t)A(t') = 2kT£St-1'). (10.2.2)

Here the overbar denotes the statistical average over a large number of rotators which have at time t the same angular velocity and same angular position. In accordance with the notation of Chapter 4 et seq., we shall use the notation 6(t) and 0(t) to denote a random variable while we shall denote a sharp (definite) value at time t by 0 and 0. We recall that just as in Chapters 2 and 4-9, the average value of a random variable £• (t) at time t +1' is calculated from the Langevin equation regarded as an integral_equation and interpreted in the Stratonovich sense by expressing £• as an equation of motion for a sharp value £; = x at time t.

In order to specialise Eq. (10.2.1) to the step-on field, we write

F(t) = F0U(t), (10.2.3) where U(t) is the unit step function and F0 its amplitude. Here, we require to calculate the average (ji cos 6) when inertial effects are ignored. The conventional method [2] of doing this is to write the Smoluchowski equation for the distribution function f0,t) in one-dimensional configuration space, namely

a ,,. . a MF0 *D^f(0,t) = — —f(0,t) + ^sm0f(0,t) at 60

(10.2.4) d0 kT

[TD = £/(kT) ] and then calculate the linear response from this equation

by assuming that the field parameter £ = juF0 /(kT) is so small that terms

O (£)2 and higher may be neglected so that

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510 The Langevin Equation

£s in0 / (0 ,O«£s in0 / o (0 ) , (10.2.5) where f0 is the distribution function in the absence of the field. This allows to solve Eq. (10.2.4) easily obtaining the mean dipole moment as

(//cos0)(O = A t / ( o [ l - e-"tD ] . (10.2.6)

The angular braces in Eq. (10.2.6) represent the statistical average of cos 6 over a set of initial conditions fy (at time t = 0 when the field F0 is switched on).

The problem that presents itself when treating the model using the Langevin equation in the form of Eq. (10.2.1) above is that it is not apparent how that equation may be linearised to yield the solution for small /J.FQl(kT). Frood and Lai [7] suggested in 1975 that this difficulty may be circumvented by rewriting Eq. (10.2.1) as an equation of motion for the instantaneous dipole moment. This suggestion has proved to be of singular importance in formulating the central theme of this book as it provided the first indication [8] of how the set of differential-recurrence relations generated by a Fokker-Planck equation may be generated from the Langevin equation. Now the quantity of interest is p(t) = jucos 6\t) so that

0 = £—, (10.2.7) //sin#

0 = L l ^ — = - ^ - l j ^ . (10.2.8) (ju2-p2) (M2-P2)

Msine Msine

The Langevin equation (10.2.1) with this change of variable becomes (see Chapter 4)

Ip(t) + Cp(t) + (W2(t) + pF0)p(t) = jU2Fo-jusin0(t)Mt), (10.2.9)

which is the exact Langevin equation for the motion of the instantaneous dipole moment. As shown in Chapter 4, we may average Eq. (10.2.9) yielding

Ip + CP + (^2 + pF0)p = M2F0, (10.2.10)

where p denotes a sharp (definite) value of p (t) at time t. If we now average Eq. (10.2.10) over the initial Maxwell-Boltzmann distribution prevailing at time t = 0, we have

l(p) + C(p) + lpe2 + p2F0) = jU2F0 (10.2.11)

(the angular braces denoting the averaging over this initial distribution). Equations (10.2.10) and (10.2.11) are nonlinear. In order to linearise

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Chapter 10. Inertial Langevin Equations 511

either of them, we note that the term p F0 when written in terms of cos 20 is

p2Fo = (l + cos20)//2Fo/2. (10.2.12)

Now, the average (cos 20) will be at least [9] of order £2 so that for the

linear response we are fully justified in setting

(p2)F0=M2F0/2. (10.2.13)

Since the Debye theory pertains to the situation, where t » I/£, which implicitly means [10] that a Maxwellian distribution of angular velocities has set in, we may now write using the equipartition theorem

(l62p) = kT(p) (10.2.14)

since the orientation and angular velocity variables, when equilibrium of the angular velocities has been reached, are decoupled from each other as far as the time behaviour of the orientations is concerned [10]. The assumption that t » 11f also means that

/ ( p ) = 0 (10.2.15)

so that finally the linearised form of Eq. (10.2.11) is in the limit of long times

TD(p) + (p) = ^U(t). (10.2.16)

If we take the Laplace transform of Eq. (10.2.16), we have

L(p) = ^ , (10.2.17) lN / J 2kTs(l + sTD)

which yields in the time domain

(p)t) = ^Ut)[\-e-"*°\ (10.2.18)

Equation (10.2.18) coincides with Eq. (10.2.6) obtained from the Smoluchowski Eq. (10.2.4). We remark that rather than setting l(o) = 0 in Eq. (10.2.1) in order to obtain our results, we have set l(jp) = 0 in Eq. (10.2.11) so that the term (l02p) resulting from 10 remains to contribute to the linearised equation for (p ) .

We have illustrated our method by considering the step-on solution. If we suppose instead that the field F0 having been applied for a very long time so that statistical equilibrium has been reached, is suddenly switched off at time t = 0 then Eq. (10.2.16) is modified to

TD(p) + p) = ^[\-U(t)] (10.2.19)

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512 The Langevin Equation

SO that

and

(p) = (p0)e-"T° (10.2.20)

with p0 = jucos0o . In the linear approximation, £ = (j.F0 lkT)«1, l I " 2K 1 ,,2r.

Here, we have noted that the distribution function /0 of the initial orientations and angular velocities for a steady field F0 is given by

n -^L f0~jl/(8x\T)e 2W(l + £cos6>0).

Thus, Eq. (10.2.20) becomes (p)(t) = ^e-'/*°, (10.2.21)

which is the noninertial after-effect solution. A similar argument may be used at the calculation of the

stationary linear response in an ac electric field Fme10". Here, the linearised Langevin equation becomes

TD(p) + (p) = ^ e i " ' , (10.2.22)

so that the stationary ac response is given by

fr> = / f £ W i -1 eiM- ( 1 ° - 2 - 2 3 )

2kT 1 + IQ)TD

We shall now derive the complete set of differential-recurrence relations for the inertial response. We shall also justify the assumptions made in obtaining the noninertial response results by considering the inertial response determined by averaging the underlying Langevin equation without any approximations. 10.3 Inertial Rotation about a Fixed Axis 10.3.1 Inertial effects and nonlinear response

In this section, the inertial response is again evaluated in the context of the model of the inertial rotational Brownian motion of a fixed axis rotator as this is the simplest model illustrating the combined influence of

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Chapter 10. Inertial Langevin Equations 513

inertial and nonlinear effects. Here, the particular application of this model to the transient nonlinear dielectric relaxation of a system of planar rotators following a step change in the strong applied uniform electric field is considered only (however, with small modifications, the model can be applied to other physical phenomena such as the Josephson effect, see Chapter 5).

Following the exposition of Coffey et al. [11], we consider a system of noninteracting planar rotators undergoing rotational Brownian motion in the presence of an external uniform dc electric field and suppose that the magnitude of the field is suddenly altered at time t - 0 from Fi to Fn. We are interested in the relaxation of the system starting from an equilibrium state I with the distribution function Wh (t < 0), to another equilibrium state II with the distribution function Wlh (t —> °°). These initial and final state distribution functions are given by

Wl = Z;V["2>^)H = z-^-m^coSo ( i a 3 U )

and

Here ft is the dipole moment of a rotator, the ZN (N = I, II) are the partition functions, and

T] = Jll2kT), £N = juFN l(kT). (10.3.1.3)

This transient relaxation problem is truly nonlinear, because the alteration in the magnitude of the strong external dc field is arbitrary.

The Langevin equation for a typical element of an assembly of noninteracting rigid dipoles rotating about a fixed axis is similar to Eq. (10.2.1), viz.,

I0(t) + £0(t) + juFnsm6t) = Xt), (t>0). (10.3.1.4) Our goal is to evaluate the transient relaxation of the electric polarisation P(t) defined as

P(t) = juN0(cosd)(t), (10.3.1.5)

where TVo is the concentration of dipoles and the angular brackets mean the statistical average over the sharp values of 0 and 6. The dynamics of P (t) can be described by the normalised relaxation function:

. . (cos0)(O-(cos6>V f^= / i\ / a\ > ( ? > 0 ) - (10.3.1.6)

(cos#)j - (cos#) n

Here, the angular brackets ( ) denote ensemble averages over the

equilibrium distribution functions WN:

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514 The Langevin Equation

°° In

cos6)N = j j cos0WN(0,0)d0d0. (10.3.1.7) -oo 0

The integral relaxation time z is defined, as usual, as the area under the curve of the relaxation function:

oo

T = jf(t)dt = f(0). (10.3.1.8) o

In order to proceed, we now change the variable in Eq. (10.3.1.4) using the transformation

r(t) = e-im. (10.3.1.9)

Thus we obtain

Ir(t) + Cr(t) + W2(t)r(t) + -jUFu[r2(t)-l] = -ir(t)A(t). (10.3.1.10)

Equation (10.3.1.10) is a Stratonovich stochastic differential equation with a multiplicative noise. On noting that

0 = ^H1(?J0), 02=-±Y\H2(T?0) + 2'\, (10.3.1.11) 27 47 L J

and

where r = -iHiT]d)rl2t]), (10.3.1.12)

Hl(x) = 2x and H2(x) = 4x2-2 (10.3.1.13) are the first two Hermite polynomials [12], Eq. (10.3.1.10) can be rewritten as

rj^-Hl [rj0(t)]r(t)+ P'HX [7]0(t)]r(t) + UH2[rj0(t)] + 2r(t) dt (10.3.1.14)

+ l-Zll[r\t)-Y\ = ^r(t)A(t).

Here p' = £rjll. On averaging Eq. (10.3.1.14) and on noting that

r(t)Mt) = 0 (see Chapter 4, Section 4.2), we have

7?±(Hlr) + /rHlr + UH2r + 2r) + ll(r2-l) = 0. (10.3.1.15)

at I I Equation (10.3.1.15) is the leading term in the infinite hierarchy of differential-recurrence relations for the statistical moments governing nonlinear dielectric relaxation. In order to solve Eq. (10.3.1.15), we need to continue the hierarchy, so we require a prescription for calculating H2 r and r2. Let us show how one can obtain, for example, an equation for

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Chapter 10. Inertial Langevin Equations 515

H2 r. This is accomplished as follows. Taking account of Eqs. (10.3.1.11) and (10.3.1.12), we have

~[H2(t)r(t)] = r(t)H2(t) + r(t)H2(t) (10.3.1.16)

= 8r(t)T]20(t)0(t) ~^-[H3(t) + 4H (t)] r(t), 27

where H3(x) = 8x3 -\2x and

8r(t)7]20\t)0Xt)=8rlr(t)ti2e\t)[-£0\t)-vFusm0(t) + Mt)], (10.3.1.17)

using the Langevin equation (10.3.1.10). We now use Eq. (10.3.1.17) in Eq. (10.3.1.16) to get

T]—H2(t)rt)) + 2j3'H2(t)r(t) = 8T3rxr(t)0(t)A(t)-4fi'r(t) dt . (10.3.1.18)

+i^Hl(t)(l-r2(t))-±H3(t) + 4H1(t))r(t).

Equation (10.3.1.18) becomes on averaging d 7]—H2r + 2/3'H2r = ST]3I'1 r(t)0(t)A(t) - Ap'r dt (10.3.1.19)

+itn(H1-r2H1)-i(H3+4Hl)r/2.

We evaluate rt)0t)X(t) explicitly as follows. The Langevin equation (10.3.1.4) is always to be regarded (see Chapter 2) as the integral equation

0(t) = 0-^-[0(t)-0] + - \ A ( 0 * ' - — J sm0t')dt', (10.3.1.20) ' * o o

where 0 and 0are the sharp initial conditions. Thus

rt)0(t)Xt) = e-imA(t)0 - £rl [0^t)-0]Mt)e~'m

't nF ; (10.3.1.21) +rxe-ie(,)\ A(tWt')dt'-Z^zxv[-i0(t)]At)\ sm0(t')dt'.

o ' o

Now, by definition, Xi) is independent of 0t) and 0. Hence,

e-mO)A(t)0 = O, [0(t)-0]A(t)e-im =0, exp[-i0(t)]A(t)j s in0(O^ ' = O, o

and thus

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516 The Langevin Equation

r(t)[ •s-WCr r(t)0(t)Kt) =—l Mt)MtW = - \ 2kT£5t-t')dt' = ^ . (10.3.1.22)

Equation (10.3.1.22) can equivalently be presented as

r(t)Hl [ndt)\ Mt) = 2kTp'r. (10.3.1.23)

Thus, Eq. (10.3.1.19) can be reduced to

7i—H2r + 2j3,H2r + -(H3+4Hl)r + i^Jr2H1-Hl) = 0. (10.3.1.24) dt 2 K '

This is the second term of the desired hierarchy. In order to obtain a general term of the hierarchy, we need a

recurrence relation for Hn(rj6)rq for any n and q [Hn(x) is the Hermite polynomial of order «]. This is accomplished as follows. We have

jt(Hnr") = qHnr"-lr + r''Hn. (10.3.1.25)

The two terms in the right-hand side of Eq. (10.3.1.25) can be rearranged respectively as

qHnri-1r = -^-HnHlrc'=-^(Hn+l+2nHn_l)r« (10.3.1.26)

2?] 2T]

and

Hnr«=2nl1r«(t)Hn_l[ridt)]d(t)

= 2nr\t)Hn , [Tj0(t)]\ -/fftf(f) - — s i n 0 ( f ) + — — n-ni W J ^ H ^ 2r]kT / (10.3.1.27)

V -P'H^-^H^r ^+1

•Hn-ir «-r

Here, we have noted that [12] d dx

Hn(x) = 2nHn_1(x), Hn+l(x) = 2xHn(x)-2nHn_l(x), (10.3.1.28)

and

Hn[Tid(t)]r»t)Mt) = 2j3'kTnHn_ (10.3.1.29)

We shall not prove Eq. (10.3.1.29) here as a similar proof for a more general model, viz., three-dimensional rotators, is given in the Appendices A and B of this chapter. We only remark that Eq. (10.3.1.23) proved above is a particular case of Eq. (10.3.1.29).

Thus, we have

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Chapter 10. Inertia! Langevin Equations 517

dt (10.3.1.30)

One can readily verify that for q- 1 and n= 1 and 2, Eq. (10.3.1.30) yields Eq. (10.3.1.15) and (10.3.1.24), respectively. On averaging Eq. (10.3.1.30) over the distribution function of the sharp values 0and 6, we have the moment system:

7 ^ ( f l „ r « ) = - n / 3 ' ( H „ r « ) - | ( ( H r , + l r « ) + 2„(fl„_,r«))

-^(K^'H"^'-). (10.3.1.31)

which is often called the Brinkman equations [2,13]. As a result of the above calculation, we can now gain more

insight into the approximations which lead to the linear noninertial response considered in Section 10.2.1. First, by making the approximation, Eq. (10.2.14), viz.,

ld2p^i = kTp) (10.3.1.32)

so that (H2p) = 0 (10.3.1.33)

[and by implication all higher order terms (Hnp)], we are implicitly ignoring all contributions due to the coupling between the angular velocity terms and the external field. The neglect of the (H2p) contribution which has as a consequence Gaussian behaviour of the angular velocity also requires one to simultaneously neglect d(H1p)/dt as this is on the same time scale as (H2p). Thus, the infinite set of Brinkman equations (10.3.1.31) is now reduced to the three coupled equations:

<r> = (<T»), V(r) = -iHir)l2, / r (H 1 r ) = - / V « - ^ ( r 2 - l > ,

which on solving simultaneously become

The real part of Eq. (10.3.1.34) corresponds to the averaged Langevin

equation (10.2.11) with l(p) = 0 and / Id2p) = kT(p).

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518 The Langevin Equation

We shall now briefly indicate how the set of Brinkman equations (10.3.1.31) may be derived from the Fokker-Planck equation for the distribution function W(&,a>,t) in phase space for fixed axis rotators [here co=d]. As shown by Brinkman [13], the solution of the Fokker-Planck equation, namely,

dW 6W / /Fn . „6W C d

6t 60 I 6(0 I 6(0 (t > 0), is the Fourier-Hermite series:

r 0)W + ——\, (10.3.1.35)

/ 6(0

oo oo

« = 0 ^ = - o o

The Dn are the parabolic cylinder functions (often called the oscillator functions since they arise naturally in the solution of the Schrodinger equation for the harmonic oscillator) of integer order [14]. The parabolic cylinder functions are related to the Hermite polynomials Hn as [12,14]

Dn(y) = 2-»l2e-y2"Hn(y/j2).

Denoting the expected value of an observable A over the time dependent distribution W by

(A)= J J A(d,o))W(0,O),t)dedQ), (10.3.1.36) -oo 0

we have

aq^O~(e^nDn(^rJco)e-^)~^(Hn(w)e-i''0). (10.3.1.37)

Utilising the recurrence and the orthogonality relations of the Weber functions, one can show [2] that the aqn(t) satisfy the Brinkman equations

r]aqn (0 + n/3 \ n (t) = = [aq_u_x (t) - aq+u_x (t)] l<l (10.3.1.38)

-^[(n + l)aq<n+l(t) + aqn^(t)].

These equations govern the time behaviour of the fundamental solution (Green's function). Moreover, with the substitution of

2nnn\aq<nt)~lHnrico)e-l«e),

they reduce to Eq. (10.3.1.31).

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Chapter 10. Inertial Langevin Equations

10.3.2 Matrix continued fraction solution

519

In order to solve the two-index recurrence Eq. (10.3.1.31), we shall use the matrix continued fraction approach. This is accomplished as follows. Let us introduce column vectors

C,(0 =

L0,-2

L 0 , - l

^O.l

L0,2

andC„(0 =

c n- l , -2

Cn-1,-1

Cn-l,0

Cn-\,\

Cn-l,2

, (n>2), (10.3.2.1)

where

cm,q(t) = (Hmr")(t)-(Hmr'>)u,(m>0, q = 0,±l,±Z...). (10.3.2.2)

According to Eq. (10.3.1.31), the cmq(t) satisfy

^^.cn^) = -np'cn<q(t)--^[cnHqt) + 2ncn_^t)\ dt

ingn •[C»-W*i (')-£„_!,,_! (*)]•

(10.3.2.3)

The recurrence Eq. (10.3.2.3) can be transformed into the matrix three-term recurrence equation

V^Cn(t) = Q-nCn_l(t) + QnCn(t) + Q+nCn+l(t), (n>l) , (10.3.2.4)

dt

where C0(f) = 0 and the matrices Q~, Q n , and Q^ are defined by

Q„ = i(n-l)

-4 £, 0 0 0

-<fn - 2 £ , 0 0

0 -<fn 0 £ n 0

0 0 -<fn 2 <fn

0 0 0 - £ n 4

Q „ = - ( / i - l ) £ l ,

, (n>2) , (10.3.2.5)

(10.3.2.6)

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520 The Langevin Equation

Q : = -

• -2

• 0

• 0

• 0

• 0

0

-1

0

0

0

0

0

0

0

0

0

0

0

1

0

0 ••

0 ••

0 ••

0 ••

2 ••

, ( n > l ) , (10.3.2.7)

v J

and I is the unit matrix of infinite dimension. The exceptions are the

matrices Qf and Qj , which are given by

r

<s=i

-4

< 0

0

0

4 -2

-4 0

0

0

0

4 2

-?n

0

0

0

& 4

«~i •• -2 0

•• 0 -1

••0 0

••0 0

0 0 0 -

0 0 0 ••

0 1 0 - -

0 0 2 -

Introducing [77] the one-sided Fourier transform

F(co) = \Ft)e-iwdt (10.3.2.8)

and applying it to Eq. (10.3.2.4), we obtain the matrix recurrence relations

irjOCl(co)-rjCi(0) = QlC2((o), (10.3.2.9)

i7?(0Cn(G)) = QnCn(G)) + Q+nCn+i(<o) + Q-nCn_l((O), (n>2), (10.3.2.10)

where

C,(0) =

/ 2 ( 6 ) / / 0 ( 6 ) - / 2 ( 6 I ) / ' O ( 6 I ) 1^)11^-1^)11^) /1(^ I)/ /0(^ I)-/1(^ I I)/ /0(^ I I)

(10.3.2.11)

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Chapter 10. Inertial Langevin Equations 521

In(x) is the modified Bessel function of the first kind of order n [12], and we have noted that

(ri\ =JL- = Ll™l. (10.3.2.12) \ IN 2n I (P \

0

Here, C„(0) = 0 for n > 2 because

(Hnr")N=0, (n>2), (10.3.2.13)

for the equilibrium Maxwell-Boltzmann distribution functions in the states I and II.

By invoking the general method for solving the matrix recursion Eq. (10.3.2.4) (Chapter 2, Section 2.7.3), we now have the exact solution for the spectrum C^co) in terms of a matrix continued fraction:

Cya>) = ^ _ _ C,(0), (10.3.2.14) irjoA - Ql j Q2

iria*-Q2-Q£-irjoA - Q3 ' •.

Having determined Cx(0) from Eq. (10.3.2.14), we can now calculate the spectrum of the relaxation function ft), Eq. (10.3.1.6), for the transient nonlinear response,

cnAco) + cn Am) /( f l ,) = _2!lU—o!Ziv__/ (10.3.2.15) c0,i(0) + Co,_,(0)

and the integral relaxation time rfrom Eq. (10.3.1.8),

c01(0) + cn ,(0) T = f(Q) = ^U.—Q^1_L_ (10.3.2.16)

CojW + Co CO)

The continued fraction Eq. (10.3.2.14) converges rapidly to some definite limit. As far as practical calculations of the infinite matrix continued fraction under consideration are concerned, we approximate it by some

matrix continued fraction of finite order (by putting Q~,Q^ =0 at some n = N). Simultaneously, we confine the dimensions of the matrices

Q~,Q^ , and Qn to some finite number M. Both of the numbers N and M

depend on the field ($, £n) and damping (/?0 parameters and must be

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522 The Langevin Equation

chosen taking account of the desired degree of accuracy of the calculation.

The evolution of the relaxation time T of the electric polarisation as a function of the dimensionless damping (friction) parameter fi' and the field parameters £i and £n (characterising the strength of the dc field) is illustrated in Figs. 10.3.2.1-4 for various kinds of nonlinear transient responses, viz., step-on, step-off, and rapidly rotating field. Here, one can see that for small ( / ? ' « 1) and large ( / ? ' » 1) friction, the relaxation time rincreases monotonically with /3'~l and with f5', respectively.

kg idJln)

bgioOSO

Figure 10.3.2.1. log10 T/TJ) [Eqs. (10.3.2.14) and (10.3.2.16)] as a function of £„ and

log10 P') for the transient step-on response (£ = 0 —> £n)-

log VJJM)

bgioO?) i

Figure 10.3.2.2. log10(r/7) [Eqs. (10.3.2.14) and (10.3.2.16)] as a function of § and

log10 (/?') for the rapidly rotating field response (£n = - 5).

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Chapter 10. Inertial Langevin Equations 523

1 - | , i • ' » F i , , | , | 1 |

0.01 1 100

Figure 10.3.2.3. r/f] as a function of fi' for various transient responses. Solid lines: Eqs. (10.3.2.14) and (10.3.2.16); filled circles: overdamped limit Eq. (10.3.2.18); stars: Eq. (10.3.2.21) (low damping limit), and crosses : Eq. (10.3.2.23).

H i i 0.01 1 100

Figure 10.3.2.4. T/TJ as a function of fi' for various transient responses. Solid lines: Eqs. (10.3.2.14) and (10.3.2.16); diamonds: Eq. (10.3.2.22).

In order to explain such behaviour, we shall use the nonlinear transient response theory presented in Chapter 2, Section 2.14.1 for an arbitrary system with dynamics governed by a one-dimensional Fokker-Planck equation, viz., an exact analytical equation for the integral relaxation time r of the transient decay function, Eq. (2.14.1.9). Since the relaxational dynamics of planar rotators in the high damping limit fi'» 1) is described by the one-dimensional (Smoluchowski) equation for the probability density function W(6,t) of the orientations of rotators [2]:

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524 The Langevin Equation

2^j,w<-8-')=^W^me, W(0,t), (10.3.2.17)

the relaxation time t for nonlinear transient responses can be calculated for arbitrary values of £i and £n from the analytical equation:

7 _ %D _ IP'T] 2/r

J x¥(0)^(0)

d0, (10.3.2.18) (COS^)II-(COS^)I o Km

where 2/3'rj = C, lkT) = TD is the Debye relaxation time,

^(0) = ](cosx-(cos0)n)w°(x)dx, ^(0) = j(w^(y)-Wl°(y))dy, 0 0

W^(0) (iV = I, II) are the equilibrium distribution functions [which are stationary solutions of Eq. (10.3.2.17)] given by

W°N0) = e^°s9l[2xI^N)\,

and casO)N=IltfN)/I0tfN), (N=l,ll). (10.3.2.19)

According to Eq. (10.3.2.18), the relaxation time is directly proportional to the damping coefficient /? '

In the opposite low damping limit fi'« 1), one may introduce the energy of the dipole

e = TJ202 - £ n cos 0

and the time w (phase) measured along the trajectory in phase space as action-angle variables (see, Chapter 1, Section 1.5.12). The energy varies very slowly with time. Consequently, it may be considered as a slow variable while the phase becomes a fast variable. Here, the dynamics of the system are also described by the one-dimensional Fokker-Planck equation for the probability density function W(e,t) [15]:

dt t]

d_ de ' 2 + -de-

-rj202 W(e,t),(t>0), (10.3.2.20)

where r/202 = e + £n cos 0(e) and the double overbar denotes averaging over the fast phase variable w. Equation (10.3.2.20) was derived by Praestgaard and van Kampen [15] by averaging the underlying Fokker-Planck equation over the fast phase variable (more details of the derivation of Eq. (10.3.2.20) can be found in the recent review of Coffey et al. [16]). Thus, according to Eq. (2.14.1.9) from Section 2.14.1, the

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Chapter 10. Inertial Langevin Equation 525

relaxation time r of the nonlinear transient response in the low damping limit, / ? ' « 1, is given by

T, "f mWle (10.3.2.21) f _ 2/?((cos0)n -(cosfl),) _4 e+^cos^e) < ( f )

where

^ ( f )= J (cos0Oc)-(cos0)n)wj?(jc)dx, -ft

-ft

W°(f) (N = I, II) is the stationary solution of Eq. (10.3.2.20), and

(cos#)w is given by Eq. (10.3.2.19) [equations for cos#(f) and W„(£)

are derived in the Appendix C)]. Here, the relaxation time is inversely proportional to the damping coefficient /? ' Such behaviour of the relaxation time r at fi'« 1 and $'» 1 suggests that T may be approximated for all /?'by the following equation:

*" = rHD+?-LD> (10.3.2.22)

where the subscripts "HD" and "LD" stand for high and low damping, respectively. As is apparent from Fig. 10.2.3.4, Eq. (10.3.2.22) describes the behaviour of r for all ranges of the damping parameter j3', even for /?'= 0 and <fn =0 (free rotation), where the limiting value of Tis

oo

TLD = rFR = J eH"2?1)2dt = T]yf?r . (10.3.2.23) 0

This value is independent of /? ' (see curves 1 in Figs. 10.3.2.3 and 10.3.2.4). Equations (10.3.2.18) and (10.3.2.21) thus provide us with an independent check of the matrix continued fraction solution as well as allowing us to evaluate the relaxation time rfor various particular cases.

The real and imaginary parts of the one-sided Fourier transform of the relaxation function ft) are illustrated in Figs. 10.3.2.5 and 10.3.2.6. Here, it is clearly seen that two processes appear in the spectra of nonlinear transient responses. One Debye-like relaxational process dominates the low frequency part of the spectra and is due to the slow orientational relaxation mode of the dipole in the dc external electric field Fn. For (3''» 1, the characteristic frequency of this low-frequency mode corresponds to the inverse of the relaxation time r. Such behaviour

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526 The Langevin Equation

implies that at long times tlr» 1) the relaxation function/(O may be approximated by a single exponential f(t)« e~tlx with one-sided Fourier transform spectrum given by

/«»)«- (10.3.2.24)

where the relaxation time rmay evaluated from the exact Eq. (10.3.2.16) or from the approximate Eq. (10.3.2.22) (see Fig. 10.3.5.6).

"*-,

Step-off response (£, = 5, P'= 0.2)

.1

Figure 10.3.2.5. Step-off response: Relf(a)/Tj\ and -Im[/(co) I'i\ vs. corj for § = 5,

fi'= 0.2, and various values of §n: 5i = 0 (curve 1); £n = 2 (curve 2); §, = 3 (curve 3); and

§i = 5 - K(K-* 0, linear response; curve 4); filled circles: (10.3.2.26).

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Chapter 10. Inertial Langevin Equations 527

COT]

Figure 10.3.2.6. Step-on response for § = 1 -> £„ = 5: Re[/(a>)/77] and -Im[f (co) / T]~]

vs. COT] for various values of /?": /?'= 10 (curve 1); /?'= 1 (curve 2); and /?'= 0.1 (curve 3);

filled circles: Eq. (10.3.2.24) with tl 77 = 0.475 calculated from Eq. (10.3.2.18).

The most striking feature in the nonlinear response following the step like alteration of the external field is the fact that if the dc external electric field Fn is sufficiently strong, then a pronounced resonant like mode occurs in the high frequency end of the spectrum (see Figs. 10.3.2.5-6). This region corresponds to the THz (far-infrared) band of

frequencies. The pronounced high frequency resonant peaks are due to the fast librations of the dipoles in the strong dc external electric field Fn. For £u » 1 and small damping fi'~ 0), the characteristic frequency of librations ffifc, can be estimated from Eqs. (C3) and (C6) in Appendix C and is given by 0)L ~ T]~lyj2^u (this frequency is weakly dependent upon

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528 The Langevin Equation

t/t]

Figure 10.3.2.7. Step-on transient response for § = 1 —> £n = 5: f(t) as a function of t/T] for various values of ft'. Curve 1: fl'= 10 (overdamped limit); curve 2: /?'= 1; and curve 3: /?'=0.1; filled circles: single exponential decay e'"T with r/ n~ 0.475 calculated from Eq. (10.3.2.18).

/?0- 0 ° increasing /? ' although both processes persist, the amplitude of the high-frequency process decreases progressively. We may thus conclude that a sudden strong alteration in a uniform electric field applied to an assembly of non interacting permanent dipoles will always give rise to pronounced resonant type effects in the far-infrared region. Such behaviour is likely to be of importance in assessing the role played by the reaction field in the dynamical version of the Onsager model for explaining the far-infrared absorption in molecular liquids (see Section 10.6).

The resonant-like behaviour corresponds in the time domain to the oscillations in the rise transient for large fields, which were noted in computer simulations by Evans [17] and explained by Coffey et al. [18]. This oscillatory behaviour is seen in Fig. 10.3.2.7, where the response function f(t) is plotted. Here, the calculation has been carried out numerically (using the Fast Fourier Transform algorithm) from the following equation

f(t) = ^Rc\]eimf(co)dco\.

On the other hand, no resonance behaviour is observed in the step-off response when £n = 0, where the system simply exhibits inertia-corrected behaviour. In the step-off response when & = 0, for all f5\ the relaxation

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Chapter 10. Inertial Langevin Equations 529

function/(0 [Eq. (10.3.1.6)] is independent from ^ and is given by the free rotation diffusion Eq. (3.4.1.7) of Chapter 3, namely,

m = e-kTir2^'"-l+0"\ (,>0), (10.3.2.25) which is in complete agreement with the numerical calculation from the matrix continued fraction solution (see curves 1 in Figs. 10.3.2.5). The one-sided Fourier transform of the relaxation function fit), Eq. (10.3.2.25), may be expressed in terms of Kummer's function Ma,b,z) [12] (cf. Eq. (3.4.1.9) of Chapter 3, Section 3.4)

~fico) = —^—M\\,\ + 2/3'-2i\ + iO)TD),2p'~^- (10.3.2.26) 1 + ionD

L J

The approach developed above allows us to evaluate also the linear response characteristics of the system of planar rotators to infinitesimally small changes in the strength of the strong dc field ¥h i.e. for £n = £ - K, at K —> 0 (see curves 4 in Fig. 10.3.2.5). In these cases, the relaxation function fit) from Eq. (10.3.1.6) coincides with the normalised longitudinal dipole equilibrium correlation function C^(t);

(cos0(O)cos0(f))T -(cos0(O))2

lim f(t) = qit) = X , ^ - ^ -2-4- .(10.3.2.27) *-*° (cos2 0(0)) -(cos0(O))j

Having determined the one-sided Fourier transform of C^(t), one can

calculate the linear dynamic longitudinal susceptibility ^(fi?):

^-^- = C„(0)-ico\ C,it)e~imdt.

The linear response of the system under consideration has been studied in Refs. [19-21]. As far as a comparison with the known numerical solution presented in Ref. [21] is concerned, our continued fraction approach is applicable to fields of much higher amplitudes (up to ^ ~ 30).

In evaluating the transient nonlinear dielectric response we have dealt with two-dimensional linear dipoles. In principle, this simplification is unimportant, however, it allows only a qualitative description of dielectric relaxation. The quantitative theory of linear and nonlinear response of dielectric fluids requires analysis of molecular reorientations in three dimensions. Although including rotation in space causes the theory to be more complicated (as one needs to solve recurrence equations with more than two indices), the approach developed in this section may again be used. In the next two sections, we explain in detail a method of evaluation of the relaxation functions for rotation in space,

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530 The Langevin Equation

which provides the basis for linear as well as nonlinear response theory. For simplicity, we shall treat below only linear response problems for thin rods (linear molecules) and symmetric top molecules (linear response of asymmetric top molecules as well as nonlinear response problems can be analysed in a similar manner).

10.4 Inertial Rotational Brownian Motion of a Thin Rod in Space

10.4.1 Derivation of recurrence equations

The problems involving inertial effects so far have considered rotation about a space-fixed axis. If the axis of rotation is no longer space-fixed (moving axes) then the separation of variables method of solution which arises from the Fokker-Planck equation is rather difficult to use when inertial effects are included [22,23] as many intricate transformations of that equation are required even in the linear response approximation in order to effect that separation. In addition the moving axes inertial response presents new features mainly arising from the nonlinear nature of the Euler equations which do not occur in the corresponding space-fixed axis rotator response. It is again worthwhile therefore to seek an approach based on direct averaging of the Langevin equation. It is the purpose of this section to show how the three-dimensional inertial response may be found by direct averaging of the Langevin equation just as was accomplished above for the plane rotator inertial response. We shall illustrate our procedure for the sake of mathematical simplicity by referring to the rotating needle (linear rotator) model [22,23] of a polar molecule where the angular velocity vector is normal to the dipole vector.

We consider the rotational Brownian motion of a thin rod, or rotator, representing the linear polar molecule (see Chapter 2, Section 2.6.1). In the molecular coordinate system oxyz rigidly connected to the rotator, the angular velocity <o and the angular momentum M of the rotator are defined as in Refs. [20,23]

co = (CDX,O) ,coz) = (t?,0sinz?,0cost?) (10.4.1.1)

and

M = (lcox, l(Oy, 0) = ( M l(p sin A 0), (10.4.1.2)

where / is the moment of inertia, & and (p are the polar and azimuthal angles, respectively (these angles are related to the Euler angles a and /? introduced in Chapter 7 by t?=/?and <p= a+ Jtl2, see Fig. 7.6.1.1). In the absence of external fields, the rotational Brownian motion of the rotator is governed by the vector Euler-Langevin equation [23]

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Chapter 10. Inertial Langevin Equations 531

—M(?) + o(0xM(f) + C<o(t) = k(t), (10.4.1.3) dt

where £(o(t) and X(t) are the frictional and white noise torques due to the Brownian motion of the surroundings, respectively. The white noise torque has the following properties

lfi) = 0, Xj(t)Am(t') = 2kT(8jmSt-t'), (j,m = x,y,z), (10.4.1.4)

where the overbar means the statistical average over an ensemble of rotators that all start at the instant t with the same sharp values of the angular velocity and the orientation. The Aj(t) also satisfy Isserlis's theorem (see Chapter 1, Section 1.3).

We proceed by noting that Eq. (10.4.1.3) rewritten for the vector components in the molecular frame becomes [23]

Iti)x(t) + £cox(t)-10))t) cot tf(t) = Ax(t), (10.4.1.5)

I(by(t) + C<0y(t) + Icox(t)(Oy(t)cotiKt) = Ay(t). (10.4.1.6) Equations (10.4.1.5) and (10.4.1.6) combined with, the definition of the angular velocity components, Eq. (10.4.1.1):

&t) = coxt), <p(t) = (Dy(t)/sm&(t),

constitute a system of nonlinear Stratonovich stochastic differential equations.

We now introduce the functions

fn'm(t) = PT [cost?(f)K [cox(t),coy(t)], (10.4.1.7)

where 0 < m < / ; /, n = 0, 1, 2, ..., P™(z) are the associated Legendre

functions [12] and the functions s™(cox,G)y) are given in terms of finite

series of products of Hermite polynomials Hn(z) in the components

a>x, G)y of the angular velocity as

2m+M Sn [cox,coy) = ± r2X(

M(^H2n_1(!+M r)cox)Hlq(Vcoy). (10.4.1.8) =o q\(n-q)

Here M = 0 or 1, rj = ^II2kT), and the coefficients of the series r2m+M (n> 0) a r e determined by the recurrence relations

r2m("'9) = l " - ? + - ) ( l-^—j-)'2m-l("'^) + ( " - ^ ) ^ 7 Y ' 2 m - l ( " ^ + l)>

r2m+M,q) = (l + -)r2m(n,q)-^-r2m(n,q-l) (10.4.1.9) V ml m

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532 The Langevin Equation

with r0(n,q) = rl(n,q) = l so that

r2(n,q) = n-2q, rin,q)-n-Aq, etc.

The above recurrence relations and the properties [12] of the Hermite

polynomials guarantee that the functions s™(a)x,0) ) are orthogonal, viz.,

j" J s^(ax,a>y)s^(ajx,ioy)e~t' x+ y]daxd(oy~Snn,8mm,, —oo —oo

and that they form a complete set in angular velocity space. We desire an infinite hierarchy of differential-recurrence

equations for the averaged values of f„'m(t) over its realisations in phase space (here configuration - angular velocity space). This is accomplished by evaluating

^77^77 ,. Pr[cos^t+T)]s^\o)x(t+T),coy(t+T)\-F>r(c^t?)s:(cox,coy)

&x

dt *->o T (10.4.1.10)

where p t+T t+T I t+T

(t + T) = CQx-?-\ (Ox (t')dt' + j (O) ( 0 COt Wt')dt' + - J Xx (t')dt', t I t

p t+T t+T -• t+T

6)yt + r) = coy-^\ coyt')dt'- j CQxt')coyt')cot$t')dt'+- j Ay(t')dt' t t t

(10.4.1.11) are integral forms of Eqs. (10.4.1.5) and (10.4.1.6). We remark that as usual the time t is assumed of such short duration that the angular velocities do not significantly alter during rand neither does any external conservative torque. Nevertheless r is supposed sufficiently long that the chance that the rapidly fluctuating stochastic torques Aj(t) take on a given value at time t + x is independent of the value, which that torque possessed at time t. As before, here z?, cox, and a>y and &(t), cox(t), and 0)y(t) in Eqs. (10.4.1.10) and (10.4.1.11) have different meanings, namely, &(t), 0)x(t), and coy(t) are stochastic variables (processes) while &, cox, and coy are the sharp values at time t (recall that the time ris infinitesimally small).

In order to derive the hierarchy of equations for average values of

fln'

mt), we first note that

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Chapter 10. Inertial Langevin Equations 533

L /'.- (r) = Pp L s»+s™ L p»>. dt " dt n n dt l

For m = 0, we can evaluate the two terms in the right-hand side of the above equation as follows:

dt sQ

n[cox(t),(Oyt)]—Pl [cos0(0]

= ax (t)s°n[a)x (t), toy (t)]Pl [cos z?(?)]

i-p/tcos^WZ-^ „ 2rj £>q\(n-q)\ -2q+l ( f) + 4 ( " ~ l)H2n-2q-\

= ±[sln(<ox,e>y) + 4s1

n_l(a>x,a>y)]Pl(cos&) (10.4.1.12)

and

/[a*t*r)]4#fil(0,ai(0]

?=o<?!("-^)!L J

1 q^qKn-qV- V (10.4.1.13)

where

HJH(t) = Hn[jja>j(t)] (j = x,y)

and 0)x(t) and 6^(0 are given by Eqs. (10.4.1.5) and (10.4.1.6). In

order to simplify Eqs. (10.4.1.12) and (10.4.1.13), we have used Eq. (10.3.1.28), the identity [12]

— P, (cost?) = P/(cos 0 ) , (10.4.1.14)

and the relations from the Stratonovich calculus

4 WW), a>j (t)Wn [m (t)] = (Cn/Tj)F(#, (Oj )Hn_x (TJO), ), (10.4.1.15)

where F is an arbitrary function and ij -x,y (i &j). Equation (10.4.1.15) follows from Isserlis's theorem and is proved in Appendices A and B (originally this equation was derived for n = 1 and F = 1 in Ref. [49]). On combining Eqs. (10.4.1.12) and (10.4.1.13), we obtain

n7F=-wyF+2fi5+7F'2- ao.4.1.16)

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534 The Langevin Equation

In like manner, we derive for m > 1 and M = 0, 1:

s^+M[a>x(t),ay(m^P^+M[COsiKt)]

= a>x(t)stm+M[cox(t),6)y(t)]^P?m+M[coSm]

= A-[p?m+M+l(Cos0) - (/ + 2m + M)(l - 2m + M - \)p^+M~l(C0St?)]

x ^ W ( n g ) ^ ^ + 2(2n _2q + M ] ^

(10.4.1.17) and

P?m+M [ c o s i X O ] s2nm+M [cox (t), coy ( 0 1

= - • £ - (2n + M)P?m+M (cos z?)5„2m+M [ffl., <ov ] *7

! r/fm+M+\cosz?) + (/ + 2m + M)(/-2m+M-l)/fm + w~1(costf)l 4(2m + M)^L J

^ (2n-2q + M)r2m+M(n,q)-2(n-q)r2m+M(n,q + l) llX r ,on„,u„,l XL q\(n-q)\ Hm-2q+M-x \_H2q+2 + 2(2q + l)H2q j .

(10.4.1.18) Here, Eqs. (10.3.1.28), (10.4.1.15), and the following relations of the associated Legendre functions have been used [12,24]:

2^-Plm=Pl

m+l-(l + m)(l-m + l)Plm-1, ( m > l ) , (10.4.1.19)

2incot#P/B=-P/"+ 1-(/ + m)(/-ro + l)Jf~1, ( m > l ) , (10.4.1.20)

(2/ + l)sin^P/m=P,m,+1-P,"!

1+1

1 >~l M (10.4.1.21) = (l-m +1)(/ - m + 2)P™~1 -(l + m-1)(/ + m)P^.

Noting Eqs. (10.4.1.17)-(10.4.1.18), we have

dt _ (10.4.1.22)

a i i 1\// o i i \ j2m-l,-l rl,2m-2 . j2m-l,-l rl,2m-2

+ 2m- l ) ( / - 2m + 2) dn+1 /n+1 + d„>0 /„ and

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Chapter 10. Inertial Langevin Equations 535

.2m,+1 rl,2m+\ _ " rl,2-m r\ o'rl,lm . L rt,lm+i . jZm,+l rl,ln n — fn =-2nPfn +^fn +dn,-l fn-l

-(I + 2m)(l -2m + l) j2m,-l rl,2m-l , j2m,-l rl,2m-\ dnfl fn +dn,-\ t n-\

(10.4.1.23)

where

j2m,-l r2m(n'<l) _,_ 1 r2m(n,q-l) _n-m + \

m Jhm-M^) n

-+ m rn lm-\ (n,q)

d„ n =\n-q + ln,0

f 1 +

V 2m

2q + l '2m-\ (n,q)

'2m-2 (n,q)

(n-g)(2q + l)r2m_l(n,q + l) _

2m-I r2m-2(n><l) n + m,

j2m,±l) V - l 2m

'2m (n,q) +2q + l r2m(n,q + l) r2m±l(n-l'l') 2 m r 2 m ± l ( " - ! ' < ? )

d2m-u±D =n-q + \)(l± 2q \ r2m_xn,q) nM 4 -2m-l)r2m_l±l(n + l,

(2n-2q + 3)q r2m_x(n,q-\)

\n + m

+

2m-l±l

4(2m -1) r2m_1±l (n + l,q) 4[n-m + 2

q)

Thus, Eqs. (10.4.1.22) and (10.4.1.23) yield d ^fiM-M =<2n+M)Pfl

n:lm-M

at r rl,2m+\-M ,± fl,2m+\-M

+ Jn-l+M + . Jn+M

-Q + 2m-M)l-2m+\+M) ^n-m+l + M)f^-l-M Hn+m)fln^

M

(10.4.1.24) where m > 1 and M = 0 or 1, and fi'-^rj/1 as for planar rotators.

Equations (10.4.1.16) and (10.4.1.24) are the required hierarchy

of differential-recurrence relations. All the f^'m in Eqs. (10.4.1.16) and

(10.4.1.24) are functions of the sharp values # o^, and ax?, which are themselves random variables with the probability density function W. Therefore, in order to obtain equations for the moments, which govern the relaxation dynamics of the system, we must also average Eqs. (10.4.1.16) and (10.4.1.24) over W. However, if a system is in equilibrium (as in the present problem) all such averages are either constant or zero indicating that one must first construct from Eqs. (10.4.1.16) and (10.4.1.24) a hierarchy of equations for the appropriate equilibrium correlation functions.

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536 The Langevin Equation

As far as the majority of applications is concerned, the quantities of interest are the equilibrium orientational correlation functions Ct (t) for the Legendre polynomials Pi defined as (in a physical system, these are the observables rather than the sharp averages)

Q (t) = (P, [cos tf(0)]P, [cos tf(r)]) • (10.4.1.25) These characterise the orientational relaxation in liquids (the angular brackets denote the equilibrium ensemble averages). Having determined the correlation function Q(t), one can also evaluate the corresponding orientational correlation time T; defined as the area under the normalised correlation function Cz(f)/C/(0), viz.,

1 °° T,= f CAt)dt. (10.4.1.26)

Q(0) J0

By way of illustration, we shall obtain below exact analytical solutions in terms of ordinary continued fractions for the one-sided Fourier transforms of the first and second order equilibrium orientational correlation functions:

Q (t) = Pl [cos tf(0)]P, [cos tf(f)]) (10.4.1.27)

and

C2(0 = (P2[cos^0)]/52[cosz?(0]) • (10.4.1.28)

(These correlation functions are used for the interpretation of dielectric and infrared absorption and Raman and Rayleigh scattering measurements [25].) We also propose a general method for the evaluation of the spectra of higher order correlation functions in terms of matrix continued fractions.

10.4.2 Evaluation of Cx

One can readily derive differential-recurrence equations for the correlation function for the first Legendre polynomial (/ = 1)

ci'm(0 = (cost?(0)7Pw) [so that ^ ( O - q C f ) ]

by multiplying Eqs. (10.4.1.16) and (10.4.1.24) by cost?(0) and by averaging the equations so obtained over the equilibrium distribution function W0 at the instant t = 0. Using Laplace transformation, these equations can be written as the recurrence relations, namely,

(rjs + 2n/3')cl;\s)-2cl-l_l(s)-cln'

l(s)/2 = Sn0rjc^(0), (10.4.2.1)

[7js + (2n + l)/3']cln\s) + 2(n + \)clr;

0(s) + (n + l)cl;°l(s)/2^0. (10.4.2.2)

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Chapter 10. Inertial Langevin Equations 537

Here, we note that all the clf(0) vanish with the exception of n = 0, viz.,

CQ'°(0) = 1/3 (this follows from the orthogonality property of the associated Legendre functions [12]).

The solution of Eqs. (10.4.2.1) and (10.4.2.2) may be obtained as a scalar continued fraction as follows. First of all, Eqs. (10.4.2.1) and (10.4.2.2) can be rearranged as a three-term recurrence equation:

[Tjs -qn(s)]cl-°(s) - qlisYcZ(s) - q ^ c ^ s ) = S ^ 0 (0), (10.4.2.3) where

-, x 4n + , , (n + l)/4 .._ . „ .,

g - ( f ) g - ( 2 n - l ) f + ^ ^ - ( i n + s * ( 1 0 A 2 4 )

q(s) = —f 2nj3' ^ . (10.4.2.5) 2n-\)p +J]s (2n + l)0' + rjs

Thus, the solution of Eq. (10.4.2.3) is the infinite continued fraction C1Q°(S) _ TJ

C°0(0) ,S-qo(s) *W> Tjs-qM- ql(S)q~2(S)

7]s-q2(s)-

or

ri-°rm ...... 1 l x_ <*°(0) , - f l B 7 + _ l P'+ian p'+iconf i 2 ( 2

- + 2j3'+iaxj+—; —— x p +ian; ' — " '-—

On using the equality (10.4.2.6)

n n2 1

A A2 B + nIA A + n/B Eq. (10.4.2.6) can be further rearranged to yield

h°(()-) ...... 1 co' (°) icor+ P+ICOJ+-

2/?+i(W]+-3/?+icor+

4p+ictJi]+ 5j3'+icor+...

(10.4.2.7)

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538 The Langevin Equation

Having determined the spectrum C^ico) = Co°(iaJ), one can evaluate the

orientational correlation time Zi defined by Eq. (10.4.1.26) for / = 1, viz.,

r, = — ^ f C, (t)dt = Z^P_ (10.4.2.8) Q(0)J

0 4°(0) and the complex susceptibility xi.®) = XX®) - iz"(®>) • This i s given by linear response theory as

4£!)=1_^^), (1o.4.2.9)

where xXO) = N0//2 l^kT is the static susceptibility, N0 is the number of dipoles per unit volume, and fi is the dipole moment of a molecule. Equations (10.4.2.7) and (10.4.2.9) can be reduced (after appropriate changes of parameters) to the result of Sack [22] for the complex dielectric susceptibility.

In the high damping limit (J3'» 1), the inertia-corrected (Rocard) equation for the complex susceptibility [22,25] is recovered fromEqs. (10.4.2.7)-(10.4.2.9),

(10.4.2.10)

(10.4.2.11)

XXO) l + iG)TD-J]2G)2 '

along with the following Taylor series expansion for zi: 2 2 2 3 11 4 5 ,

T1=TD \+y--f +-r-—?' +-r+• Here TD = 7]/3' = £/2kT is the Debye relaxation time and y= \l(2/3'2) is the inertial (Sack's [22]) parameter. Equation (10.4.2.10) will be a good approximation to the continued fraction solution for ^<0.05 [22]. The small value of y (or, equivalently, the large value of P) indicates that equilibrium of the angular velocities is almost attained before a dipole has time to change its direction appreciably. For larger values of y, the higher order terms in the continued fractions became progressively more important, but the classical model is then no longer a good approximation in view of discrete spacing of rotational levels and a quantum-mechanical treatment must be used [19]. Equation (10.4.2.11) is in complete agreement with the results of Refs. [23] and [26]. The high damping limit is the case of greatest interest in the explanation of dielectric relaxation data of molecular liquids. For example, for liquid chloroform at 25 °C

/ = 2.7xl0"38gcm2 ,TD=6.4xl0"12s, p i = 1.9xlOI3s_1

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Chapter 10. Inertial Langevin Equations 539

leading to y ~ 4xl(T3. Experimental measurements [27] by Vij and Hufnagel for dilute solutions of polar liquids (acetone in very dilute solutions of cyclohexane, acetonitrile, and methyl iodide) indicate that the magnitude of the excess absorption over the Debye plateau is very much reduced in comparison to that for the neat liquid and that the return to transparency at high frequencies is governed by the Rocard equation (10.4.2.10). If either of these equations are used to model the absorption spectrum of neat polar fluids with physically meaningful values of the parameters, it is found that the return to transparency at high frequencies is too slow in comparison with the observed high frequency part of the far-infrared spectrum; furthermore the theory cannot explain the excess (Poley) absorption [25] in the far-infrared region, (see Section 10.6).

10.4.3 Evaluation of C2

In like manner, one may obtain the system of recurrence equations for the corresponding correlation functions for the second Legendre polynomial

cl'm(t) = (p2[cosM0)rtm(t)) so that c2fi (t) = C2(t)

(the spectrum of C2(t) pertains to Raman and Rayleigh scattering [25]).

By multiplying Eqs. (10.4.1.16) and (10.4.1.24) by P2[cos*?(0)] and by averaging the equations so obtained over the equilibrium distribution function Wo at the instant t = 0, we have for / = 2

(TJS + 2n/3') c2n°(s) - 2c2

nti (s) - c2-1 (s)/2 = Tjc2fl(0)Sn0, (10.4.3.1)

[sri + (In + 1)/T] c2'1 (s) + 3(n +l)cn2;0, (s) 12

(10.4.3.2) +6(/i + l ) c f ( s ) - c2'2(s) - c2

n'2(s)/4 = 0,

(s7j + 2nfi')c2-2(s) + nc2'\s) + 4(n + l)c2'1(s) = 0. (10.4.3.3)

Here, we note that all the c2m(0) vanish for any n and m save n = 0 and

m = 0,viz., CQ'°(0) = 1 / 5 .

Just as the first order response, Eqs. (10.4.3.1)-(10.4.3.3) can be

rearranged [by eliminating c2'°(s) and c2'2(s)] as a three-term

recurrence equation for c2'\s), which has an exact continued fraction

solution for CQ'1(S) . This exact solution in terms of an infinite continued fraction combined with the relation

Sc2'°(s) = c2

0'°(0) + c2'\s)/(27])

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540 The Langevin Equation

[that is, Eq. (10.4.3.1) for n = 0] yields

£ (s) _ V c2o'°(0) TJS + -

TJS + /3' + A T/S + 2F h

Tjs + a2--]h

Tjs + a3-(10.4.3.4)

where

and

an=(2n + \)/3 +

b=-

2nP'+T]s 2n + \)P' + r]s

16(n + l)(n + 2)

[2(n + l)j3' + Tjsf

Having determined the spectrum of the correlation function

C2(<y) = Co'°(z'a>)/co'0(0), one may evaluate the orientational correlation

time Tz, viz., r2=c0

2'°(0)/c02'°(0). (10.4.3.5)

In the high damping limit ( / ? ' » 1), Eqs. (10.4.3.4) and (10.4.3.5) yield the Taylor series expansion for ^:

(10.4.3.6) 2 3

, _ 32 , 368 , 23464 4 l + 5 r r + — r 7 +• 3 9 135

Equation (10.4.3.6) is in agreement with the results of Refs. [23,26].

10.4.4 Evaluation of Cl for an arbitrary I

In order to proceed, we must derive differential-recurrence equations for the equilibrium correlation function

c'nm(t) = (p/[cosz?(0)]/n2'm(0) so that c£°(0 = Ct(t).

This is accomplished by multiplying Eqs. (10.4.1.16) and (10.4.1.24) by Pz[cost?(0)] and averaging the equations so obtained over the equilibrium distribution function W0 at the instant t = 0. In order to evaluate the Z* order equilibrium orientation correlation function Ct(t), the Z+l independent Eqs. (10.4.1.16) and (10.4.1.24) must be considered. Hence, it is simpler to represent the solution in terms of matrix continued fractions. In order to solve the hierarchy of moment

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Chapter 10. Inertial Langevin Equations 541

equations so obtained, we introduce a column vector C„ (t) comprising of / + 1 elements:

fJ.o

C„(0 =

(0 C'n-l(t)

1,1 Vn-1

l,m

c-l.(0y

Then the hierarchy of equations for c„(t) can be transformed into the matrix three-term differential-recurrence equation

rJCn(t) = Q-nCn_l(t) + QnCn(t) + Q:Cn+l(t), (n>l ) , . (10.4.4.1)

where C0(t) = 0 and Q„, Q*,and Q~ are (/+ l)x(/+ 1) square matrices with elements to be determined from Eqs. (10.4.1.16) and (10.4.1.24). The initial conditions are given by

QCO): 0

v

and C„(0) = 0 for all n>2,

where C/(0) = (2/ + l) ' . On taking the Laplace transform of Eq.

(10.4.4.1), we have the matrix three-term recurrence relation

(TjsI-Qn)Cn(s)-Q+nCn+l(s)-Q-nCn_l(s) = SnlrjCl(.0). (10.4.4.2)

The exact solution for the Laplace transform C, (s) is then given by the matrix continued fraction

c, (*)=//-7 ^ - Q i - Q r

I -C,(0), (10.4.4.3)

Tsl-Q2-Q2 I

77*I-Q3 -. where I is the unit matrix and the fraction lines denote matrix inversion.

For example, explicit solutions for / = 1 and 1-2 can be given as follows. For /= 1, the scalar recurrence relations (10.4.2.1) and (10.4.2.2) may be recast as the matrix recurrence relations (10.4.4.2), where

Cn(s) = cn-l

1,1 V-n-1

(S)

cZ, (s) C,(0) =

C,(0)

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542 The Langevin Equation

Qn = (o 2

o o Q„ = Q:= ( 0

•nil

o^ 0

-2(n-l)/T 1/2

-2n -(2n-V)p'j For 1 = 2, the scalar recurrence Eqs. (10.4.3.1)-(10.4.3.3) can be recast as the matrix three-term recurrence (10.4.4.2), where this time

-n-l

Cn(s) =

(s)

C,(0) =

(C2(0)\

0

I o J (0

Qn =

Qn =

2

0

-An

2p\n-V)

-6n

0

Q:=

0

-3n/2

0

0 *

1/4

0

0

0

0

1/2 0 '

-p\2n-X) 1

- ( n - l ) -2fi'(n-l)/

In both cases (/ = 1 and / = 2), the matrix continued fraction solutions are rendered by Eq. (10.4.4.3). The calculation shows that the matrix continued fraction solutions given by Eq. (10.4.4.3) and the ordinary continued fraction solutions Eqs. (10.4.2.7) and (10.4.3.4) coincide [both (ordinary and matrix) continued fraction solutions can be used as an independent check of calculations].

The real parts of Q and C2 vs. logw(a>?j) and log10(/f) are shown in Fig. 10.4.4.1. Two limiting cases may be used to check the numerical calculations. In the low damping limit (/?'—> 0), the C, id)) reduce to those of the free rotation model [28]:

C[R(ia>) _ 1

G(0)

where

ico '0,0

2,

-.2

- 2 1 m=l

d0,m 2,

(Onlmfe^'^E^-WIm)2

(10.4.4.4)

£,(z) = j exp(-f)

dt

is the first order exponential integral function [12] and the dlM M'(&) are

given in Chapter 7, Section 7.6.1. For / = 1 and / = 2, Eq. (10.4.4.4) yields [28]

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Chapter 10. Inertial Langevin Equations

C™(ifi» = We-^E, ( -7 /V)

543

(10.4.4.5)

and

£ f (/©) = — i - + i^-(on2e-^IAEA-7V/4). (10.4.4.6)

40) 16 K ' In the opposite (high damping) limit, yff'—» °°, the form of the spectra Q becomes Lorentzian (see Fig. 10.4.4.1) and coincides with that predicted by the Debye model of noninertial rotational diffusion [23]:

C?(ia>)_ 1

Q(0) ico+l(l + l)/(2TD) (10.4.4.7)

f ' ' 2

bgi0Re[C2]-:

k>gio(j8')

•og I O ^ ) 0

Figure 10.4.4.1 log10Re[C!] and log10Re[c2] vs. log10(coTj) and log,0(yS').

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544 The Langevin Equation

where rD = £l2kT) is the orientational correlation time for the Debye model [23].

Thus, using the Langevin equation method, we have obtained exact analytical solutions (in terms of ordinary continued fractions) for the one-sided Fourier transforms of the first, Ci(t), and the second, C2(t), order equilibrium orientational correlation functions for the inertial Brownian motion of a linear molecule freely rotating in space. Moreover, we have proposed a general method for the calculation of higher order correlation functions Ctt) for any / in terms of matrix continued fractions. The distinct advantage that the dynamical (Langevin) method is that it avoids the derivation of the Fokker-Planck equation for the distribution function W(&,6)x,a> ,t), which for linear molecules in space

is [22]

dW

dt - + m

dW + 0)v COt &

dtf y (o„

V

dW

da)r cor

dW

d(0, y J

dd)r

(OxW + kT dW

+ •

X J dd)„ 0)yW +

kT dW

I da y )

(10.4.4.8)

We shall show now that the same method may also be applied to the rotating sphere and symmetric top models, the only difference being that it is more tedious to evaluate the Hermite polynomial averages as they now contain the z component of the angular velocity, i.e., coz.

10.5 Rotational Brownian Motion of a Symmetrical Top

10.5.1 Derivation of recurrence equations

In the molecular coordinate system oxyz rigidly connected to the top, the angular velocity co and the angular momentum M are defined as [22,23]

<o = (G)x,0)y ,coz) = (&,<p sin &,i/r + <p cos &) (10.5.1.1)

and

M = (lO)x,IO)y,Izo)z), (10.5.1.2)

where / and Iz are the components of the moment of inertia tensor, the angles •& and <p are defined in Section 10.4.1 and y/ is the angle characterising rotation about the axis of symmetry of the top. In the absence of external fields, the rotational Brownian motion of the top is governed by the vector Euler-Langevin equation (10.4.1.3), which rewritten for the vector components in the molecular frame becomes

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Chapter 10. Inertial Langevin Equations 545

Icbx(t) = -6)y(t)[lzO)z(t)-Icoy(t)cot^(t)]-Ccox(t) + Ax(t), (10.5.1.3)

Icby(t) = a)x(t)[lzcoz(t)-Iay(t)cotm)]-Co>y(t) + Ay(t), (10.5.1.4)

Izd)z(t) = -Czooz(t) + Az(t). (10.5.1.5) Equations (10.5.1.3)-(10.5.1.5) combined with

•dt) = (Oxt), <p(t) = a)y(t)/smi9(t), ii/t) = coz(t)-coyt)coti%t) (10.5.1.6)

[which follows from the definition of the angular velocity components Eq. (10.5.1.1)], constitute a system of nonlinear stochastic differential equations.

We now introduce the functions [cf. Eq. (10.4.1.7)]

/ , a (0 = P\m] [cos*?(0K* [cox(t),coy(t),(Oz(t)] , (10.5.1.7)

where l,n,k = 0, 1, 2, ...; - l < m < l, 2m-M I,. „ , . \ Sn,k K ' ^ ' ^ J

-Hk (W)± h:ZnftH2n_2q+M_em (r)cox)H2q+£m (na,y). (10.5.1.8)

' , " q\(n-q)\

Here Tj = Jl/(2kT), TJZ = Jlz l(2kT), £m=0 for m>0 , em=l for

m < 0, M - 0 or 1 and the coefficients r2m+M (n,q) are determined by the recurrence relations

hm (»,q) = n-q-\— V 2.

f x 2^ + 1

V 2m-1 2m - 1

r2m+i(n,q) = l + q/m)r2m(n,q)-(q/m)r2m(n,q-l),

r-2m(n> <?) = (n-q) ^ 2m-\)~ (2m-l)(w.?) + ^ — r r - ( 2 m - l ) ( n ^ + 1)

,'-(2m+i) («, ?) = | 1 + - | ^ - 1 r_2m (n'^~J^ f

2m-\

1

ra-g + 1

\ r-imin,q-\),

with m > 0 and ro(n,<30 = /-±1(/i,9) = l .

The orthogonality properties of the functions s^. (a)x,a)y,eoz), viz.,

f J J slkcox,<»rcoz)sik\a>x,a>^

result from the orthogonality of the Hermite polynomials and the recurrence equations for rm (n, q).

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546 The Langevin Equation

We desire the hierarchy equations for the averaged values of

/„''"(?) over its realisations in phase space. In order to derive this

hierarchy, we first note that

dt n'k ' dt n'k n'k dt l

We can evaluate the two terms in the right-hand side of the above equation for arbitrary / and m = 0 as follows:

*l A Pi = G>x (t)s°n k K (0, G)y (t), COz (tWl [COS t?(f)] dt

1 = ^ P i H k t ^ ! AHX

ln-2^+^n-q)H*2n_lq_x)Hlq 2rj q=0q\n-q)\

= (1/2^/(4+4^) (10.5.1.9)

and

P, [cos i%m4$k K (0,0)y (0, coz t)] = P, [cos (OlZ , / 7 ., dt 0q\(n-q)\

(n - q)H*n_lq_xt)Hy2qt)Wk t)d)x(t) + qHx

2n_2qt)H?q_x(t)Hzk (t)d)y (/)

+ ^ ^ 2 , - 2 , ( W , t)HU it)a>z (0

= P, [COS flOlZ „4 ? 7 J ( » - 9 )^ -2 , -1 ( 0 ^ J", (0

-fl^(0 -fi>7 (0-«v0)cot*? <f 1 ( 0 —^(0 + / x I

+qHx2n_2qt)Hy2q_xt) coAi) -^-a)At)-coAt)coi& - 7 ^ ( 0 + - ^

^mn-laitWUmUit) 2/7

A«(0+M)

= -(2nj3 + kj3z)Plslk, (10.5.1.10)

where H^(t) = Hn[t](Oj(t)], (j = x,y,z), and <&,(*), ^ ( 0 , and <&,(/)

are given by Eqs. (10.5.1.3)-(10.5.1.5). In order to simplify Eqs.

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Chapter 10. Inertial Langevin Equations 547

(10.5.1.9) and (10.5.1.10), we have used Eqs. (10.3.1.28), (10.4.1.19)-(10.4.1.21), and the relations from Stratonovich calculus

Cn ^(0F[^0,©/0 ,^(0W„[^(0]=—^(«? , f l ) ! / , f l i )^ l , - 1 (^) , (10.5.1.11)

where F is an arbitrary function and i,j,k-x,y,z (i&j^k). On combining Eqs. (10.5.1.9) and (10.5.1.10), we obtain

d VTf%=-[2»P+W'z'B]f$ + 2f&Jc+f%/2, (10.5.1.12)

at where fi = J]£ 11, J3'z = 7]CZ II, and B = lz II.

In like manner, we have for m & 0

slTM[coMo>y(t),coz(t)]^lj2m-%osm] at

=—[^2m"M|+1(cost?)-(/+|2m-M|)(/ + l-|2m-M|)^2m-Ml-1(cos^)]

(10.5.1.13) and

/ j 2 m - M l [ cos^ ) ] | ^ - M [^ (0 ,« /0 ,^ (0 ]=- [ (2«+M) / 5 + ^]^ 2 m " W | ^ 2 r M

" n-q)2+£m/(q + l))r2m_M(n,q+l)-2n-2q+M-em)r2m_M(n,q)

£> q\n-q)\

xj l _ [/j2m-Ml+1(cos^+(/+|2m-M|)(/+l-|2m-M|)/j2m"M|"'(cos^

X ^2n-2 9 +A/ - l -£ m

7 277 U H2n-2q+M-l-£m

H2q+H-£m [Hk+l + 2^t-l) f-'2

(10.5.1.14) Here Eqs. (10.3.1.29), (10.4.1.15), (10.4.1.21)-(10.4.1.23), and (10.5.1.11) have been used. Noting Eqs. (10.5.1.13) and (10.5.1.14), we have

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548 The Langevin Equation

vjt&2m'M) =-[(2«+M)/r+^/B]e,"+ffl1-M)

+j/«f iT 1 " M ) - ( 1 -^(2 m -M), - 1 ) (^2m-M)( / -2m+l + M)

(n + ni\ f'.±(2'»-l-M) (n-m + l + M) l,±(2m-l-M) (10.5.1.15)

,2" where m>0 . For linear molecules (Iz = 0, f3[ = 0), Eqs. (10.5.1.12) and (10.5.1.15) yield Eqs. (10.4.1.16) and (10.4.1.24) of Section 10.4. Equations (10.5.1.12) and (10.5.1.15) are the required hierarchy of differential-recurrence relations. By way of illustration, we shall obtain exact solutions of this hierarchy in terms of matrix continued fractions for the spectra of the first and second order equilibrium orientational correlation functions, Eqs. (10.4.1.27) and (10.4.1.28).

10.5.2 Evaluation of Cx and C2

Just as linear molecules, one may readily derive differential-recurrence equations for the equilibrium correlation function for the first Legendre polynomial (/= 1)

c]$(t) = ( c o s ^ O ) / ^ ( 0 ) [so that c$(f) s Cxt) ]

by multiplying Eqs. (10.5.1.12) and (10.5.1.15) by costf(O) and by averaging the equations so obtained over the equilibrium distribution function W0 at the instant / = 0. These equations can be written as a system of algebraic recurrence relations in the frequency domain using Laplace transformation, namely,

[rjs + 2nP' + k/3'z /B]c% = rjcl^(0)Sn+k0 + 2 c " u + c^k 12 , (10.5.2.1)

\jis + 2n + l)P' + kP'zIB\c^ ^7 , (10.5.2.2)

= -(l/2)(n + l)cif1Jt -2(n + l )#° -4B(cln;k\l+2kcl'Jc

l_l)/2,

[ ^ + (2n + l)yff' + M ; / f l ] c ^ 1 = > / f i ( c ^ + 1 + 2 ^ _ 1 ) / 2 . (10.5.2.3)

Here, we note that all the c*'° (0) vanish with the exception of n + k - 0,

viz., CQ'O(O) = 1/3. Equations (10.5.2.1)-(10.5.2.3) can be reduced to Eqs.

(3.18a)-(3.18d) of Morita [29] (obtained from the Fokker-Planck

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Chapter 10. Inertial Langevin Equations 549

equation) by introducing new functions Aa'•**«,*' and Cnk (in Morita's

notation), viz.,

Ai,* ~" (-1>

n+k

n\2An 2k)\\rfnr)\ ;l,0

,2/i.Jfc Ln,k '

B. nJc (-D n+k

[cn,kj (n + l)\2An+22k)UTfn+1rfz

Ln,k

cn,k

In order to solve the hierarchy of recurrence Eqs. (10.5.2.1)-(10.5.2.3), we introduce a supercolumn vector C„ (t):

C„(0 = ' -n- l

,1.-1 " n - l

(t)

(0

( l,m Cn,0

comprising 3 subvectors cn'm (?) =

(0

#i . i<0

* 0 The subvectors c|!'

m(0 have the dimension n + 1 . The three-index recurrence Eqs. (10.5.2.1)-(10.5.2.3) for cx

nm

k(f) can then be transformed into a matrix three-term differential-recurrence equation:

^C„(0 = Q;Cn_1(0 + QnC„(0 + Q:Cn+](0., ( n > l ) , (10.5.2.4)

where C0(0 = 0, the matrices Qn, Q^, and Q~ are given by

Q:=

0 q~

0 0

o P ;

0

-p«

0

, Q : =

0 0

0

o P :

0

-P:

0

Qn = 2(r„-I„)

0

V2 oi 0

and the submatrices p*, q*, rn , and q^ are defined as

0^

0

0

<=-

n 0

0 n - l

0 ti\

0 0

P„"=VB

V

ro i

0

0

0 0

0

0

n-l

0

1 0

0 ^

0

0

n-l

*q„~=2

/«x(n+l)

f\ o

0 1

0 0

y0 0 ' 0 1 •

+ 4B 0 0

0 0

,(10.5.2.5)

/nx(n-l)

0 0\

1 0

0 1

,(10.5.2.6)

JroinA) /nx(n+l)

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550 The Langevin Equation

rn-\ 0

0 n-2

rfM(n -1,0)

0 0

0

/ " ( n - 2 , 1 )

0^ 0

0

(10.5.2.7)

'row

0

0

•M fM(0,n-l))

(10.5.2.8)

Here

fM(n,k) = -(2n + M)j3'-kj3'z/B

and I„ is the unit matrix which has dimension nxn. On taking the Laplace transform of Eq. (10.5.2.4), we have the

matrix three-term recurrence relation

(^l3„-Q„)C I I (*)-Q:CB + 1 ( i ) -Q;CB_ 1 ( J ) = (yi,17C1(0), (10.5.2.9)

where the initial conditions are '1 /3^

C,(0) = 0 and C„(0) = 0 for all n> 2.

The exact solution for the Laplace transform C, (s) is then given by the matrix continued fraction

C,(s) = /7 I, -C,(0). I ~1V

^ i 3 - Q , -Qi - 6-—f Q2 J?sl6-Q2-Q2—^—Q-

7]sl9-Q3 •. (10.5.2.10)

For linear (7Z = 0) and spherical top (7 = Iz) molecules, the results of numerical calculations based on Eq. (10.5.2.10) were compared with those of Sack [22] (Sack obtained the solutions in terms of ordinary continued fractions). For linear molecules, this is Eq. (10.4.2.7). For spherical tops, Sack's continued fraction solution [22] is (in our notation):

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q(*) = STJ +

Chapter 10. Inertial Langevin Equation 551

77/3

1 5/4

sn+p - ««> + 2ft S« + 2P' + -,a> l 7 / 3

STJ + 3P +-6STJ + 4P') STJ + 4P' + ...

(10.5.2.11) [Eq. (10.5.2.11) is derived in Appendix D]. The numerical calculation indicates that the matrix Eq. (10.5.2.10) and ordinary continued fraction solutions, Eqs. (10.4.2.7) and (10.5.2.11), yield the same results.

In like manner, we can derive differential-recurrence equations for the equilibrium correlation function

<£T(0 = (p/[cost?(0)]/n2f(0) so that c$(r ) = C2(t).

This is accomplished by multiplying Eqs. (10.5.1.12) and (10.5.1.15) by P2[costf(0)] and averaging the equations so obtained over the equilibrium distribution function W0 at the instant t = 0. We have

[TjS + 2n0' + k/rt/B]c™ =rjc™(0)Sn+k0 + 2c2'\k +cn2j 12 ,(10.5.2.12)

[j1s + (2n + l)/' + kp"z/B]c2njl=-3(n + l)c2

n?u/2 — , . (10.5.2.13)

-6(" + l)c„2>° + < t u /4 + c2;*2 -yfBfiti + 2*e„2£1)/2,

(10.5.2.14)

[W + 2nP + kP[IB]c% =-nc2i - 4 ( « + l)c2 l ,u ->fB(c%2

+l +2kc2nf_x),

(10.5.2.15)

[Tjs + 2n/r+k%/B]c%2 = -nc%1 -4(n + l)c„2l^ + V%2>2+1 + 2*e2j_1).

(10.5.2.16)

In order to solve the hierarchy of moment equations so obtained, we introduce a supercolumn vector

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552 The Langevin Equation

C„(0 = 2,2

c2,~2

'< f t (O x

compris ing of 5 subvectors c 2 'm ( f ) = ,,2,m "n-1,1 (0

.<#T(oj

„2,m ; with m = 0, ± 1 , ± 2 . Then the hierarchy of equations for cn'k (?) Eqs .

(10.5.2.12)-(10.5.2.16) can be transformed into the matrix three-term differential-recurrence Eq. (10.5.2.4), where

Q; =

q„

o

Pn

V n

0

0

- P n

0

0

v~

Qn =

V

Qn

6(r„-I„)

0

0

0

Q;

f o

3q:

0

0

0

0

0

PI

0

0

I „ / 2

qi 0

rn

0

0

- P :

0

0

0

0

0

0

0

2P„"

0

0

Qn

0

rn

0

< 0

0

2p„+

0

0

0

-2p„~

0

0 0

0

0 I

q° n

0

q° 0

0

<

-2P;

0

Here the submatrices p * , q * , r„ , and q f are given by Eqs . (10.5.2.5)-

(10.5.2.8) and the submatrices v* are defined as

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Chapter 10. Inertial Langevin Equations 553

v~=-4 n

0

n-\

0

0

0^

0

2

0,

• < - \

(\ 0 •

0 1 •

v0 0 •

• 0 0s)

• 0 0

• i oj /nx(n+l)

The exact solution, for the Laplace transform C^s) is then given by the matrix continued fraction [cf. Eq. (10.5.2.10)]:

QU)=-^ I 5 - Q , - Q : I

-C,(0) (10.5.2.17) 10

Tslio - Q 2 -Q2 I, Q2

7^15 -Qa with initial conditions

C,(0) =

'1/5^

0

0

0

v ° y

and Cn(0) = 0 for all n> 2.

The calculations have shown that for /? ' f5'z > 0.005, the matrix continued fractions involved Eqs. (10.5.2.10) and (10.5.2.17) converge very rapidly, thus 10-60 downward iterations in evaluating these matrix continued fractions are enough to arrive at not less than 6 significant digits in the majority of cases (with decreasing /Tand jB\, the number of iterations increases). For very small /? ' j5\ (< 0.001), the numerical procedure may become unstable. However, in this very low damping limit, the C^iaJ) for a>f3'> 10 is very close to that of the free rotation

model C™(ia>) (J3', f3'z = 0), which can be evaluated from the following analytic equation [29]

C,(ifl>)«C™(iffl)

/ 1.

= --CrM + 4t]^r^^m-l\\4m\coS-l(x)]\ F

m . J L Jl (O

where b = 117. - 1 ,

m=\

^VT^7 m

dx,

(10.5.2.18)

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554 The Langevin Equation

1 n

C™(°°) = >/!+£ J P' X) dx, F( z) = z l - ^ e x p ( z 2 ) [ l - e r f ( z ) ] , 0-y/(l + fex2)3

erf (z) is the error function [12], and the dlM M>(&) are given in Chapter

7, Section 7.6.1; one has, for example, [24]

< 0 [cos"1 (*)] = Pt (x), < ± 1 [cos"1 (*)] = ±-j=Jl-x2 ,

<*o.±i [cos_1 (*)] = ± J | W l - x2 , ^ [ c o s ^ U ^ ^ - J - a - x 2 ) , etc.

;'.^?'. 5-'i .-S& £ A

logI0(Re[Ci(w)]) _2

l°8i(<j3')

bg10Re[C>(w)]-> - . ' . v i - :•••••

l*irf£')

log10(wi;)

Figure 10.5.2.1. l o g ^ n e ^ ] and log10Re[c2] vs. logl0(corj) and log10(/?') for

spherical top molecules (I = Iz) and /?' = /?z'.

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Chapter 10. Inertial Langevin Equations 555

In the opposite (high damping) limit, /?'—> <», the form of the spectrum

Cl becomes Lorentzian and coincides with that of the Debye model of

noninertial rotational diffusion, Eq. (10.4.4.7). The real parts of C, and

C2 vs. log10(<W77) and log10(/T) for spherical top molecules (I = IZ)

and isotropic diffusion (/?' = /3'z) are shown in Fig. 10.5.2.1. The matrix continued fraction solution we have obtained is also

fruitful in analysis using symbolic calculations in the MATHEMATICA program. For example, in the high damping limit ( / ? ' » 1) and isotropic diffusion fi'z = fi), we obtain from Eqs. (10.5.2.10) and (10.5.2.17) the following Taylor series expansion for the orientational relaxation times zx = C1(0)/C,(0) and zx = C2(0)/C2(0) of the symmetric top:

T2 _

TD/3

= 1 +

-1+-

l + B + B2

l + B

5 + 5B + B2

l + B

7

-7-

2(2 + 5B + 4B2+B3+3B5) ,

3(l + B)2(2 + B)

2(32 + 80S+16B2 + 46B3 +12B4

3(l + B)2(2 + fi)

... (10.5.2.19)

+ 3S5) , / ' •••

(10.5.2.20) Here y=ll(2/3'2) is the inertial (Sack's [22]) parameter and TD = £l(2kT). For spherical tops (B = 1) and linear rotators (B = 0), Eqs. (10.5.2.19) and (10.5.2.20) reduce to known results, respectively [23] [cf. Eqs. (10.4.2.11) and (10.4.3.6)]

and

r \ 1 3 5 p.

Tl 1 2 1 —=i+7--r+-, TD 3

r2 , 1 1 83 ,

TD/3 2 6

Tl =l + 5y 32y2 + ... TD/3 3

. (10.5.2.21)

(10.5.2.22)

Thus, we have derived an infinite hierarchy of differential-recurrence Eqs. (10.5.1.12) and (10.5.1.15) for the statistical moments governing the inertial Brownian motion of a symmetric top molecule freely rotating in space by using the Langevin equation method. We have also obtained exact analytical solutions (in terms of matrix continued fractions) of this hierarchy for the one-sided Fourier transforms of the first, C\(f), and the second, C2(0. order equilibrium orientational correlation functions. The problem at hand provides a convenient benchmark solution for the method as it may also be solved using other methods [cf. Sack's Eq. (10.5.2.11)]. The method of solution of

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556 The Langevin Equation

orientational relaxation problems, which we have proposed, is quite general because it is based on the concept of the equation of motion of an observable which in turn is based on the dynamical (Langevin) theory of the Brownian motion. An attractive feature of the dynamical method is that it allows one to compute directly from the dynamical equations of motion the spectra of the equilibrium correlation functions Ci(t) taking account of the effect of molecular inertia on orientational relaxation in liquids. The major advantage that the dynamical (Langevin) method has over the Fokker-Planck equation is that it avoids both the derivation of that equation for the distribution function W(&,aix,a) ,aiz,t) in phase

space, which for the problem in question is [29]

dW dW f a I. + 01.—— + \ fil,COtt? LG1

dW dW

d f „, kTdW Q)rW +

/ dOl dai. +-

0c\ 'xj dai„ v 7 d0)yj

+ ^ (

Ldco, aizW+

kTdW

and also the complex mathematical manipulations associated with the separation of variables in that equation, which combine to obscure the physics underlying the problem. The method developed is essential for the solution of problems involving spatial rotation in a potential as recurrence relations involving more than two numbers will always occur.

The results of our calculations for the spectra C (ico) and C2 (ioi) and the orientational relaxation times t\ and Ta show that these quantities depend on the frictional anisotropy (/?7/?Q and the shape parameter B = Iz II and agree in all respects with those obtained using the Fokker-Planck equation [23,29,30]. As far as comparison with experimental data is concerned, the Langevin-Fokker-Planck model is suitable for the explanation of the rotational motion of small molecules (such as CO, N2O, CF4, and so on) in liquids [29-31]. Here, the model reasonably describes experimental data on infrared absorption, Raman scattering, nuclear magnetic relaxation, etc. (see, e.g., [31], where a detailed comparison with experiments is given), however, it is not applicable [25] to liquids comprised of larger molecules, where the rotational motion is more hindered and has a librational character. The failure of the Langevin-Fokker-Planck model as well as all other inertia corrected Debye type models to account for the high-frequency molecular librations in neat liquids even though they explain the return to transparency at high frequencies is to be expected in view of the assumption made in the

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Chapter 10. Inertial Langevin Equations 557

theory that all electrical interactions between dipoles may be neglected. In the next section we show how the Langevin equation approach can be extended to other models of orientational relaxation such as the itinerant oscillator, which are concerned with the rotational Brownian motion in external potentials, which simulate intermolecular interactions. These models simultaneously exhibit both resonance and relaxation behaviour, which heuristically explains both the low-frequency (relaxational) and high-frequency (librational) behaviour of dipole systems as we now describe.

10.6 Itinerant Oscillator Model of Rotational Motion in Liquids

10.6.1 Introduction

The itinerant oscillator introduced by Hill [32] and Sears [33] is a model for the dynamical (rotational or translational) behaviour of a molecule in a fluid embodying the suggestion that a typical molecule of the fluid is capable of vibration about a temporary equilibrium position (cage), which itself undergoes Brownian motion. Sears [33] used a translational version of the model to evaluate the velocity correlation function for liquid argon, while a two-dimensional rotational version was applied in [32] and [34] to explain the relaxational (Debye) and far-infrared (Poley) absorption spectrum of dipolar fluids: the Brownian motion of the cage gives rise to the Debye absorption; the librational motion of the molecule gives rise to the resonance absorption in the far-infrared (FIR) region. The treatment of [32] and [34] based on the small oscillation (harmonic potential) approximation was further expanded upon by Coffey et al. [35-38], where a preliminary attempt was made to generalise the model to an anharmonic (here a cosine) potential. Yet another recent application of the model is to relaxation of ferrofluids (colloidal suspensions of single-domain ferromagnetic particles) [39]. A three-dimensional rotational version excluding inertial effects, termed an "egg model" has been used by Shliomis and Stepanov [40], to simultaneously explain the Brownian and Neel relaxation in ferrofluids, which are due to the rotational diffusion of the particles and random reorientations of the magnetisation inside the particles, respectively [39]. The analysis in the context of ferrofluids yields similar equations of motion to these given by Damle et al. [41] and van Kampen [42] for the translational itinerant oscillator. This formulation has the advantage that, in the noninertial limit, the equations of motion decouple into those of the molecule and its surroundings (cage). Such a decoupling of the exact equations of motion is also possible in the inertial case if we assume a massive cage.

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558 The Langevin Equation

Figure 10.6.1. Itinerant oscillator model, i.e., a body of mass M moves in a fluid and contains in its interior a damped oscillator of mass m. The displacement of the body is X relative to the fluid and the displacement of the oscillator is x.

The first objective of this section is to describe a method of solution of the Langevin equations of motion of the itinerant oscillator model for rotation about a fixed axis in the massive cage limit, discarding the small oscillation approximation; in the context of dielectric relaxation of polar molecules this solution may be obtained using a matrix continued fraction method. The second objective is to show how using the Langevin equation method, the model may be easily adapted to explain the resonance and relaxation behaviour of a ferrofluid [43].

10.6.2 Generalisation of the Onsager model — Relation to the cage model

The far-infrared (FIR) absorption spectrum of low viscosity liquids contains a broad peak of resonant character with a resonant frequency and intensity which decrease with increasing temperature [25]. This phenomenon is known as the Poley absorption [44]. It takes its name from the work of Poley [44] who observed that the difference em - nfr

between the high-frequency dielectric permittivity em and the square of the infrared refractive index nfr of several dipolar liquids was proportional to the square of the dipole moment of a molecule leading him to predict a significant power absorption in these liquids in the FIR region. A detailed review of the problem and of various theoretical approaches to its solution is given in Ref. [25], where the interested reader can find many related references.

As far as other physical systems are concerned, the FIR absorption has also been observed in supercooled viscous liquids and glasses (for a detailed review, see Ref. [45]). A similar resonance phenomenon seems to occur [45] in the crystalline structures of ice clathrates, where dipolar (and nondipolar) molecules are trapped in the small-size symmetrical and rigid cages formed by tetrahedral hydrogen-bounded water molecules. The encaged molecules undergo both torsional

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Chapter 10. Inertial Langevin Equations 559

oscillations and orientational diffusion. Thus, like liquids and glasses, the spectra of ice clathrates exhibit a resonant type peak in the FIR region, a broad relaxation spectra due to the motion of the encaged molecules at intermediate frequencies (in this instance occurring in the MHz region unlike the microwave region characteristic of simple polar liquids) and a very low-frequency broad-band relaxation process due to orientational motion of the water molecules confined at the lattice sites. Yet another phenomenon which appears to be related to those described above is the peak in the low-frequency Raman spectra (in the region of acoustic phonons of the corresponding crystals) for a variety of glasses which appears to be an intrinsic feature of the glassy state [45]. Since the scattering intensity associated with this feature could be explained by the Bose-Einstein statistics, the peak in the low-energy Raman spectra has been called the Boson peak. Referring to the opening part of our discussion, dielectric loss spectra of several viscous liquids and glasses also exhibit a peak in the THz frequency range (particularly, in the 40 -80 cm-1 range of the FIR spectra). That peak has again been identified as the Boson peak, because the scattering function for the light is approximately proportional to the magnitude of the dielectric loss, £'.

Both the low-frequency Raman scattering peak and Poley FIR absorption peak are due to the underlying molecular process which may be regarded as analogous to the torsional oscillations of molecules confined to cage-like structures [45]. The compelling advantage of such a representation of the relaxation process is that it leads to the modelling of the process as the rotational Brownian motion of a rigid dipolar molecule undergoing torsional oscillations about a temporary equilibrium position in the potential well created by its cage of neighbours. It follows from the foregoing discussion that the schematic description of the process corresponds to the itinerant oscillator or cage model of polar fluids originally suggested by Hill [32,46]. Hill considered a specific mechanism whereby at any instant an individual polar molecule may be regarded as confined to a temporary equilibrium position in a cage formed by its neighbours, where the potential energy surface may in general have several minima. The molecule is considered to librate (i.e. execute oscillations about temporary equilibrium positions) in this cage. Using an approximate analysis of this model based on the Smoluchowski equation (Chapter 1) for the rotational diffusion of an assembly of noninteracting dipolar molecules proposed by Debye [1], Hill demonstrated that the frequency of the Poley absorption peak is inversely proportional to the square root of the moment of inertia of a molecule. Hill's model was later generalised in a number of papers (see, e.g., [34-36,46-48]) to take full account of inertial effects.

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560 The Langevin Equation

It is the purpose of this section to illustrate how Hill's model [32,46] may be solved exactly in the context of the Langevin equation approach in the limit of a very large cage and small dipole using the existing results for the free Brownian rotator and the matrix continued fraction method for the fixed centre of oscillation cosine potential model considered in Section 10.3.2. Furthermore, we shall remark on the similarity of the cage or itinerant oscillator model to the problem of generalising [51,52] the Onsager model of the static permittivity of polar fluids to calculate the frequency dependent complex susceptibility. Thus, we shall pose our discussion of the Hill model in the context of the generalisation of the Onsager model which suggests that the origin of the Poley absorption may lie in the long range dipole-dipole interaction of the encaged molecule with its neighbours. We remark that in its original static form [25], the Onsager model constitutes the first attempt to take into account the contribution of the long range dipole-dipole interactions to the static permittivity of a polar fluid. The key difference between the dynamic Onsager model and its static counterpart is that when a time varying external field is applied, the reaction field produced by the action of the orienting dipole on its surroundings will lag [51,52] behind the dipole. The net effect of this is to produce a torque on the dipole. If the inertia of the dipole is taken into account, the torque will naturally give rise to a resonance absorption peak with peak frequency in the FIR region thus explaining the Poley absorption (as well as the microwave dielectric loss peak). The whole process is analogous to the behaviour of a driven damped pendulum in a uniform gravitational field with a time varying centre of oscillation [2]. In order to proceed, we first describe the problem of generalising the Onsager model to include the frequency dependence of the relative permittivity and how this problem may be linked to the cage or itinerant oscillator model.

The Onsager model [2,25,51,52] consists essentially of a very large spherical dielectric sample of static relative permittivity e placed in a spatially uniform electric field which is applied along the polar axis. The effect of the long range dipole-dipole coupling is taken account of by imagining that a particular (reference or tagged) dipole of the sample is situated within an empty spherical cavity at the centre of the sample (now a shell or cage). The polarising effect of the dipole then creates a reaction field which exerts a torque on the dipole if the dipole direction is not the same as the reaction field direction as is so in the time varying case [51,52]. In calculating the polarisation of the sample when placed in the uniform external field, it is assumed that the shell may be treated macroscopically (i.e., by continuum electrostatics) while the orientation

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Chapter 10. Inertial Langevin Equations 561

of the (tagged) dipole in the cavity is treated using classical statistical mechanics. The treatment in the static case then yields the famous Onsager equation (whence the static permittivity may be calculated) which was subsequently generalised to a macroscopic cavity by Kirkwood and Frohlich [53]. We remark that in the static case the reaction field is a uniform field which is parallel to the dipole direction so that it cannot orient the dipole.

In order to attempt a generalisation of the Onsager model to a time varying applied field of angular frequency CO for the purpose of the explanation of the FIR absorption peak, we shall suppose, following a suggestion of Frohlich [54] that the surroundings of a tagged dipole, that is the shell or cage, may be treated as an inertia-corrected Debye dielectric [1,2] so that the sample now has permittivity ea>). Thus the electrical interactions between cage dipoles are ignored; the only interaction taken account of is that between the cage and the tagged dipole. We shall also suppose that a weak external uniform applied field parallel to the polar axis is switched off at an instant t = 0 so that we may utilise the methods of linear response theory. The Langevin equation of motion of the surroundings is then [50]

7,(b, (0 + eM [ra, (0 - co^ (0] + £ > , (0 a (10.6.2.1)

+R(0 x W ) • R(01 = - ^ (0 + K (0,

while the Langevin equation of motion of the dipole u in the cavity is

IMaM(t) + CM[<oM(t)-<os(t)] + ii(t)x^V[p(t)-R(t)] = XM(t), (10.6.2.2)

where we note that by Newton's third law dV dV

fix + R x = 0. (10.6.2.3) o>H o'R

Note that we do not have to introduce the dipole moment of the surroundings explicitly as their influence is represented by R (t). In the Langevin Eqs. (10.6.2.1) and (10.6.2.2), which reflect the balance of the torques acting on the cage and dipole, respectively, 7V is the moment of inertia of the surroundings (i.e., the cage) of the cavity which are supposed to rotate with angular velocity cos., £s.cos., and >,s are the stochastic torques on the surroundings which are generated by the heat bath, IM is the moment of inertia of the cavity dipole and d^V x\i is the torque on \i. The torque arises because in a time varying applied field ]i will not be parallel to the reaction field R (t) unlike in a static field. The

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562 The Langevin Equation

terms ^ (<oM - cos,) and kM represent the dissipative Brownian torques

acting on fi due to the heat bath, eo^ is the angular velocity of the

dipole. Equations (10.6.2.1) and (10.6.2.2) are recognisably a form of the equations of motion of the itinerant oscillator model with a time dependent potential. In the above equations, the white noise torques X,v

and kM are centred Gaussian random variables so that they obey the

Isserlis theorem and have correlation functions

^\t)^\t') = 2kTCAAt-t'), (10.6.2.4)

$)(t)AlJ)(t') = 2kT£MSijS(t-t'), (10.6.2.5)

where i, j = 1,2,3 refer to distinct Cartesian axes fixed in the system of mutually coupled rotators represented by Eqs. (10.6.2.1) and (10.6.2.2). It is also assumed that ks and k^ are uncorrected, the over bars denoting the statistical average over the realisations in a small time \t -1'\ of the Gaussian white noise processes kt and X^. We also note that the system is in governed by the time-dependent Hamiltonian H given by

H =±Isat+±IMa$+V vR). (10.6.2.6)

The dynamics of the dipole \i are governed by the kinematic relation [23]

ji(f) = a>A(f)X|i(0- (10.6.2.7) Our objective is to calculate from the system of Eqs. (10.6.2.1)—

(10.6.2.7) the autocorrelation function of an assembly of encaged dipoles

(n(0)-n(0))0

hence the complex susceptibility xt®) of such an assembly is given by the linear response theory formula

^ ^ - = l-ia>[CJt)e~ia,dt. (10.6.2.9) X (0) M

In Eq. (10.6.2.8), the zero inside the angular braces which denote equilibrium averages, refers to the instant at which the external uniform field F0 is switched off and the subscript zero outside the braces indicates that the average is to be evaluated in the absence of that field.

The stochastic differential Eqs. (10.6.2.1) and (10.6.2.2) cannot be integrated to yield Eq. (10.6.2.8) in explicit form as they stand. The

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Chapter 10. Inertial Langevin Equations 563

reasons are: they are mutually coupled via the term ^fliQ/l in Eq. (10.6.2.1) and £M&S in Eq. (10.6.2.2), moreover, we have no knowledge of the functional form of the time dependent reaction field R (t) save that it should still be spatially uniform and that by quasi-electrostatics its Fourier transform R(<2>) should be of the order of magnitude [25]

- ^ ^ M ( « ) , dO.6.2.10)

where a is the radius of the cavity. In the Onsager model, the radius of the cavity a is determined from

v = -Jca*Na, (10.6.2.11) 3

so that 47Ta3 /3 is the mean volume per molecule, N0 is the number of molecules in the spherical sample. It is apparent since we do not know the functional form of R (t), that no further progress can be made unless we make an assumption concerning the time varying amplitude of R(f). Thus we shall assume that the amplitude g(0)) of the reaction field factor, Eq.(10.6.2.10), is only a very slowly varying function of the frequency (corresponding to a quasi-stationary function of the time) so that we may replace it by a constant. Thus, the time dependence of R (t) arises solely from the time varying angle between u and R. We shall now demonstrate how the variables may be separated in Eqs. (10.6.2.1) and (10.6.2.2) by making the plausible assumption that the moment of inertia of the tagged dipole is much less than that of the surroundings or cage of dipoles. If this is so, the dipole autocorrelation function C^ (t) will automatically factor into the product of the autocorrelation function of the surrounding inertia corrected Debye dielectric and the autocorrelation function of the orientation of the dipole relative to its surroundings. We again remark that the treatment closely resembles that of the egg model of orientation of a single domain ferromagnetic particle in a ferrofluid proposed by Shliomis and Stepanov [40].

10.6.3 Dipole correlation function

By addition and subtraction of Eqs. (10.6.2.1) and (10.6.2.2), we have

V»„ (0 + ' A (0 + £>,(') = K (0, (i 0.6.3.1)

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564 The Langevin Equation

IfiSlR(t) + £M£lR(t) + p(t)x—V[li(t)-R(t)] = ku(t), (10.63.4)

/ i i\ dV , , n , (10.6.3.2)

In the limit 1^ «Is, Eqs. (10.6.3.1) and (10.6.3.2) become

7,<b, (0 + £ > , ( 0 = K(0. (10.6.3.3)

dfi where the relative angular velocity of the dipole and cage is

ft^e^-o, (10.6.3.5)

so that the equations of motion decouple in the relative angular velocity ilR and <os.

Thus, the cage and dipole orientation processes may be considered as statistically independent in the limit of a very large cage and small dipole. Hence, the autocorrelation function of the tagged dipole from Eq. (10.6.2.8) factors [50] into the product of the autocorrelation function of the inertia-corrected Debye process corresponding to the behaviour of the tagged dipole and the longitudinal,

(costf(0)cost?(0)0, and transverse,

(sin tf(0) cos 0(0) sin 0 cos <p(t)0, (sin 0(0) sin <p(0) sin 0(f) sin </>(t))Q,

autocorrelation functions of the motion of the dipole in the cosine potential

V = -MR cos 0(f)- (10.6.3.6) The calculation of orientational autocorrelation functions from the free rotator Eq. (10.6.3.3) which describes the rotational Brownian motion of a sphere is relatively easy because Sack [22] has shown how the onesided Fourier transform of the orientational autocorrelation functions (here the longitudinal and transverse autocorrelation functions) may be expressed as continued fractions, Eq. (10.5.2.11). The corresponding calculation from Eq. (10.6.3.4) for the three dimensional rotation in a potential is very difficult because of the nonlinear relation between (oM

and (i arising from the kinematic Eq. (10.6.2.7). A considerable simplification of the problem can be achieved in

two particular cases: (a) in the noninertial limit and (b) for rotation about a fixed axis. In case (a), a general method described in Chapter 7, Section

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Chapter 10. Inertial Langevin Equations 565

7.2 may be used to write differential-recurrence relations for the longitudinal and transverse autocorrelation functions. The method is valid even when the form of R ( 0 is not explicitly given. Moreover, if the quasi-stationary assumption for the amplitude of R (t) holds, the autocorrelation function from Eq. (10.6.2.8) may be calculated numerically. In case (b), the kinematic relation Eq. (10.6.2.7) describes rotation in a plane:

p(t) = a)fl(t)kx]i(t) (10.6.3.7)

and Eqs. (10.6.3.3) and (10.6.3.4) reduce in the limit IM«IS to the

itinerant oscillator equations

Ijm + CsW) = K<t)> (10.6.3.8)

/ / ( O + C^CO + Z ^ O sin 0(f) = y f ) , (10.6.3.9)

where

0 = £lR=jM-0s, vr = a>,=fa. (10.6.3.10)

The tagged dipole autocorrelation function is [55]

CM(t) = 2pst)pe(t) = 2(cosiK0)cos jK0)0 <C0S A ^ ( 0 ) 0 , (10.6.3.11)

where the cage autocorrelation function ps(t) is that for the free Brownian rotation (see Chapter 3, Section 3.4)

p , ( 0 = (cos^(0)cos^(0)0 =-er-re~"lnD)-"TD , (10.6.3.12)

7 = kTIslg, TD = CKkT) , (10.6.3.13)

and the dipole autocorrelation function pe (0 is

p9(t) = (cosA6(t))Q, A0 = 0(t)-6(O),

comprising the sum of the sine and cosine (longitudinal and transverse) autocorrelation functions of the angle 6t) between the dipole and the reaction field. The angle y/t) is taken as the instantaneous angle of rotation of the cage. The autocorrelation function Eq. (10.6.3.12) is the inertia-corrected Debye result for an assembly of noninteracting fixed axis rotators giving rise to the low-frequency (microwave) Debye peak and a return to transparency at high frequencies without any resonant behaviour. The autocorrelation function pe(i) has been examined in detail by Coffey et al. [35-37] in the small oscillation (itinerant oscillator) approximation. Invariably, pe(t) gives rise to a pronounced FIR absorption peak in the frequency domain. The characteristic frequency

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566 The Langevin Equation

Q)Q of this peak on making the quasi-stationary assumption R = const for R (t) is given by

<y02=£/(2772) (10.6.3.14)

and the complete expression for CM') is

CM(t) = e n"TD+r re e L \ •* w •*> ^ ''J. (10.6.3.15)

The damped natural angular frequency a\ is given by

o^2=o)^-j3,2/4 = /2-fi'2/4 (10.6.3.16) and

t' = tlrjt Tj = yllM/2kT), F = £Mfj/IM, £ = fiR/(kT). (10.6.3.17)

It is apparent that Eq. (10.6.3.15) represents a discrete set of damped resonances and Debye relaxation mechanisms as may be seen by writing the complex susceptibility in series form [2]. We remark in passing that an equation, very similar in mathematical form but not identical to the small oscillation solution, Eq. (10.6.3.15), may be obtained by applying Mori theory [56] to the angular velocity correlation function of the tagged dipole. Here, the notion of a cage is dispensed with and instead the angular velocity is supposed to obey a generalised Langevin equation incorporating memory effects which is truncated after the third convergent (3-variable Mori theory [25]). The solution of that equation in the frequency domain is then represented by a continued fraction. Moreover, the Mori 3-variable theory has been extensively compared with experiment by Evans et al. [25]. We now return to the main theme of our discussion, namely the behaviour of the complex susceptibility as yielded by Eq. (10.6.3.11) when the small oscillation approximation is discarded.

10.6.4 Exact solution for the complex susceptibility using matrix continued fractions

The complex susceptibility yielded by Eq. (10.6.3.11) combined with Eq. (10.6.2.9) when the small oscillation approximation is abandoned, may be calculated using the shift theorem for Fourier transforms and the matrix continued fraction solution for the fixed centre of oscillation cosine potential model treated in detail in Section 10.3. Thus, we shall merely outline that solution as far as it is needed here. We have on considering the orientational autocorrelation function of the surroundings ps(t) and expanding the double exponential

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Chapter 10. Inertial Langevin Equations 567

Ps(t)= — llU-Le'(Hnlr)"TD • (10.6.4.1)

Now, in view of the shift theorem for Fourier transforms as applied to the one-sided Fourier transform defined as [77]

oo

3/(0=/(/©) = J e'imft)dt o

so that

Se'a'f(t) = f(io + a), (10.6.4.2)

we have the following expression for the Fourier transform of the dipole autocorrelation function given by Eq. (10.6.3.11):

2Zps(t)pe(t) = erfjt^pe(i(D + <uD'+nlrrD)). (10.6.4.3) «=o n\

The Fourier transform pd(ico) is to be determined from Eq. (10.6.3.9) in terms of matrix continued fractions. Equation (10.6.4.3) is useful for the calculation of the complex susceptibility Xn which is given by Eq. (10.6.2.9) because the spectrum CM(i(0) now involves only the quantity

pe(i(0) which is determined from Eq. (10.6.3.9). That equation in turn is clearly the same as Eq. (10.3.1.4), which has already been solved using the matrix continued fraction technique. We summarise as follows. The Green function or transition probability corresponding to Eq. (10.6.3.9) is determined from the differential-recurrence relation which may be derived either from the Fokker-Planck equation (by means of a Fourier expansion of the transition probability) or by averaging the Langevin equation over its realisations in a small time At given a sharp set of values

of 6 and r at time t. This recurrence relation is given by Eq. (10.3.1.30), which for the present problem reads

7 ^ W * ) = -P'nHne^ - | ( / / n + 1 ^ + M H ^ ) 1 (10.6.4.4)

where n = 0,1,2,..., and q - 0,±1,±2,.... Our interest, as dictated by Eq. (10.6.3.11), is in the equilibrium

average of cosA#(0 so that it is advantageous to obtain a hierarchy

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568 The Langevin Equation

which will allow us to calculate (cosA$(t))Q directly. Hence, let us

introduce the functions

cn,q(t) = (Hn(r?e(t))e-i^(,)-em)(t). (10.6.4.5)

Multiplying Eq. (10.6.4.4) by e'ie [6 = 0(0) denotes a sharp initial

value] and averaging over a Maxwell-Boltzmann distribution of 6 and 6 which procedure is denoted by the angular braces, we automatically have a differential-recurrence relation for c (t), i.e., the equilibrium averages given by Eq. (10.6.4.5), namely

Vcn,q (0 = -P'ncn* (0 - »tf<Vn,, (0 + 2ncn_li9 (*)] / 2

-inGK-iji+i ( 0 - c„-i,,_i (01 / 2 . In particular, in the present problem, we are interested in

^ ( O = (cosA0(O)o = | (c 0 , 1 (0 + c0,_I(0) • (10.6.4.7)

Thus, by Eqs. (10.6.2.9) and (10.6.4.3), the complex susceptibility is

(10.6.4.6)

Z(&) = 1-icoe' (-r)n

Z'(0) ' c0il(0) + c0>_,(0)^S n\ (10.6.4.8)

x[c0i, (ifi>+To1 + «(?T0)_1) + CQ-! (ifl) + To1 + n(?rD)_ 1)].

Now, the scalar five-term recurrence Eq. (10.6.4.6) can be transformed into the matrix three-term recurrence equation

7C„(0 = Q;Cn_1(0 + QnC„(0 + Q X + i ( 0 + R^2> (n>\), (10.6.4.9) where the column vectors Cn (/) are defined as

f • \

C0(0 = 0, ^ ( 0 =

L 0, -2

C 0 , - l

C0,l

C0,2

V : J

c„(0 =

c n - l , - 2

C n-1 , -1

Cn-1,0

C n- l , l

Cn-1,2

( n > 2 ) . (10.6.4.10)

The matrices the column vector R and Q„ ,Q„, and Q* are given by

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Chapter 10. Inertial Langevin Equations 569

R = # , ( # ) 2/0(#)

( • "\

0

-1

0

1

0

• J

f.

Q : = - : l T"

... -2

... o

... o

... o • • • 0

0

-1 0

0 0

0

0

0

0 0

0

0

0

i 0

0 •

0 •

o •

o • 2 •

(10.6.4.11)

Q ; = - I ' ( « - 1 )

• -4

• S • 0

• 0

• 0

<? -2

-4 0

0

0

<? 0

-<f 0

0

0

£ 2

0 •

0 •

0 •

£ • 4 •

V

(10.6.4.12)

Q n = - ( n - l ) / ? l , and I is the unit matrix of infinite dimension matrices Q^ and Qj , which are given by

(10.6.4.13)

Q;= i

2

... _2

... o

... o

... o

0

-1

0

0

0 0 0 —

0 0 0 —

0 1 0 -

0 0 2 •••

Q 2 = -

... _4

... _£

... o

... o

... o

£ -2

S 0

0

0

0

# 2

-<f

0 • • •

0 •••

0 • • •

£ ... 4 ...

By applying the one-sided Fourier transform to Eq. (10.6.4.9), we obtain the matrix recurrence relations

ir]coCx (id)) - T]CX (0) = Q\ C2 (ico), (10.6.4.14)

irjaCn ico) = Q„C„ ico)+QjCB+1 ico)+QTC^ ico) + RSn2 lico,\ 0.6.4.15) where the initial value vector Cx (0) is expressed in terms of the modified Bessel functions of the first kind

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570 The Langevin Equation

C!(0) = /<>(#)

hit) (10.6.4.16)

V • J

Here, we have used the fact that C„(0) = 0 for n > 2 because (Hn) = 0 ( n > l ) for the (equilibrium) Maxwell-Boltzmann distribution function. By invoking the general method for solving the matrix recursion Eqs. (10.6.4.14) and (10.6.4.15), we have the exact solution for the spectrum Cj (id)) in terms of a matrix continued fraction:

C1(ia;) = A1(»7C1(0) + Q1"A2R/to), (10.6.4.17)

where the continued fraction An, n = 1, 2, is defined as (see Chapter 2, Section 2.7.3)

A = • I

— . (10.6.4.18) irjoA - Q „ - Q X + i Q n + i

The foregoing equations allow us to evaluate the complex susceptibility.

10.6.5 Results and comparison with experimental data

Before analysing the numerical results, we first remark that the presently available small oscillation (itinerant oscillator) approximation for complex susceptibility %ai) [55] has the advantage of being very easy to use for the purpose of comparison with experimental observations as the solution in the time domain is available in closed form [Eq. (10.6.3.15)]. Moreover, if the inertial parameter y is sufficiently small (< 0.1) and I„a$ <*: kT, the complex susceptibility for small oscillations, i.e., Eq.

(10.6.3.15) combined with Eq. (10.6.2.9), may be closely approximated in the frequency domain by the simple expression:

Xia>) 1 - + -

kT iCOTr

„6)n iCOTD +1

/ 1-

0) 2\

m

Cn ico

'o J ln <

-1

Z'(0) (ianD + \)icoyvD +1) v ^ 0

(10.6.5.1) Equation (10.6.5.1) was originally given in Ref. [35] for the single friction itinerant oscillator model. The first term in Eq. (10.6.5.1) is essentially the Rocard equation of the inertia-corrected Debye theory of dielectric relaxation and so is due to the cage motion, the second damped

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Chapter 10. Inertial Langevin Equations 571

harmonic oscillator term represents in our picture the high-frequency effects due to the cage-tagged dipole interaction.

In Fig. 10.6.5.1, we show plots of the imaginary part tf\(d) of the complex susceptibility from the exact solution, Eq. (10.6.4.8), for j3' = 2

and y = 0.01 and illustrate the effect of varying the reaction field parameter £ The real party(<y) is shown in Fig. 10.6.5.2. Super-imposed on these are /cd) and y(6>) as yielded by the harmonic oscillator approximation, Eq. (10.6.3.15), that is, the conventional itinerant oscillator model. This is compared both with the approximation to the harmonic oscillator solution yielded by Eq. (10.6.5.1) and the exact matrix continued fraction solution, Eq. (10.6.4.17). These results are also shown in Figs. 10.6.5.1 and 10.6.5.2. The advantage of the comparison is that it demonstrates the extent to which one is justified in using the small oscillation solution, Eq. (10.6.3.15), or its approximation, Eq. (10.6.5.1). It appears that Eq. (10.6.5.1) provides a reasonable approximation for a large reaction field parameter. The above behaviour is entirely in accord with intuitive considerations as one would expect the harmonic or simple pendulum assumption to become less and less accurate as the reaction field parameter is decreased.

Figure 10.6.5.1. Imaginary part of the complex susceptibility tfXai) vs. normalised frequency r/co for various values of the reaction field parameter £ Solid lines correspond to the exact cosine potential solution Eq. (10.6.4.17), circles correspond to the small oscillation solution, Eq. (10.6.3.15), and dashed lines correspond to the approximate small oscillation solution, Eq. (10.6.5.1).

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572 The Langevin Equation

0.01 0.1 1

TJO)

Figure 10.6.5.2. The same as in Fig. 10.6.5.1 for the real part of the complex susceptibility;f/(<y).

In Fig. 10.6.5.3, we demonstrate the effect of varying the inertial parameter yfor fixed dipole friction parameter /?'and large reaction field parameter £ It appears that if y < 1, then the small oscillation approximation given by Eq. (10.6.5.1) again provides a very close approximation to the exact solution based on numerical evaluation of Eq. (10.6.4.17) for large £,. The approximate Eq. (10.6.5.1) begins to fail at y = 1 and higher. This is to be expected as one is now dealing with the region in which the Rocard approximation for the complex susceptibility of a free rotator [which is the first term of Eq. (10.6.5.1)] is no longer valid. In Fig. 10.6.5.4, we fix ^and a reasonably high value of £and we demonstrate the effect of varying /? ' Again the agreement with the exact matrix continued fraction solution Eq. (10.6.4.17) is good for relatively high values of /?'(see curve 2 and 3). For relatively small values of fi', however, there is some discrepancy in the intermediate frequency region. At high frequencies, the approximate Eq. (10.6.5.1) cannot predict the comb-like peak structure of the exact solution. (The peaks occur at the harmonics of the fundamental frequency.) This to be expected because Eq. (10.6.5.1) results from an expansion of the correlation function (a double transcendental function with sines and cosines in its argument) which is truncated at the fundamental frequency term so that the comblike structure may not be reproduced. Nevertheless, Eq. (10.6.5.1) provides a reasonable description of the low-frequency absorption and the fundamental frequency peak in the FIR region as is obvious by inspection of Fig. 10.6.5.4. We remark that a similar comb-like structure occurs in the theory of quantum noise in ring laser gyroscopes [57].

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Chapter 10. Inertial Langevin Equations 573

* 0.0

Figure 10.6.5.3. Effect of varying the inertial parameter y: the real, x'(<*>)> and imaginary, X"oi), parts of the complex susceptibility vs. 7ja>. Solid lines correspond to the exact solution yielded by Eq. (10.6.4.17) and circles correspond to the approximate Eq. (10.6.5.1).

«

Figure 10.6.5.4. Effect of varying the friction parameter f': the real, ^(ftj), and imaginary, x"ai), parts of the complex susceptibility vs. r)(0. Solid lines correspond to the exact cosine potential solution Eq. (10.6.4.17) and circles correspond to the small oscillation solution Eq. (10.6.5.1).

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574 The Langevin Equation

In Fig. 10.6.5.5, a comparison of typical experimental data for liquid CH3CI at various temperatures [58] with the theoretical dielectric

loss spectrum e"(co) = (e0 -nfr)x"G))lx'(®) predicted by Eq. (10.6.4.17) is shown (here, we ignore the internal field correction [25]). The moment of inertia which was used in the calculation is the reduced moment of inertia Ir defined by

7 - i _ r i + /-i

where Ib and Ic are the principal moments of inertia of a symmetric top molecule about molecular axes perpendicular to the axis of symmetry. For the CH3C1 molecule, Ib=Ic= 57 10"40 gem2 [59]. The use of the reduced moment Ir allows one to obtain the correct value for the dipolar integral absorption (Gordon's sum rule [25]) for two-dimensional models.

The experimental values of Ae = £0-nfr were taken from Ref. [58]. The dipole-cage interaction energy is regarded as a constant and the parameter £ which is the dipole-cage interaction energy represented in units of thermal energy, was calculated by means of the equation £ = 193<f0/7 .

The model parameters ^ , y, and /Twere adjusted from the best fit of experimental data (in the present case, that is methyl chloride, the value of §) consistent with the best fit is ~ 8). It is apparent that the overall fit provided by utilising the exact solution embodied in Eq. (10.6.4.17) is superior to that provided by utilising the small oscillation solution Eq. (10.6.3.15) or the approximate Eq. (10.6.5.1). Previous fits of the model to the experimental data have been carried out using the approximate Eq. (10.6.5.1) (see, for example, [60]). Moreover, the solution for small oscillations Eq. (10.6.3.15) has been compared with the experimental data in Ref. [61]. The main difficulty in fitting the small oscillation solution to the experimental data is that usually one may achieve a good fit in one region of the spectrum only at the expense of a poorer fit in another region of the spectrum. The use of the exact (cosine potential) solution, which unlike the small oscillation solution includes the energy (temperature) dependence of the frequency of oscillations, certainly removes this difficulty as a reasonable fit can be achieved in all regions of the spectrum simultaneously, as is apparent from the plots shown in Fig. 10.6.5.5. We emphasise, however, that for practical purposes, the approximate small oscillation formula Eq. (10.6.5.1) can provide a reasonable qualitative description of the experimental results if the field parameter £is sufficiently large. The approximate formula Eq. (10.6.5.1) also has the advantage of being very easy to use in practice.

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Chapter 10. Inertial Langevin Equations 575

1

0.1

0.01 0.1 1 10

s ^j 0.1]

o.bi oil i 10

10

I

0.1

0.01 0.1 1 10 / ( T H z )

Figure 10.6.5.5. Dielectric loss £•"(/) vs. frequency/(/"= col 27t). Solid lines correspond to the exact matrix continued fraction solution and circles are experimental data for liquid CH3C1 [58] (Ir = 28.5 10"10 gem2 and £ = 193£0 IT with £0 = 8 ).

We have shown in this section how the equations of motion of the cage model of polar fluids originally proposed by Hill [32] may be solved using matrix continued fractions. Furthermore, the model leads to a microwave absorption peak having its origin in the reorientational motion of the cage and a pronounced FIR absorption peak having its origin in the oscillations of the dipole in the cage-dipole interaction potential. The equations of motion of dipole and cage may be justified in terms of a generalisation of the Onsager model of the static susceptibility of polar fluids to include the dynamical behaviour. A conclusion which may be drawn from this is that the origin of the Poley absorption peak is the interaction between the dipole and its surrounding cage of neighbours which causes librations of the dipole. The model appears to reproduce satisfactorily the main features of the microwave and FIR absorption of CH3CI. In addition, it is shown that the simple closed form expression Eq. (10.6.5.1) for the complex susceptibility which was originally derived in the context of the small oscillation version of the cage model (that is the itinerant oscillator model) can also provide a reasonable approximation to

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576 The Langevin Equation

the cage model susceptibility for large values of the reaction field parameter £

The results given here may also be extended to rotation in space. As far as the cage motion is concerned the complex susceptibility for small y will still be governed by the Rocard equation since the equations of motion again factorise. However, the solution for the dipole correlation function is much more complicated because of the difficulty of handling differential-recurrence relations pertaining to rotation in space in the presence of a potential. We remark that activation processes which involve crossing of the cage dipole over an internal potential barrier may also be incorporated into the present model by adding a double-well (cos26|) term to the potential in Eq. (10.6.3.6) (see Chapter 11). This addition may give rise to a Debye-like relaxation process at very low frequencies with relaxation time governed by the Arrhenius law, the prefactor of which may be calculated precisely using the Kramers theory of escape of particles over potential barriers outlined in Chapter 1. We remark that a cos2£? term in the potential has also been considered by Polimeno and Freed [62] in their discussion of a many-body stochastic approach to rotational motions in liquids. Their discussion is given in the context of a simplified description of a liquid in which only three bodies are retained, namely a solute molecule (body 1), a slowly relaxing local structure or solvent cage (body 2) and a fast stochastic field as a source of fluctuating torque. In general, one may remark that this approach which is based on projection operators and numerical solution of many body Fokker-Planck-Kramers equations is likely to be of much interest in the context of the extension of the model described here to rotation in space.

10.7. Application of the Cage Model to Ferrofluids

We have already mentioned that a longstanding problem in the theory of magnetic relaxation of ferrofluids is how the solid state or Neel mechanism of (longitudinal) relaxation of internal rotation of the magnetic dipole moment with respect to the crystalline axes inside the particle, the associated transverse modes (which may give rise to ferromagnetic resonance), and the mechanical Brownian relaxation due to physical rotation of the ferrofluid particle in the carrier fluid, may be treated in the context of a single model comprising these relaxation processes. (We recall that the slowest mode of the longitudinal relaxation process describes the reversal of the magnetisation over the potential barrier created by the internal anisotropy of the particle which of course

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Chapter 10. Inertial Langevin Equations 577

will be modified by an external applied field. We have seen that the time to cross the barrier is known as the Neel relaxation time and follows the Arrhenius law).

We have also mentioned that the question posed above was answered in part by Shliomis and Stepanov [40]. They showed that for uniaxial particles, for weak applied magnetic fields and in the noninertial limit, the equations of motion of the ferrofluid particle incorporating both the internal and the Brownian relaxation processes decouple from each other. Thus, the reciprocal of the greatest relaxation time is the sum of the reciprocals of the Neel and Brownian relaxation times of both processes considered independently, that is those of a frozen Neel and a frozen Brownian mechanism! Here, the joint probability of the orientations of the magnetic moment and the particle in the fluid, i.e., the crystallographic axes, is the product of the individual probability distributions of the orientations of the axes and the particle so that the underlying Fokker-Planck equation for the joint probability distribution also factorises as do the statistical moments. Thus, the internal and Debye processes are statistically independent. If the applied field is sufficiently strong, however, no such decoupling can take place. The Shliomis-Stepanov approach [40] to the ferrofluid relaxation problem, which is based on the Fokker-Planck equation, has come to be known in the literature on magnetism as the egg model. Yet another treatment has recently been given by Scherer and Matuttis [63] using a generalised Lagrangian formalism: however in the discussion of the applications of their method they limited themselves to a frozen Neel and a frozen Brownian mechanism, respectively.

Here, we reexamine the egg model (which is of course a form of the itinerant oscillator model) noting the ratio of the free Brownian diffusion (Debye) time to the free Neel diffusion time and discarding the assumption of a weak applied field. The results will then be used to demonstrate how the ferrofluid magnetic relaxation problem in the noninertial or high mechanical friction limit is essentially similar to the Neel relaxation in a uniform magnetic field applied at an oblique angle to the easy axis of magnetisation (see Chapter 9, Section 9.2). Unlike in the solid state mechanism however, the orientation of the field with respect to the easy axis is now a function of the time due to the physical rotation of the particle in the ferrofluid. The fact that the behaviour is essentially similar in all other respects to the solid state oblique field problem suggests that a strong intrinsic dependence of the greatest relaxation time on the damping (independent of that due to the free diffusion time) arising from the coupling between longitudinal and transverse modes

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578 The Langevin Equation

occurs. Alternatively, the set of eigenvalues which characterise the longitudinal relaxation, now depends strongly on the damping, unlike in axial symmetry. This precession aided (so called because the influence of the precessional term is proportional to the inverse of the damping coefficient) longitudinal relaxation is absent in the weak field case [40]. Here, the equations of motion decouple into those describing a frozen Neel (pure Debye or Brownian) and a frozen Brownian (pure Neel) mechanism of relaxation, respectively. Thus, the Neel or longitudinal relaxation is governed by an axially symmetric potential. Hence, no intrinsic dependence of the greatest relaxation time on the damping exists. Moreover, the longitudinal set of eigenvalues is independent of the damping, the damping entering via the free (Neel) diffusion time only. It follows that, in the linear response to a weak applied field, the only effect of the fluid carrier is to further dampen, according to a Debye or Rocard (inertia-corrected Debye) mechanism, both the longitudinal and transverse responses of the solid state mechanism. Furthermore, it will be demonstrated that in this case the transverse relaxation process in the ferrofluid, which may give rise to ferromagnetic resonance, may be accurately described by the solid state transverse result for axial symmetry because the Brownian relaxation time greatly exceeds the characteristic relaxation times of all the transverse modes. The latter conclusion is reinforced by the favourable agreement of the linear response result with experimental observations [43] of the complex susceptibility of four ferrofluid samples.

In order to illustrate how precession aided relaxation effects may manifest themselves in a ferrofluid, it will be useful to briefly summarise the differences in the relaxation behaviour for axially symmetric and non axially symmetric potentials of the magnetocrystalline anisotropy and applied field, when the Brownian relaxation mode is frozen. Thus only the solid state (Neel) mechanism is operative; that is, the magnetic moment of the single domain particle may reorientate only with respect to the crystalline axes. Our starting point is the Landau-Lifshitz or Gilbert equation for the dynamics of the magnetisation M of a single domain ferromagnetic particle, namely (see Chapter 7, Section 7.3)

2rwM = /?(tf~1M5[MxH]+[[MxH]xM]) (10.7.1)

(all the definitions are given in Sections 1.17 and 7.3). Equation (10.7.1) is the Langevin equation of the solid state orientation process. The field H may be written as

H = H e / + H „ ( 0 . (10.7.2)

Here

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Chapter 10. Inertial Langevin Equations 579

H , / = - ^ 7 = - — (10-7-3) dV _ dU

dM dm is the conservative part of H, which is determined from the free energy density V (U = v V is the free energy, m-Msvm is the magnitude of the magnetic moment m of the single domain particle.) The random field H„(/) is white noise and must also obey Isserlis's theorem (Chapter 1, Section 1.3). By introducing the dipole vector

e = M IMs=mlm, (10.7.4) we find that Eq. (10.7.1) assumes the form of a kinematic relation involving the angular velocity <oe of the dipole vector e:

e = aexe = ((oL+<aR)xe. (10.7.5) Here

<0, = - H = - - ^ T ( H e / + H „ ( 0 ) (10.7.6) 1 + a2 l + a

is the angular velocity of free (Larmor) precession of m in the field He/

superimposed on which is the rapidly fluctuating Hn(t) and

< » « = - ^ ( e x H ) (10-7-7) l + a

is the relaxational component of <oe. Equations (10.7.6) and (10.7.7) differ from Eqs. (8) and (9) of Shliomis and Stepanov [40] because they contain the noise field and the factor 1/(1 + a2) since the Gilbert equation is used rather than the Landau-Lifshitz equation. The kinematic relation, Eq. (10.7.5), and the coupled Langevin equations, Eqs. (10.7.6) and (10.7.7), are stochastic differential equations describing the motion of the dipole vector e relative to the crystallographic axes, that is, the internal or solid state relaxation. Differential-recurrence relations for the statistical moments governing the dynamical behaviour of e may be deduced from Eqs. (10.7.5)-(10.7.7) as described in Chapter 7. If we now, following Ref. [40] and allowing for the factor (l + a2), introduce the magnetic viscosity

<u = -^-(l + a2), (10.7.8) 6ayx

Eq. (10.7.7) becomes

6 / / v m w / ? = - m x | ^ + m x H n ( 0 . (10.7.9) dm

Equation (10.7.9) will be the key equation in our discussion of precession aided Neel relaxation.

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580 The Langevin Equation

The discussion given above holds for arbitrary free energy U. We now specialise our discussion to a particle with uniaxial anisotropy, so that

U = -mH0(e-h)-Kvm(e-n)2 (10.7.10) where h = H0/ H0 is the amplitude of the external uniform magnetic field (the polarising field), A">0 is the constant of the effective magnetic anisotropy, and n is a unit vector along the easy magnetisation axis. The fact that h is not necessarily parallel to n means that the axial symmetry characteristic of uniaxial anisotropy will be broken. Thus, Eq. (10.7.9) becomes

6jUvm(oR -mH0[exh]-2Kvm(e-n)[exn] = mxHn(t) + 'kR(t), (10.7.11) say. Next, we write U explicitly as

U = -mH0 cos 0 - Kvm cos2 # , (10.7.12) where

cos 0 = cos z?'cos z? +sin i? sin ??'cos( #>-$/) . (10.7.13) Here $,(p are the polar angles of e with respect to the easy axis, which is the polar axis; $,<p' are the polar angles of the external field direction h again with respect to the easy axis which in the solid state problem are constants independent of the time. Analytic expressions for the greatest relaxation time Tin the bistable potential given by Eq. (10.7.12) in the intermediate to high damping case (IHD) (where oris such that the energy loss per cycle of the motion of the magnetisation at the saddle point energy trajectory, AE>kT) may be obtained using Langer's theory [64,65] (see Section 1.18.2) of the decay of metastable states applied to the two degrees of freedom system specified by z? and q>, since in the solid state case •&' and <p' are fixed. Likewise, the Kramers energy controlled diffusion method [65,66] may be used to obtain T in the very low damping (VLD) case, where AE«kT (see Chapter 1,Section 1.13.2). If H0 is applied parallel to the easy axis n, the asymptote for r is given by Eq. (8.3.22) (Chapter 8, Section 8.3.2). In the case (H0 = 0), Tis given by Brown's asymptotic expression [67] for simple uniaxial anisotropy (see Chapter 1, Section 1.18.1):

T-IO^.e*, (cr>2), (10.7.14) 2d"3'2

where a= f3K. Thus, r normalised by TN is independent of a, so that the mode coupling effect completely disappears.

Let the crystallographic axes now rotate with angular velocity a>„ corresponding to physical rotation of the ferrofluid particle due to the

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Chapter 10. Inertial Langevin Equations 581

stochastic torques imposed by the liquid and the aligning action of H (we now have 5 degrees of freedom, viz., the dipole angles *?, >as before and the Euler angles •&', (p', y/, which instead of being constant are now functions of the time due to the physical rotation of the easy axis). The relative angular velocity of the dipole and easy axis is then &R - <on, so that Eq. (10.7.11), describing the motion of m must be modified to

6Mvm((oR-<on)-mH0[exh]-2Kvm(e-n)[exn] = kR(t). (10.7.15) The corresponding mechanical equation of motion of the particle is by Newton's third law (the particle is treated as a rigid sphere, / is the moment of inertia of the sphere about a diameter)

Im„+6pvm(mn-mR) + gwn-2Kv„(e-n)[nxe] = ka(t)-kR(t), (10.7.16) where g = 677'v is the mechanical drag coefficient of the particle in the fluid, v is its hydrodynamic volume, 7/ is the viscosity of the fluid and the white noise torque kn (t) arising from the fluid carrier obeys

<X„(0> = 0, (X^\t)Xifi\t')) = 2kTCS^S(t-f), (10.7.17)

where the indices a, J3=\,2,3 refer to the orientation of the crystallographic axes relative to the laboratory Cartesian axes. The kn(t) again obey Isserlis's theorem, and we shall suppose that kn(t) and kR(t) are uncorrected. Thus, by addition of Eqs. (10.7.15) and (10.7.16), we have the mechanical equation [cf. Eq. (7.2.2), Section 7.2]

I&n +g<an = mH0[exh] + Xn(t). (10.7.18) Equations (10.7.15) and (10.7.18) in general are coupled to each other inextricably by the external field term e x h. If that vanishes, however, so that U depends only on (en)2, they become:

6jUvm(o>R-<on)-2Kvm(e-n)[exn] = XR(t), (10.7.19)

Ie>n+g<on=Xn(t). (10.7.20) The equations thus separate into the equation of motion of m relative to the easy axis, Eq. (10.7.19), and the equation of motion of the easy axis itself, Eq. (10.7.20). The mechanical equation (10.7.20) is governed by two characteristic times: the Brownian diffusion or Debye relaxation time

vB=gl2kT) (10.7.21) and the frictional time

TJ1 = I/g. (10.7.22)

Thus, the dynamical behaviour of Eq. (10.7.20) is governed by the inertial parameter

a = Tn lrB = 2IkTg'2 (10.7.23)

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582 The Langevin Equation

(corresponding to Sack's inertial parameter y[22] for polar molecules introduced in Section 10.4). The parameter a has been evaluated by Raikher and Shliomis [68] for typical values of ferrofluid parameters and is of the order 10~5. Hence, one may entirely neglect inertial effects unlike in polar dielectrics, where the inertial effects become progressively more important at high frequencies. If a —> 0, we have the noninertial response. This is treated by Shliomis and Stepanov [40] who were able to factorise the joint distribution of the dipole and easy axis orientations in the Fokker-Planck equation into the product of the two separate distributions. Thus, as far as the internal relaxation process is concerned, the axially symmetric treatment of Brown [67] applies. Hence, no intrinsic coupling between the transverse and longitudinal modes exists, i.e., the eigenvalues of the longitudinal relaxation process are independent of a. The distribution function of the easy axis orientations n is simply that of a free Brownian rotator excluding inertial effects.

The picture in terms of the decoupled Langevin equations (10.7.19) and (10.7.20) above [omitting the inertial term I(an in Eq. (10.7.19)] is that the orientational correlation functions of the longitudinal and transverse components of the magnetisation in the axially symmetric potential,

U = -Kvm cos2 & , (10.7.24)

are simply multiplied by the liquid state factor, e~'lTB , of the Brownian (Debye) relaxation of the ferrofluid stemming from Eq. (10.7.20). As far as the ferromagnetic resonance is concerned, we shall presently demonstrate that this factor is irrelevant. The longitudinal and transverse relaxation in the axially symmetric potential Eq. (10.7.24) was studied in detail in Sections 7.4.2, 7.4.4, and 8.4.2. We summarise these results as follows: the longitudinal and transverse magnetic susceptibilities characterising the solid state (Neel) process are approximately described by

Zi\(a>) _ 1

and

where

Z±(af) _ (l + iflw2) + A

(10.7.25)

Xx_ (l + iG)T2)(l + icot±) + A (10.7.26)

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Chapter 10. Inertial Langevin Equations 583

are the longitudinal and transverse static susceptibilities,

"l + (P2)0/2 2cr + (P2)0(a- 6) are the effective relaxation times,

2V^crerf r(Vcr) 4cr 2 or-r^

and the Neel relaxation time T is rendered reasonably accurately by Brown's uniaxial anisotropy asymptote, Eq. (10.7.14), if a > 2. The transverse susceptibility, Eq. (10.7.26), which is derived using the effective eigenvalue method, yields an accurate result for the transverse response provided a > 5 (see Section 8.4.2, Chapter 8). The effective eigenvalue solution, Eq. (10.7.26), fails for a <5 since at small to moderate barrier heights, a spread of the precession frequencies of the magnetisation in the anisotropy field exists. Here, the hierarchy must be solved exactly using matrix continued fractions. In Eq. (10.7.26), T± is the effective relaxation time of the autocorrelation functions of the transverse components of the magnetisation [which are linear combinations of the autocorrelation functions of the spherical harmonics YXi^,<p) and Y\-\$,<P) ] of the dipole moment relaxation mode, while Vi is the

effective relaxation time of the quadrupole moment relaxation mode [which is a linear combination of the autocorrelation functions of Y2l(d,<p) and Y2_l(t,<p) ]. The effective relaxation times rx and ^ decrease monotonically with a, each having asymptotes TN/CT, < T » 1

(see Section 8.4.2 of Chapter 8). Now, according to linear response theory

00

Z7(O)) = fr(0)-iO)j f7(t)e'ia"dt = fr(0)-iofr((O), (MI.-L). (10.7.27) 0

where f^(t) and f±(t) are the longitudinal and transverse after-effect functions; in the frequency domain (the tilde denoting the one-sided Fourier transform) they are according to Eqs. (10.7.25) and (10.7.26)

1(0)) = -^—, (10.7.28) 1 + ion

and Z,T, (1 + iOJT., )

/ l W = , , , , • \ A" ( 1 0 J - 2 9 )

(1 + 10)T2 ) (1 + KQT± ) + A

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584 The Langevin Equation

The complex susceptibility of a ferrofluid in a weak applied field may be written directly from Eqs (10.7.28),(10.7.29), and the Langevin equations (10.7.19) and (10.7.20) [taking note of Eq. (10.7.23)] using the shift theorem for one-sided Fourier transforms, Eq. (10.6.4.2). Thus

*(ffl) = 7 T 7 F ' (10-7-30)

1 + Z<WTj| , , ^r, [i6/r2 + i + AT,T 2 / ( r ,T 2 ) l

Z±(a>)=, w / 2 X 2 J , (10.7.31) (l + ifflT2)(l + iflff,

±) + ATxT2 /(T±T2) where the Neel relaxation time rand the effective relaxation times T± and Ta are modified to

TTR (10.7.32)

(10.7.33)

l'l

Tx

T,

=

_

T + TB '

*±TB

TL+TB

VB

T2+TB

(10.7.34)

In a weak measuring field, the particle anisotropy axes are oriented in a random fashion. Hence, the susceptibility (averaged over particle orientations) is given by

%co) =[*||(fl» + 2^j.(a»]/3. (10.7.35)

In ferrofluids, where the Neel mechanism is blocked, T »tB , and so we will have in Eq. (10.7.32)

T„=TB. (10.7.36)

Furthermore, in Eqs (10.7.33) and (10.7.34) we may use the fact [68] that r± and t2

a r e nionotonic decreasing functions of a, and also that usually [69], for the ratio of the free diffusion times,

rN lxB ~ 10"2 (10.7.37) in order to ascertain which times may be neglected in Eq. (10.7.31). Thus, we deduce that in Eq (10.7.32) for all a

T L S ^ a n d T j S T j . (10.7.38) Hence, we may conclude, recalling the exact transverse relaxation solution for %± (Section 8.4.1) that the solid state effective eigenvalue solution embodied in Eq. (10.7.26) can also accurately describe the ferromagnetic resonance in ferrofluids, except in the range of small <J (< 5). Then, the exact solid state solution based on matrix continued

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Chapter 10. Inertial Langevin Equations 585

fractions must be used. The conclusion appears to be in agreement with that of Fannin [70] and Fannin et al. [71] who have analysed experimental data on ferrofluids using Eq. (10.7.26). We also remark that the very large a (high barrier) limit of Eq. (10.7.31) (Landau-Lifshitz limit) agrees with the result of Scaife [72] who analysed the problem using an entirely different method. In the high damping limit, a » \ Eq (10.7.31) reduces to a pure relaxation equation in complete agreement with Shliomis and Stepanov [40], namely their Eqs. (31) and (32).

We shall now very briefly describe inertial effects arising from the term I(an in Eq. (10.7.20). If inertial effects are included, the correlation functions pertaining to longitudinal and transverse motions will still be the product of the correlation functions of the rotational Brownian motion of a particle [40] and the solid state correlation functions, however the composite expressions will be much more complicated for an arbitrary inertial parameter a, Eq. (10.7.23). The reason is that the orientational correlation functions for the Brownian motion of a sphere may only be expressed as the inverse Laplace transform of an infinite continued fraction in the frequency domain [22].

We remark that our treatment will apply not only to uniaxial anisotropy but to an arbitrary nonaxially symmetric potential U(e-n,t) of the magnetocrystalline anisotropy. Since the potential is a function of e • n only, the autocorrelation functions of the overall process (e.g., cubic anisotropy) will be the product of the individual orientational autocorrelation functions of the freely rotating sphere and the solid state mechanism. However, unlike uniaxial anisotropy even though the correlation functions still factorise, substantial coupling between the transverse and longitudinal modes (which now have a dependent eigenvalues) will exist. The reason is that the nonaxial symmetry is now an intrinsic property of the particles. This phenomenon should be observable in measurements of the complex susceptibility of such particles.

We commence by recalling Eqs. (10.7.15) and (10.7.18). In the noninertial limit, Eq. (10.7.18) becomes

gmn-mHQ[exh] = ka(t), (10.7.39)

and on eliminating a>„ in Eq. (10.7.15) with the aid of Eq. (10.7.39)

< ^ - ^ W 0 [ e x h ] - ^ ( e - n ) [ e x n ] = | ^ + - ^ , (10.7.40) 3// 6//vm g

where

C 1 = ( 6 / / v m ) - 1+ r 1 - (10.7.41)

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586 The Langevin Equation

Equation (10.7.40) is of the same form as Eq. (10.7.11) describing the motion of the dipole moment with a frozen (rj'—>°°) Brownian mechanism in the presence of a field H0 at an oblique angle to the easy axis. It immediately follows that the effect of the fluid on the solid state or internal mechanism of relaxation is to alter the magnetic drag coefficient of the solid state process. The corresponding change in the dimensionless damping coefficient or of the solid state process is

af = a(l + 6/ivm/g) = al + TN/TB). (10.7.42)

Here the ratio rN/tB represents the coupling between the magnetic and mechanical motions arising from the nonseparable nature of the Langevin Eqs. (10.7.39) and (10.7.40). Thus, the correction to the solid state result imposed by the fluid is once again of the order 10~2. Hence, we may conclude, despite the nonseparability of the equations of motion, that the Neel relaxation time of the ferrofluid particle should still be accurately represented in the IHD and VLD limits, respectively (see Chapter 1, Section 1.18.2). Furthermore, Eq. (10.7.40) should be closely approximated by the solid state relaxation equation

(aR -mH0(eXh) (e • n)(eXn) = -^-l. (10.7.43) 6,«vm 3// 6//vm

Hence, just as the solid state problem (see Chapter 9), one would also expect the following effects to occur in a ferrofluid for a large dc bias field superimposed on which is a small ac field:

(a) A strong dependence of T/rN on a (i.e., a frictional dependence of the smallest nonvanishing eigenvalue) unlike in the weak field case (axial symmetry), which is a signature of the coupling between the longitudinal and transverse modes.

(b) Suppression of the Neel and barrier crossing modes in favour of the fast relaxation modes in the deep well of the bistable potential created by Eq. (10.7.12), if the reduced bias field hc exceeds a certain critical value [73]. This gives rise to a high-frequency Debye-like relaxation band [73].

(c) A very high-frequency FMR peak due to excitation of the transverse modes having frequencies close to the precession frequency.

In the longitudinal response however, guided by the relaxation time Eq. (10.7.32), it will be necessary to have particles with t< xB so that the characteristic or dependence of the response is not masked by the Brownian process due to the damping imposed by the fluid.

We now return to the mechanical equation, Eq. (10.7.18), of motion of the particles, which for small damping (if the small inertial term is retained) predicts a damped oscillation of fundamental frequency

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Chapter 10. Inertial Langevin Equations 587

[69] O)0 = ^JmH0/I, which would appear in the spectrum as a high frequency resonant absorption, as has been verified [11] in the theory of dielectric relaxation. In ferrofluids, however, it has been estimated [69] that fields of order of magnitude 107 Oe are needed for oscillatory effects, which is higher than any, which may be obtained under terrestrial conditions. Hence, one may rule out this resonant mode of the motion.

It remains to discuss the influence of H0 on the mechanical relaxation modes. An estimate of this may be made by recalling that Eq. (10.7.18) is basically the equation of motion of a rigid dipole in a strong constant external field. Moreover, if inertial effects are neglected, it has been shown in Section 7.5 that the longitudinal and transverse effective relaxation times decrease monotonically with field strength from TB, having asymptotic behaviour

Tn~TB/£, T±~2TB/£,

where £ = vMsH0/(kT)»l. Thus, the principal effect of the external field in Eq. (10.7.39) is to reduce the Brownian relaxation time.

In order to support our theoretical discussions, we now present room temperature complex susceptibility data [43] for a ferrofluid consisting of magnetite particles suspended in isopar M (the surfactant is oleic acid). The particles have a median diameter of 9 nm and a bulk saturation magnetisation of 0.4 T. Since detailed experimental data for ferrofluid susceptibilities in a strong oblique polarising field (of intensity 10 T and higher) superimposed on which is a weak ac field, are not yet readily available, we shall confine ourselves to an illustration of how the weak ac field susceptibilities, Eqs. (10.7.30), (10.7.31), and (10.7.35), which incorporate the effect of the fluid carrier, compare favourably with experiment. The susceptibility profile has the same resonant form as that predicted by Eqs. (10.7.30), (10.7.31), and (10.7.35). Plot (a) of Fig. 10.7.1 shows the susceptibility components obtained over the wide frequency range of 50 MHz to 10 GHz, whilst plot (b) shows the fit obtained using Eqs. (10.7.30), (10.7.31), and (10.7.35). As magnetic fluids have a distribution of particle shape and size, these parameters are accounted for by modifying the above equations to include a normal distribution of anisotropy constant, K, and a Nakagami _distrihution [70,71] of radii, r. Here, the fit was obtained for a mean K = 10 J/m with a standard deviation of 6103 and a Nakagami distribution of radii r with a width factor J3= 4 and a mean particle radius, 7 ~ 4.5 nm and a saturation magnetisation of 0.4 T. The value used for a, the damping parameter, was 0.1, a figure within the range of values normally quoted [71] for a.

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588 The Langevin Equation

f(Hz)

Figure 10.7.1. Normalised plots of a) x'i03) a nd x'i®) > ar>d b) corresponding fits using Eqs. (10.7.30), (10.7.31), and (10.7.35).

The approach to the combined mechanical and magnetic motions of a ferrofluid particle and their mutual behaviour, which is based on rearrangement of the Langevin equations and a consideration of the various characteristic time scales, indicates how the physical effects of the fluid carrier on the magnetic relaxation may be explained without elaborate and detailed solution (which will always involve super matrix continued fractions) of the various equations describing the system. The relative orders of magnitude of the time scales involved determine which of the existing independent internal and Brownian mode solutions may be applied to the ferrofluid particle in any given situation. These considerations hold even in the strong field case, where the variables cannot be separated in the underlying Langevin equations. In particular, for zero or very weak external fields, we have shown that the high frequency behaviour may be accurately modelled by the solid state result Eq. (10.7.26), as the Brownian relaxation time TB simply cancels out of Eq. (10.7.31) due to the relative orders of magnitude of the various time scales. These are dictated by the ratio of the free diffusion times and the monotonic decrease of the effective relaxation times with barrier height . This appears to be the explanation of the success of Eq. (10.7.26) in explaining the experimental results. We further remark that no assumptions beyond that of the effective eigenvalue truncation of the set of differential-recurrence relations have been made to obtain this result, since the Langevin equations will always decouple for zero or weak fields. Furthermore, for <r< 5, where the effective eigenvalue solution is not an accurate representation of the exact transverse susceptibility

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Chapter 10. Inertial Langevin Equations 589

solution (see Section 8.4) - that solution may always be found from the underlying set of differential-recurrence relations by using matrix continued fractions. The experimental data on the linear response, that is the weak ac field susceptibility, which we have presented support our conjectures concerning the application of the solid state transverse response result, Eq. (10.7.26), to magnetic fluids.

Appendix A: Statistical Averages of the Hermite Polynomials of the Angular Velocity Components for Linear Molecules

In the derivation of the infinite hierarchy for statistical moments in Section 10.4, we have stated without proof that

and

where

Ax(t)Hn (nx(t))Hm (ay(t))p,k [cos#(o] = 2nkT/3'Hn_x ax)Hm(ay)Pt (cos 0)

Xy (t)Hm (Qy (t))Hn Qx (0) P/ [costf(0]

= 2mkT/3'Hm_x (Qy)Hn ( Q j t f ( c o s 0 ) ,

(Al)

(A2)

Q,x=j]cox, Q.y=rjcoy, (A3)

(Ox and (Uy are the angular velocity components obeying the Euler-Langevin Eqs. (10.4.1.5) and (10.4.1.6)

In order to prove Eqs. (Al) and (A2), we shall use a theorem proved in Appendix B below, namely

Qrxt)Axt)G[&y(t),m~\ = nkTI3'Q!l-xG(Q.y,€) (A4)

and

Qny (t)Ay (t)F [Qx (0,0(0] = nkTP'Q!1;' F£lx,&) (A5)

F and G are arbitrary functions). Noting the definition of the Hermite polynomials [12]

/ / n W = ( 2 x r - ^ ^ ( 2 x r 2+ . . . + ^ ^ , (A6)

1! (n/2) (for the odd order polynomials, the last term in Eq. (A6) is equal to zero), Eq. (Al) can be presented as follows

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590 The Langevin Equation

\t)Hn(axt))nm(^yt))if[^7Xt)\

=4(0 j[x\(or - ^ V ^ r 2 + - - - + t ^ Hm («y w)^ [cos*)]

=2W/r»[2Q,r - ( n " 1 )1

( " " 1 ) [ ^ ] " " 3 + . . J H m ( ^ ) / f (cos^

=2*r^/iffn_1(QI)i/IB(fty)if(cos^. (A7)

Similarly

^(f)H„ («y(r))ffB (nx(0)^* [«***)]

= V0j[20,«r -^^[2n,(r)f2 + ... + < I ^ H . (0,(0)* [«*«*)]

= 2kT/3'mHm.l ay)Hn (flz)if(cos0). (A8)

Appendix B: Averages of the Angular Velocities Components

We first require to show that

Q.nx(t)Ax(t) = nkTj3'Q.x'1 (Bl)

and

anyt)Xyt) = nkT/3'Q.n;1. (B2)

In order to prove this, we will use the method of mathematical induction. Let us consider Eqs. (Bl) and (B2) for n = 1. Thus, we need to evaluate averages of the type

£lxmx(t), £ly(t)Xy(t).

In order to accomplish this, we recall [49] that the Langevin equations (10.4.1.5) and (10.4.1.6) can be written in terms of the new variables Q.x

and Q.y, as rflx t) = Q.) (?) cot W) - fitlx (?) + 4 (0 l(2kT), (B3)

TjQy(t) = -Slx (my(t) cot tf(?) -F&.yt) + Xy (?) /(2kT), (B4)

and can be regarded as the integral equations,

ax(t) = Qx -r]'lp'\Q.x(t')dt'+T]-l\tfy(t')com')dt'+^— \Xx(t')dt\ (B5) o o 2kTo

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Chapter 10. Inertial Langevin Equations

ny(t) = ay-Tj-1/3'fay(t')dt'

- i /

591

(B6)

-v-l\nx(t')ayt')com')dt'+±-\Ayt')dt'.

o lkl o The white noise torques Xx(t), Ay(t) are by definition both independent

of the angle &(t) and the initial angular velocities Q.x, Q.y. Thus, just as

for planar rotators in Section 10.3, we have

Ax(t)Qx(t) = —l—j Axmx(t')dt' = ^j S(t-t')dt' = kTj3'. (B7) 2klT] 0 7] 0

In like manner

Ay(t)&y(t) = kT/3'. (B8)

Now, we assume that Eqs. (Bl) and (B2) are true for n = K:

Q.ft)lt(t) = kTpKilf-x (i = x,y) (B9) and we shall prove that it is true for n - K + 1, i.e., we need to show that

Q.f+\t)ll(t) = kTp'(K + \)Q.f. (BIO) Let us introduce the K + 2 Gaussian variables

xi(t) = x2(t) = ... = xK+i(t) = ai(t)-ai, xK+2(t) = Ai(t) (Bii) such that Xt(t) - 0 . Therefore, from Gardiner [76]

(2N + 2)\ r x ( o x ( r ) x ( r ) X ( t ) i i i Xit)Xmt)\ ,K = 2N

K = 2N-\

(B12) where the subscript 'syni means the symmetrised form of the products [76] (this is simply another statement of Isserlis's theorem (Chapter 1, Section 1.3). Substituting Eq. (Bl 1) into Eq. (B12), we arrive at

Xl(t)X2(t)...XK+2(t) = - (N + 1)!2A

0,

(2N+2)! iiN

(JV + 1)!2

0,

N+l

Using the formula Eqs. (B7) and (B8) and [76]

[*M0-m 4(0[*M0-m , K = 2N,

1 , W J (2N-IVA '

K = 2N-\. (B13)

(B14)

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592 The Langevin Equation

we can rewrite Eq. (B13) as follows

[W)-n,F1W=\Kr/r(-2N+imt)-a'fN' K=2N' (Bis) [ 0, K = 2N-l.

Now, on using the binomial formula (a-b)n=an-nan-lb + ... + (-l)"bn

and Eq. (Bl) for n< K,we can rearrange the left hand side of Eq. (B15) as follows,

-kip

Df+1(0-(^+i)^(^ +...-K-i)JC+1rVr+1]4(o=of+1(04(0

(K + ljKQf^Qt --(K + VKiK-yQ?-2^2 + ...+(-lf(K + l)OiK

= af+\t)At(t)-kTjB\K + l) af-[a,.(0-n,f (B16)

Note that for K=2N-\, the last term in the parentheses vanishes. Combining Eqs. (B15) and (B16), we obtain

Q.f+\t)Ai(t) = kTpXK + \)Q.f (B17)

both for K = 2N and K = 2N - 1 as required. In like manner, one can prove that

Q.nx(t)Ax(t)F[i(t)] = nkTj3'Q.nx-lF(d), (B18)

Q"y(t)Ay(t)F[m] = nkTjB'Q.n;lF#), (B19)

Q.nx(t)Ax(t)G[m),ny(t)] = nkT/3'ax-lG(#,£ly), (B20)

and

any(t)Ay(t)G[^t),nx(t)] = nkTpan

y-lG(^nx), (B2i)

where F and G are arbitrary functions. Let us show, for example, how one can evaluate

Ax(t)nx(t)F[i%t)] and Ay(my(t)F[#(t)]. The white noises /L* and Ay are by definition independent of the orientation and the initial (sharp) angular velocities Q.x and Q.y. Thus, we have

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Chapter 10. Inertial Langevin Equations 593

(B22)

Xx(t)&xt)F[dt)\ = —-F[#t)\\ \(t)Xxt')dt' ZT]kl 0

= \ 2kTCS(t-t')dt' = kT/1'F(#).

In like manner

Ay(Wy(t)F[#(t)] = kT/3'Fd). (B23)

Let us now evaluate averages of the form

Ax(t)Q.2x(t)F[m] and Xy(t)Q.2y(t)F[iKt)] • (B24)

The Xs are centred Gaussian random variables so that they obey Isserlis's theorem (Chapter 1, Section 1.3). On squaring Eqs. (B5) and (B6) and forming the averages indicated in Eq. (B24), it is evident that only the terms linear in X in the squared equations will make a contribution to the averages as all even powers of X will give rise to odd powers in the average which will vanish by Isserlis's theorem. Further, Xxt) is independent of Q.x and $ so that

Ax(W2

x(t)F[tKt)]

= Xx(.t)\ax + \ tfyit'^om'W-n-'p] nx(t')dt' + - \ - \ Xx(t')dA F[tf(t)

TJK1 0

= 2kTj3'Q.xF(i)

and

xyt)a)t)F[m]

=^(r) |n y - j ax(myt')Com')dt'-7Txp\ ay(t')dt'+-\-\ xy(t')dA F[m] [ 0 0 2T?kTQ J

:-J-F[tf(o]<yoj xyt')xyt)dt'=2kTpayF#). Tjkl 0

Similarly, in the case of

Ax(Wx(Wf (t)F[iKt)] and Xyt)Q.yt)Q.Mx (t)F[#(t)], where M is an arbitrary whole number, the terms that will give a non-zero contribution are

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594 The Langevin Equation

Ax(t)ax(m>« (t)F[tf(t)] = kTp'a^F^),

Xy (t)Qy (t)Q.Mx (t)F [tf(0] = kT/3'Q.Mx F&),

and so on.

Appendix C: Evaluation of cos#(£) in the Low Damping Limit

In the low damping limit ( / ? ' « 1), the energy is not conserved but will vary very slowly with time (quasi-stationarity) and the dynamics of the system at t > 0 is described by the one-dimensional Fokker-Planck equation (10.3.2.20) or by the corresponding Langevin equation for the energy e(t) (averaged over the fast phase w):

4 ^ ) = -—e( t )+^cos0[e( t )]+tkl-Co S0[e( t )] \+-L 0[e(t)U(t), at T] [ 2 de ) kT which is a stochastic differential equation of the Stratonovich type with a multiplicative noise term.

In the free rotation or undamped limit (i.e., at /?'= 0 when the Langevin force vanishes), the energy of the dipole

e = rj202-^Ncos0 (CI) is a constant of the motion and so the dynamics of the dipole are described by the deterministic nonlinear differential equation [11]

-a T]—COS0(O

dt = [£ + £ w cos0(O][ l -cos 2 0(o] . (C2)

This equation may readily be derived from Eq. (CI) by noting that

02 = (d cos 01 dt)2 /(l - cos2 0). Equation (C2) has a solution in terms of the Jacobian doubly

periodic elliptic function sn(«|m) [11]:

cos0(t) = l-l-e2)sn2(t^N(\-e3)/(2Tj) + wrnJ, (C3)

where

m = -\-e2

1 ^ ^[l-cos^O)]/^)

W-c3 0

and e2 and e3, (e2 > e3) are given by

- 1 (£>ZN)

TN

j (i-mx2)(i-x

2) All

dx, (C4)

( - & < * < & ) ' ^

'-el£N (£>£N) -1 (-£N<e<£N)'

(C5)

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Chapter 10. Inertial Langevin Equations 595

In order to evaluate the relaxation time TLD from Eq. (10.3.2.21), one

needs equations for cos#(£) and W^(e). In order to obtain these equations, we make the transformation of the phase variables [e,e] -> e,w and recall that [14]

rasn (w|/n) = l-E(m) 2K1

nq -cos

nKu

K(m) (C6)

Km) K2m)^[ \-q2n

where K(m) and E(m) are the complete elliptic integrals of the first and second kind, respectively [12], and q = e-*

K(l-m'>/Kw. Thus, we have from Eqs. (C3)-(C6)

2K(m)

— I" cosOdw COS0( f ) : 1

: 1 -( l - g 2 )

m 1 -

Em)

K(m)

2Km) 0

E(m)

K(m)

2E(m)

1, (-%N<e<%N)

+ 1, (e>£N)

(C7)

m X'(m) where the dependence of the modulus m on £is determined by Eqs. (C4), and (C5) and is given by

me) j 2 ( i+e /^r , ( f>^)

1(l + £/^)/2, (-£„ <£<£„)'

Thus, the average of cos# over e is given by

/cos#\ = J cos0(e)W^(e)de,

where -4*

< ( f ) = -2ff[m(£)]<T

(C8)

(C9) K^K^Nl-e3)l0(^N)

Equation (C8) yields the correct equilibrium value for / cos#) , Eq.

(10.3.2.19), viz.,

COS0 N IQ^N)'

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596 The Langevin Equation

Appendix D: Sack's Continued Fraction Solution for the Sphere

Sack [22] obtained a solution for the dielectric (first-order) response in terms of a scalar continued fraction [his Eq. (3.19)]. Unfortunately, no details of the derivation have been given; moreover, a misprint in his equation exists. This has led to some confusion in the literature as Sack's solution was used as printed, see, e.g., [23,26]. Here, a detailed derivation of Sack's continued fraction solution is given.

In the notation of Sack [22], the dynamics of an assembly of spherical top molecules is governed by the recurrence equations:

ico'a0 + 2bx = ico', (Dl)

(ico' + n)an + (n + 2)bn+l - ybn_x = 0, (D2)

ico + n + -2y b„+ n + 2 —

n + 3 (n + 3)(ico' + n + \)

where y = l/(2/3'2) and co' = Ico/C-

One can formally solve Eqs. (D1)-(D3) for a0 as

ico'

««+ i-r««-i=0,(D3)

a n = •

ico +-ico' +1 + -

Y-

(D4)

2ico' + 2)

where a2 /a0 must be evaluated from the following continued fraction

V2

ico' + n-l + - 2Y

(n + 2)(ico' + n)

*n-1

ico +n +

Y\ n + 1-n + 2

(n + 2)

- + -

y-\ n + 3-n + A

*n+2

. , , 2y/(n + 2) . , , 2yl(n + A) ico +n-l + -L-± ico +n + l + -J-7

1

ico -Yn ico + n + 2 (D5) (n > 2). One may see by inspection that the continued fraction (D5) differs from that given by Sack [22]. However, Eq. (D5) can be rearranged as follows

yn Z = . , , 2W(n + 2) yn(n + 3)/(n + 2) '

ico +n-\ + -L- - + -' (D6)

ico +n ico' + n + Z, n+2

where Zn is defined as

Page 622: The Langevin Equation Coffey_Kalmykov_Waldron

Chapter 10. Inertial Langevin Equations 597

Z „ = •

ico' + n-l + - 2y y-\ n + l-

n + 2 J «„-2 (D7)

(n + 2)(ico' + n) Thus, we have from Eqs. (D4) and (D7)

ico' % ico'+Z2

ico'

ico +-2y

\+ico'+- (3-1/2)/ 2(2+ico')

2+ico'+-Ay

3+ico'+ Y—r+-3(4+ico')

(5-l/3)r

A+ico'+-6y

5+ico'+- V - + ... 4(6+ico')

(D8) that is Sack's result with the corrected misprint. Noting that

Q ico) = a0 /(3ico), Eq. (D7) yields Eq. (10.5.2.11).

References

1. P. Debye, Polar Molecules, Chemical Catalog, New York, 1929, reprinted by Dover, New York, 1954.

2. W. T. Coffey, M. W. Evans, and P. Grigolini, Molecular Diffusion, Wiley, New York, 1984. (Also available in Russian: Mir, Moscow, 1987).

3. W. T. Coffey in Dynamical Processes in Condensed Matter, Adv. Chem. Phys. 63, edited by M. W. Evans, Wiley, New York, 1985, pp. 69-252.

4. A. Einstein, Investigations on the Theory of the Brownian Movement, edited by R. Fiirth, Dover, New York, 1956.

5. M. C. Wang and G. E. Uhlenbeck, Rev. Mod. Phys. 17, 323 (1945). 6. P. Langevin, C. R. Acad. Sci. Paris 146, 530 (1908). 7. D. G. Frood and P. Lai, unpublished work (1975). 8. W. T. Coffey, J. Chem. Phys. 93, 724 (1990). 9. W. T. Coffey, Chem. Phys. 143, 171 (1990). 10. S. Chandrasekhar, Rev. Mod. Phys. 15, 1 (1943). 11. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov, J. Chem. Phys. 115, 9895 (2001). 12. M. Abramowitz and I. Stegun, Eds., Handbook of Mathematical Functions, Dover,

New York, 1965. 13. H. C. Brinkman, Physica 22, 29 (1956). 14. E. T. Whittaker and G. N. Watson, Modern Analysis, 4th Edition, Cambridge

University Press, London, 1927. 15. E. Praestgaard and N. G. van Kampen, Mol. Phys. 43, 33 (1981).

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598 The Langevin Equation

16. W. T. Coffey, D. A. Garanin, and D. J. McCarthy, Adv. Chem. Phys. 117, 483 (2000).

17. M. W. Evans, J. Chem. Phys. 77,4632 (1982). 18. W. T. Coffey, C. Rybarsch, and W. Schroer, Chem. Phys. Lett. 99, 31 (1983). 19. E. P. Gross, J. Chem. Phys. 23, 1415 (1955). 20. R. A. Sack, Proc. Phys. Soc. B 70, 402 (1957). 21. H. Risken and H. D. Vollmer, Mol. Phys. 46, 55 (1982). 22. R. A. Sack, Proc. Phys. Soc. B 70, 414 (1957). 23. J. R. McConnell, Rotational Brownian Motion and Dielectric Theory, Academic

Press, London, 1980. 24. D. A. Varshalovich, A. N. Moskalev, and V. K. Khersonskii, Quantum Theory of

Angular Momentum, World Scientific, Singapore, 1998. 25. M. W. Evans, G. J. Evans, W. T. Coffey, and P. Grigolini, Molecular Dynamics and

Theory of Broadband Spectroscopy, Wiley, New York, 1982. 26. J. G. Powles and G. Rickayzen, Mol. Phys. 33, 1207 (1977). 27. J. K. Vij and F. Hufnagel, J. Phys. E. Sci. Instrum. 22, 749(1989); J. K. Vij and F.

Hufhagel (personal communication, 1990). 28. R. E. D. McClung, Adv. Mol. Relaxation Interact. Processes 10, 83 (1977). 29. A. Morita, J. Chem. Phys. 76, 3198 (1982). 30. R. E. D. McClung, J. Chem. Phys. 75, 5503 (1980). 31. G. L6vi, J. P. Marsault, F. Marsault-Herail, and R. E. D. McClung, J. Chem. Phys.

73, 2435 (1980). 32. N. E. Hill, Proc. Phys. Soc. bond. 82, 723 (1963). 33. V. F. Sears, Proc. Phys. Soc. Land. 86, 953 (1965). 34. G. Wyllie, J. Phys. C: Solid St. Phys. 4, 564 (1971). 35. J. H. Calderwood and W. T. Coffey, Proc. R. Soc. Lond. A 365, 269 (1977). 36. W. T. Coffey, P. M. Corcoran, and M. W. Evans, Proc. R. Soc. Lond. A 410, 61

(1987). 37. W. T. Coffey, P. M. Corcoran, and J. K. Vij, Proc. R. Soc. Lond. A 414, 339 (1987). 38. W. T. Coffey, P. M. Corcoran, and J .K. Vij, Proc. R. Soc. Lond. A 425, 169 (1989). 39. M. I. Shliomis, Sov. Phys. Usp. 17, 153 (1974). 40. M. I. Shliomis and V. I. Stepanov, in Relaxation Phenomena in Condensed Matter,

Adv. Chem. Phys. 87, 1, Ed. W. T. Coffey, Wiley, New York, 1994. 41. P. S. Damle, A. Sjolander, and K. S. Singwi, Phys. Rev. 165, 277 (1968). 42. N. G. van Kampen, Stochastic Processes in Physics and Chemistry, North-Holland,

Amsterdam, 2nd Edition, 1992. 43. W. T. Coffey and P. C. Fannin, J. Phys. Cond. Matter 14, 3677 (2002). 44. J. Ph. Poley, J. Appl. Sci. B 4, 337 (1955). 45. G. P. Johari, J. Non-Crystalline Solids, 307, 114 (2002). 46. Dielectric Properties and Molecular Behaviour, N. E. Hill, W. E. Vaughan, A. H.

Price, and M. Davies, D. Van Nostrand Co., London, 1969, p. 90. 46. G. A. P. Wyllie, in Dielectric and Related Molecular Processes, vol. 1, Specialist

Periodical Reports, Senior Reporter, M. Davies, The Chemical Society, London, 1972, p. 21.

47. W. T. Coffey, J. Phys. D: Appl. Phys. 11, 1377 (1978). 48. W. T. Coffey, Chem. Phys. Lett. 52, 394 (1977).

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Chapter 10. Inertial Langevin Equations 599

49. W. T. Coffey, J. Chem. Phys. 95, 2025 (1991). 50. W. T. Coffey, /. Chem. Phys. 107, 4960 (1997). 51. B. K. P. Scaife, Principles of Dielectrics, Oxford University Press, London, 1989,

revised edition, 1998. 52. B. K. P. Scaife, Complex Permittivity, The English Universities Press, London,

1971. 53. H. Frohlich, Theory of Dielectrics, Oxford University Press, London, 1958. 54. H. Frohlich, personal communication to W. T. Coffey, 1987. 55. W. T. Coffey and M. E. Walsh, J. Chem. Phys. 106, 7625 (1997). 56. H. Mori, Prog. Theor. Phys. 33, 423 (1965). 57. W. T. Coffey, Yu. P. Kalmykov, and E. S. Massawe, Phys. Rev. E 48, 699 (1993). 58. A. Gerschel, T. Grouchulski, Z. Kiziel, L Psczolkowski, and K. Leibler, Mol. Phys.

54,97 (1985); ibid., 58, 647 (1986). 59. G. Herzberg, Molecular Spectra and Molecular Structure II. Infrared and Raman

Spectra of Polyatomic Molecules, D. Van Nostrand Co., New York, 1945. 60. P. M. Corcoran and J. K. Vij, Mol. Phys. 63, 477 (1988). 61. M. E. Walsh and P. M. Dejardin, / Phys. B 32, 1 (1998). 62. A. Polimeno and J. H. Freed, Adv. Chem. Phys. 83, 89 (1993). 63. C. Scherer and H. G. Matuttis, Phys. Rev. E 63, 1 (2001). 64. J. S. Langer, Ann. Phys. (N.Y.) 54, 258, 1969. 65. W. T. Coffey, D. A. Garanin, and D. J. McCarthy, Adv. Chem. Phys. 117, 483

(2001). 66. H. A. Kramers, Physica (Utrecht) 7, 284 (1940). 67. W. F. Brown, Jr., Phys. Rev. 130, 1677 (1963). 68. W. T. Coffey, Yu. P. Kalmykov, and E. S. Massawe, in Modern Nonlinear Optics,

Part 2, Eds. M. W. Evans and S. Kielich, Adv. Chem. Phys. 85, 667 (1993). 69. Yu. L. Raikher and M. I. Shliomis, in Relaxation Phenomena in Condensed Matter,

Ed. W. T. Coffey, Adv. Chem. Phys. 87, 595 (1994). 70. P. C. Fannin, Adv. Chem. Phys. 104, 181 (1998). 71. P. C. Fannin, T. Relihan, and S. W. Charles, Phys. Rev. B 55, 21 (1997). 72. B. K. P. Scaife, Adv. Chem. Phys. 109, 1 (1999). 73. W. T. Coffey, D. S. F. Crothers, Yu. P. Kalmykov, and S.V. Titov, Phys. Rev. B 64,

012411 (2001). 74. P. C. Fannin, T. Relihan, and S. W. Charles, /. Magn. Magn. Mater. 162, 319

(1996). 75. P. C. Fannin, S.W. Charles, and J. L. Dormann, J. Magn. Magn. Mater. 201, 98

(1999). 76. C.W. Gardiner, Handbook of Stochastic Methods, Springer, Berlin, 1985. 77. E. C. Titchmarsh, Theory of Fourier Integrals, Oxford University Press, London,

1937.

Page 625: The Langevin Equation Coffey_Kalmykov_Waldron

Chapter 11

Anomalous Diffusion

11.1 Discrete and Continuous Time Random Walks

We have seen in Chapter 1, Section 1.22, that Einstein's theory of Brownian motion is based on the notion of a discrete time walk. The walk may be described in simple schematic terms as follows: Consider a two dimensional lattice, then, in discrete time steps of length At, the random walker is assumed to jump to one of its nearest neighbour sites, displayed for example [1] on a square lattice with lattice constant Ax, the direction being random; as shown in Fig. 11.1.1. Such a process, which is local both in space and time, can be modelled [1] in the one-dimensional analogue by the master equation [cf. Section 1.22, Eq. (1.22.8)]

W/(f + Ar) = [w /_1(0 + W / + 1(0]/2. (11.1.1)

Here the index j denotes the position of the random walker on the underlying one-dimensional lattice, j + 1, j - 1 are the adjacent lattice sites. Wj (t + At) is the probability for the random walker to be at sitey at

1

.

w m

,

•>

1

H.

.

,

.

7

Figure 11.1.1. Schematic representation of a Brownian random walk. The walker jumps at each step of duration At to a randomly selected direction, thereby covering the distance Ax the lattice constant. Thus, the only random variable is the direction of the jump length vector. (Reprinted from Phys. Reports, vol. 339, R. Metzler and J. Klafter, The Random Walk Guide to Anomalous Diffusion: A Fractional Dynamics Approach, 1-77, Copyright (2000), with permission from Elsevier).

600

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Chapter 11. Anomalous Diffusion 601

time t + At given that it was at sites ; ± 1 at time t. In the continuum limit At —> 0, Ax —» 0 by expanding W, in a Taylor series in At and Ax as illustrated in Chapter 1, Section 1.22, Eq. (11.1.1) leads to the diffusion equation for the transition probability function W(x,t\x0,t0) [cf. Eq. (1.22.12) of Chapter 1]

^ = D^-, (11.1.2) dt dx2

where x0 = x(t0), i.e., at t = t0 the particle was at x0, and the diffusion coefficient D is defined as

D= lim ^ - . (11.1.3) Ax->0, Ar-»0 2Af

As shown in Chapter 1, Section 1.12, the fundamental solution (or Green function) of this equation is

W(x,t\0,0) = -rJ=e-x2/4Dt). (11.1.4) sI4xDt

We remark that in the context of dielectric and magnetic relaxation, the rotational diffusion equations for the distribution function W (z?, (p, t) of derived by Debye [2] and Brown [3] (Sections 1.15 and 1.17) are simply more general cases of Eq. (11.1.2), where the tips of the unit vectors specifying the dipole orientations execute a discrete time random walk on the surface of the unit sphere. W (i?, (p, t) then represents the probability density function of the orientations of the dipoles on the surface of the sphere. For polar dielectrics, where the only interaction is with an external applied alternating field, the discrete time random walk on the surface of the unit sphere then leads directly to the Debye Eqs. (1.15.1.3)-(1.15.1.5). These are substantially in accordance with the experimental evidence for simple polar liquids, supporting the hypothesis that, in such liquids, the underlying random processes are local, both in space and time.

On the other hand, in Section 1.22, we have seen that many other materials may exhibit non-exponential relaxation behaviour with a slowly decaying long time tail. These processes are characterised by a complex dielectric susceptibility, which often has the Cole-Cole (anomalous) relaxation behaviour, viz.,

^ - = X-—- (0<<T<1). (11.1.5) Z(0) l + dcotf

We have also indicated in Chapter, Section 1.22, how anomalous relaxation has its origins in anomalous diffusion, by which we mean,

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602 The Langevin Equation

referring to the simple one-dimensional translational random walk, that the mean square displacement of the diffusing variable x(t) scales with time like

x2)~ta, (0<cr<2), (11.1.6) where the successive a values 0<<7<1, <7=1, 1 < cr< 2, and a= 2 correspond to subdiffusion, ordinary diffusion, superdiffusion, and the ballistic limit, respectively.

We have briefly indicated in Section 1.22, that a physical explanation of anomalous diffusion may be given in terms of the continuous time random walk (CTRW). The concept of a CTRW was introduced by Montroll and Weiss in 1965 [41,47] as a way to render time continuous in a random walk without an appeal to the diffusion or continuum limit. Essentially, the CTRW model (Fig. 11.1.2) is based on the idea that the length of a given jump of a random walker as well as the waiting time between two successive jumps may be drawn from a distribution function ysx,t), which will be referred to as the jump probability density function. The jump distribution is the probability density that a random walker executes a jump from x to x + dx in a time interval dt having remained at some site for a waiting time t. This property defines [32] the CTRW. We may determine from y/(x,t) both the jump length (vector) probability density function

oo

Mx) = j \ffx,t)dt (11.1.7) o

and the waiting time probability density function

,_

r K.

<

h

->! J

I

r

4

Figure 11.1.2. Continuous time random walk (CTRW) model. Left: CTRW process on a two dimensional lattice - generalising the Brownian situation from Fig. 11.1.1. The waiting times are symbolised by the waiting time circles, the diameter is proportional to the waiting time which is to be spent on a given site before the next jump event occurs. The jump lengths are still equidistant. Right: (x,t) diagram of a one-dimensional CTRW process where both jump lengths and waiting times are drawn from probability density functions which allow for a broad variation of the corresponding random variables. (Reprinted from Phys. Reports, vol. 339, R. Metzler and J. Klafter, The Random Walk Guide to Anomalous Diffusion: A Fractional Dynamics Approach, 1-77, Copyright (2000), with permission from Elsevier).

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Chapter 11. Anomalous Diffusion 603

w(t) = J V(x,t)dx. (11.1.8) —oo

Here Xx)dx yields the probability of a jump length (vector) L in the interval x-^ x + dx, w(t)dt yields the probability of a waiting time Tw in

the interval t -^t + dt. If L and 7V are independent random variables, we have the decoupled (separable) form

y/x,t) = w(t)A(x). (11.1.9) If they are coupled, we have

y/x,t) = p(x\t)wt) (11.1.10) or

y/(x,t) = p(t\x)Mx), (11.1.11) that is, a jump of a certain length involves a time cost or, on the other hand, in a given time span, the walker can only travel a maximum distance [1]. Until we come to treat inertial effects, we shall use the decoupled version, and we shall mainly concentrate on the 'long rests' [1] or fractal time random walk model, where the mean waiting time diverges, however the jump length variance remains constant. The other uncoupled case, with finite mean waiting time and divergent jump length variance, is the 'long jump' or Levy flight model [1] (for a comprehensive discussion see [32] and for a general introduction to random walks, see Feller [51]).

We shall now demonstrate how the continuous time random walk in the diffusion limit may be used to justify the fractional diffusion equation, which has been written down without derivation and solved in Chapter 1, Section 1.22.1. In order to accomplish this, we shall use an integral equation method [1].

11.2 A Fractional Diffusion Equation for the Continuous Time Random Walk Model

The fractional diffusion equation may be derived using the integral equation for a CTRW and the properties of the Fourier transform as follows. First, we define the mean waiting time

oo

Tw) = jtw(t)dt (11.2.1) o

and the jump length variance [1] oo

I 2 = J x2 Xx)dx . (11.2.2)

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604 The Langevin Equation

These averages allow us to characterise different types of CTRW processes according to whether L2 or (Tw) is finite or divergent. For example, in the fractal time random walk, we choose finite E2 and divergent (7V). Furthermore, we remark that in general, without any decoupling assumptions, the CTRW process may be described by the integral equation [1]

oo oo

t]x,t)=\ dx'jij(x'/)i//(x-x',t-t')dt' + S(x)S(t), (11.2.3) -oo 0

where r/(x,t) is the joint probability per unit length and per unit time of a random walker having arrived at x at time t and of having arrived at x at time t', 5x)S(t) is the initial condition of the walk.

Now, the probability density function W(x,t) of the random walker being at x at time t is given by the convolution [1]

t

W(x,t) = j T]x,t')Gt-t')dt', (11.2.4) o

where

G(0 = 1-J w(t')dt' (11.2.5) o

is the probability of no jump occurring in the time interval (0,0, i.e., the survival probability at the initial site. Now, using the notation for the Laplace transformation

f(s) = ]f(t)e-s'dt, (11.2.6) o

we have from Eqs. (11.2.4) and (11.2.5), by the convolution theorem for Laplace transforms,

W(x,s) = -fj(x,s)[\-w(s)]. (11.2.7)

We now determine fj(x,s). To accomplish this, we return to our evolution Eq. (11.2.3) and take the Laplace transform over the time variable and the (two-sided) Fourier transform over the space variables. We write, in the space of wave numbers K,

oo

Jrg(x) = g(K)= \ gx)e-iKxdx. (11.2.8) —oo

We remark that the Fourier transform of A(x), viz., X(K) , is known [32] as the structure function of the random walk. Formally, it is the Fourier transform of the jump vector distribution and so is the characteristic function of the jump vector distribution. As far as the related function

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Chapter 11. Anomalous Diffusion 605

'W_(K,i) is concerned, in the theory of thermal neutron scattering, W(K,t) is known as the intermediate scattering function [42,43], i.e., the Fourier transform of the Van Hove function [42]. Again, using the convolution theorem, we have from the evolution Eq. (11.2.3)

fj(K,s)[\-pK,s)] = \. (11.2.9) Thus, using Eq. (11.2.7), we have the renewal theory equation

#(*- , , )= ' - ^ S ) \ . (H.2.10)

In general, if we had an initial condition Wo(x) other than S(x), we would have

# ( r , „ . ™ S W (U.2,,, s[l-iy(K,s)]

Equation (11.2.11) applies to coupled jump length and waiting time random variables (L, Tw).

Formally, Eq. (11.2.10) corresponds [46] in the space-time domain to the generalised master equation

dW(x,t) 7 ,. , , , , x—^- = I dx \ K(x- x ,t-t)W(x ,t)dt•'. (11.2.12) dt J j,

—oo 0

The equivalence of Eqs. (11.2.12) and (11.2.10) is immediately apparent on transforming Eq. (11.2.12) to the (K, S) domain. One can readily show [46] that both equations are equivalent if the Fourier-Laplace transform of the kernel K in Eq. (11.2.12) is given by

KK,S)= Y \ ' N

V . (11.2.13) l-w(s)

The generalised master Eq. (11.2.12) may also be rewritten in terms of the jump probability density function y/(x,t) as the integral equation (taking the fundamental solution as an example)

oo t

W(x,t)= \dx'\y/x-x',t-t')Wx',t')dt' + G(t)8(x), (11.2.14) -oo 0

where G(f) is the survival probability at the initial site given by Eq. (11.2.5). Equation (11.2.14) is essentially a form of the Chapman-Kolmogorov equation. Equations of the form of Eq. (11.2.14) have been used extensively with various truncation approximations to derive fractional diffusion equations (for a review, see Ref. [22]).

We now assume that L and Tw are statistically independent. Furthermore, we shall assume that the waiting time distribution is governed by a Poisson distribution, so that the jump time is a Poisson

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606 The Langevin Equation

process. (The particular choice of a jump time distribution is unimportant; what is important, is that the jump time has finite mean so that the conditions of applicability of the central limit theorem are satisfied in the limit of a large number of jumps). Thus

w(t) = T~le~"T. (11.2.15) Hence, the mean waiting time is

TW) = T, (11.2.16) so that the jumps occur on a single characteristic time scale. In addition, we shall assume that the jump length is a centred Gaussian distribution, so that the jump length probability density function is

A(x) = —^e-x2/^2). (11.2.17) IV2#

The first and second moments of this distribution are, of course, finite so that again the central limit theorem will apply. Thus, the Laplace transform of the waiting time distribution is

w(s) = —^— = 1-ST, (11.2.18) l + ST

if 5Tis small. This approximation corresponds to studying the behaviour of the random walk for long times such that ST« 1, which in turn implicitly corresponds to a large number of jumps. In like manner,

X(x-) = < r z V / 2 = l - E V / 2 (11.2.19)

if I V « 1. Thus, we assume that the random walker covers large distances or we focus on the large scale properties of the walk, again implicitly corresponding to a long sequence of jumps, i.e., long jump times. If we now substitute the low-frequency and small wavenumber results into Eq. [11.2.11], we have

W(K,s)= TW\K . (11.2.20) sr + l V / 2

Equation (11.2.20) represents, in the (K, s) domain, the original diffusion equation, Eq. (1.4.11), for the concentration of Brownian particles derived by Einstein (see Section 1.4) provided the limit

lira E2 /(2T) =KX r->0, E2->0

exists. This may be seen by taking the inverse transform over the s variable, viz.,

W(K,t) = W0(rc)e-1-2'c2"aT), (11.2.21) so that we have the ordinary differential equation

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Chapter 11. Anomalous Diffusion 607

—W(fc,t) = -—K2W(JC,t). (11.2.22) dt 2-V

Equation (11.2.21) may in turn be written as a partial differential equation in the domain (x, t) by recalling the operator equation for the two-sided Fourier transform, namely,

Thus, we have from Eqs. (11.2.22) and (11.2.23)

^ = 1 , (11.2.24) dt 2r dx2

which is Eq. (1.4.11) of Chapter 1 with L2/(2T) = kT/C . (11.2.25)

Thus, finite first moment Tof the jump time distribution and finite jump length second moment E2 give rise, in the limit T—> 0, E2 —> 0, to normal diffusion with the diffusion coefficient E2/(2r) —> K\. Physically, the first and the second order moments discussed above have a particular significance because in Brownian motion, the time scale is determined by the first moment of a waiting time distribution (Einstein by introducing his fixed r, see Section 1.4, actually assumes a delta function distribution of waiting times), while the second moment of the jump length distribution defines a physical length scale. This behaviour is of course a consequence of the central limit theorem (see Chapter 1, Section 1.6) which underpins the Wiener process, arising from the properties of a sum of random variables (see Section 1.8; here the sum of the elementary displacements of the Brownian particle), where each displacement has arbitrary distribution with mean zero and finite variance. There exist, however, individual distributions, where neither the first nor the second moments exist. Here, the limiting distribution is not a Gaussian distribution, but a so-called stable or Levy distribution [32]. The Gaussian distribution is a special case of these stable distributions. An everyday example of a Levy distribution is [32] the probability of a difference between the duration of successive beats of the human heart which corresponds to such a distribution rather than a Gaussian distribution, implying that large differences between beat durations occur more frequently than predicted by the normal distribution.

The most striking property of a Levy distribution is the presence of long-range inverse power law tails in the distribution function, which may lead [32] to divergence of even the lowest order moments. The tails prevent convergence to the Gaussian distribution if they pertain to a

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608 The Langevin Equation

sequence of random variables, however not the existence of a limiting distribution. For instance, both first and second moments are infinite if the characteristic exponent a< 1 (fractal dimension of the Levy flight, see Appendix) as is easily proved. In physical terms, the divergence of the first and second moments for certain Levy distributions indicates the absence of underlying physical scales. This may be interpreted [32] as the scale invariance, which is characteristic of self-similarity and fractal behaviour. An example of such a random walk is a walk, where all the steps in the walk take the same average time as in Brownian motion, however the distribution of the step lengths exhibits a Levy-type power law decay for the longest jumps. Random variables with Levy-like jump length distribution and finite jump time for jumps of any length are called Levy flights. The simplest example of a Levy flight is [32] the Weierstrass random walk which is separable, cf. Eq. (11.1.9), and in one space dimension x is characterised by

oo

E2 = J x2A,(x)dx = oo and (Tw) < oo,

in contrast to Brownian motion, where both E2 < oo and (Tw) < oo. (For a detailed account of the Weierstrass random walk, see [32]). The divergence of E2 means that, in such a walk, a typical displacement after N steps scales with the number of steps N like

X2N~N2la, (0<a<2),

in contrast to the scaling X2N ~ N of the Brownian motion (ex =2) which

arises when the step lengths are on the average finite. The anomalous exponent a in the Weierstrass random walk is the fractal dimension (see Appendix) of the set of jumps. The divergence of the second moment of the jump length distribution leads to enhanced diffusion (superdiffusion) behaviour as the displacement increases faster with the number of steps than in normal diffusion; in other words because jumps of arbitrary length all take the same time and the overall displacement is dominated by the largest jumps. Enhanced diffusion is usually encountered in systems, exhibiting chaotic dynamics and turbulence [46].

The Weierstrass random walk is a simple example of a separable CTRW. Moreover, it is Markovian due to the finite value of the mean waiting time (Tw) with waiting time distribution w(t) ~ t~(l+a) but highly non-Gaussian due to the slowly decaying jump probability. Let us now consider the reverse situation of a separable CTRW: on this occasion, we chose a finite E2 but a divergent average waiting time (Tw); such a walk is called a fractal time random walk. This walk always gives rise to

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Chapter 11. Anomalous Diffusion 609

subdiffusion because the random walker will always be trapped on average on a certain site for a long time before he may advance the distance (X2)1/2. Thus, the number of steps TV in a fixed time interval t is not linear in t as in normal diffusion but scales as [32]

N~ta, with anomalous exponent a< 1. Thus, the typical displacement after N steps would [32] be of the form

If <J> 1, the first moment of the waiting time distribution exists. This behaviour combined with the finite second moment of the displacements is again sufficient to ensure normal diffusion. The most distinguishing feature of a fractal time walk is the presence of a discrete hierarchy of jump time scales, not all of which have the same probability of occurrence. The characteristic exponent a may again be interpreted as a fractal dimension - the fractal dimension of the set of waiting times between jumps. In a useful example, Paul and Baschnagel [32] show how a fractal waiting time distribution w(t) may be generated from a hierarchy of discrete time Poisson processes representing the individual jump probabilities. This procedure allows one to calculate a explicitly. It should be very important in relation to problems involving a distribution of potential barrier heights which would give rise to a hierarchy of Arrhenius-like relaxation times. The collective behaviour could, therefore, be modelled as a fractal time relaxation process. Subdiffusion is usually encountered [46] in systems with geometrical constraints such as doped crystals, glasses, etc.

We may summarise the content of the preceding paragraphs by remarking that both CTRWs which we have discussed are separable. First, we have given the example of Levy flights, which describe enhanced diffusion. These have infinite jump length variance but finite waiting time. They are Markovian and of course non-Gaussian. The fractal time random walk, on the other hand, has a finite jump length variance but Levy-like waiting time distribution. It is non-Gaussian and non-Markovian and describes subdiffusion. So far, we have essentially avoided discussion of non-separable continuous time random walks. The best-known example of such walks is the Levy walk, where, unlike in a Levy flight, the walker cannot jump an arbitrary length in the same fixed time, but has to move with a given velocity from the starting to the end point [32]. The spatio-temporal coupling causes large jumps to take a longer time than shorter ones. Moreover, the coupling ensures a finite mean square displacement unlike in Levy flights. The coupled walk is of major importance in the context of inertial effects and fractional Klein-

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610 The Langevin Equation

Kramers equations. Indeed, in order to obtain finite jump variance in enhanced diffusion, one always has to introduce a coupled jump length probability density function y/(x,i) [46].

We have illustrated how the diffusion equation Eq. (11.2.24) used by Einstein may be recovered by means of the integral equation Eq. (11.2.3). We shall now write W(K,t) from Eq. (11.2.10) for the fractal time random walk or long rests model [1]. Here the jump length variance Z2 remains finite, however, the waiting time has a long tailed waiting time probability density function with asymptotic behaviour

w(t)~(T/tf+cr, 0<<T<1. (11.2.26)

This has representation in the s domain

wis) ~ 1 - (srf. (11.2.27) Moreover, if we retain the Gaussian step probability density function, Eq. (11.2.17) and assume that L and Tw are independent, Eq. (11.2.11) becomes, on ignoring squared terms (corresponding to long times and a large number of jumps)

W(K,s) = , y r )1 (11.2.28)

which on recalling the properties of the Mittag-Leffler function, Eaz), given in Chapter 1, Section 1.22, may be inverted into the time domain to yield

lV( i f , f )=f 0 ( i r )£ f f [ -LV(( / r ) f f /2 ] . (11.2.29)

On noting that Ea-Xata) satisfies the ordinary differential equation

4 / ( 0 = -r0Dtl-<Tf(t), (11.2.30)

at where the Riemann-Liouville fractional derivative (also discussed by Heaviside [33]) is given by

J _ f Sit') no) t-t')

[Eq. (11.2.31) is simply a consequence of Cauchy's integral formula], we have with Eq. (11.2.24) the fractional diffusion equation

-W(x,t) = Ka 0D'-° ^Wx,t) (11.2.32)

°D>~°g = rh*\ 77^dt' (1L2-31)

^-W(x,t) = Ka0Dt1"7^

at dx provided the generalised diffusion coefficient

Ka= lira £ 2 / (2 r a ) r->o,i:2->o

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Chapter 11. Anomalous Diffusion 611

exists. Equation (11.2.32) stems from a fractal waiting time or long rests model, where the system is jammed in a particular configuration for an arbitrary long interval. Such behaviour stems in turn from assuming random forces with an anomalous waiting time distribution. The fundamental solution of Eq. (11.2.32) may be obtained [1] using Fox functions [13]. We also have the Levy distribution as fundamental solution of Eq. (11.2.32) (for a particle starting at x0 = 0 at t = 0) in series form [1], viz.,

Wx,t) = 1 (-1)"

,n /2

fiKc f n=0 yKat , (11.2.33)

n!r(l-<r[ra + l]/2) ^

Equation (11.2.33) has the asymptotic expansion for \x\ » ^jKata [1]

W(x,t)~ 1

4^Kata(2 -<r)2yfc

<j\x\ \

i - g 2-cr -flf)

17/(2-0-)

ANK0 2

(11.2.34) (which has a characteristic long power law tail emphasising the fact that this is a Levy distribution). Such a distribution is often said to be of stretched Gaussian from. Both Eq. (11.2.33) and its asymptote Eq. (11.2.34) reduce to the probability density function of the Wiener process when (7—» 1, namely,

Wx,t) = -F=L=e-x2l^l). (11.2.35) y[47tKv

As far as the rotational Brownian motion considered in Chapter 1, Section 1.22 is concerned, Eq. (1.22.1.1) of that section may be obtained form Eq. (11.2.32) by simply replacing x by the angular coordinate (f> so that noting the definition of the Debye relaxation time for rotation about a fixed axis

T = CKkT), (11.2.36) one has

-W<t>,t) : oA l - < 7 -wy,t). (11.2.37)

Equation (11.2.37) leads to the anomalous (Cole-Cole-like) behaviour of the complex susceptibility, Eq. (11.1.5). Thus, it appears that the Cole-Cole relaxation behaviour arises from subdiffusion, which in turn arises from random torques with an anomalous waiting time distribution. The Debye time T serves to define an intertrapping time scale. For anomalous

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612 The Langevin Equation

relaxation in the presence of an external potential arising from an applied field, Eq. (11.2.37) becomes

dW

dt • = T ,D,

\-a 3 (W_jtV) 3 V kT 3^ d<p d<t>1

Equation (11.2.38) may also be written as

0D?W<l>,t)-QD?W0(<t>) = T-d</>

W dV

kT d0 + -d<p2

-W

Moreover [4]

0D?A = -

(11.2.38)

(11.2.39)

(11.2.40) r ( l -<r )

where A is a given constant, indicating that the initial state Wo(0) decays slowly with a long time tail unlike the exponential decay of normal diffusion, which is again a demonstration of the fractal time character of the process.

We remark that in all the work described in this section an implicit assumption has been made, namely that one may truncate Eq. (11.2.11) at small wave numbers and low frequencies in order to obtain a diffusion equation. In the theory of the Brownian motion, this procedure is acceptable because taking the low frequency and small wave number limit means retaining the term linear in Af and squared terms in Ax only. This procedure does not of itself justify the neglect of the higher order terms. The justification for this neglect lies in the Gaussian nature (the central limit theorem applies) of the processes in the limit of a large number of jumps so that Isserlis's theorem (Chapter 1, Section 1.3) is obeyed. Hence (cf. our discussion in Chapter 1, Section 1.9), all moments of higher order than the second in Ax may be expressed as even powers of (Ax) . In the diffusion limit, Isserlis's theorem then justifies the neglect of all powers higher than (Ax)2 in the Kramers-Moyal expansion ensuring that one can rigorously write down a diffusion equation. The fact that Isserlis's theorem is satisfied in the Brownian motion is also the cardinal reason why one may average the Langevin equation over a small time starting from fixed values of the variables at time t in order to generate a set of differential-recurrence relations for the statistical moments.

11.2.1 Solution of fractional diffusion equations in configuration space

We illustrate one method of solution of the configuration space fractional diffusion equation Eq. (11.2.38) by referring to the simplest case, namely Eq. (11.2.37), using the method of separation of the variables. We seek solutions of the form

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Chapter 11. Anomalous Diffusion 613

W(0,f) = *(0)F(r) . (11.2.1.1) Thus

dt 4>(<z>) <r^

Hence, noting that 4> must be periodic in (f> so that 1 d2

<S>(<t>)= -p\ ( p = 0 ,± l ,±2 , ), (11.2.1.3) * ( 0 ) rf<z>2

we have the equation describing the decay modes

j-Fp(t) = -^a0D^Fp(t), (11.2.1.4)

where

Apa = p2T-a. (11.2.1.5)

The Sturm-Liouville representation in terms of eigenvalues and eigenfunctions, which we use here, is very helpful as it allows one to separate local from nonlocal effects. We have

T~° d2

*(0) df where Ap a are the eigenvalues of the operator

V=-;^TJ*W. < 1 1 - 2 - 1 - 6 >

LF=T-a—T (11.2.1.7) dip

such that LF^p(<P) = -Ap^p(^), (11.2.1.8)

so that with Eq. (11.2.1.5)

Xp^=X/-°, (11.2.1.9)

which exemplifies how the eigenvalues of the normal distribution process are altered, in this case reduced, by the nonlocal character of the anomalous diffusion process. The eigenvalues of the local process are related to their Brownian counterparts by the prefactor rl~a. Thus, in general, the anomalous relaxation process will be governed by Eq. (11.2.1.4), which effectively describes the nonlocal behaviour, and Eq. (11.2.1.8), which effectively describes the influence of the nonlocal behaviour on the local behaviour. The latter fact is intuitively obvious because the form of the Fokker-Planck operator for Brownian diffusion is preserved albeit with a different diffusion constant. We note the special case of normal diffusion, <J= 1, where

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614 The Langevin Equation

ApX=Ap=p2lz. (11.2.1.10)

The equation governing the nonlocal behaviour, namely Eq. (11.2.1.4) has solutions

Fpt) = Ea-Xp(JtCJ). (11.2.1.11)

Equation (11.2.1.11) in the present context becomes

Fp(t) = Ea[-p2(t/Tf] (11.2.1.12)

replacing the conventional e~ pt =e~p tlT for a= 1. Equation (11.2.1.12) has initially stretched exponential (Kohlrausch) form [4], viz.,

£ f f [ -p 2 ( f / r ) f f ]~« r(1+c7) (11.2.1.13)

and long time inverse power law behaviour

Ea[-p2(t/Tr]~(p2(t/T)ffr(l-a)y\ (11.2.1.14)

and so describes nonexponential relaxation. Equations (11.2.1.9) and (11.2.1.11) have been derived for one-

dimensional diffusion in the absence of a potential. However, they still hold in the presence of a potential as we now demonstrate. We have illustrated the reduction of the local decay rates of the system, due to nonlocal anomalous relaxation behaviour, by considering the simplest possible case, namely, free rotation about a fixed axis. Similar conclusions also pertain to the rotation in a potential, all one has to do is to replace LF in Eq. (11.2.1.7) by the more general operator of Eq. (11.2.38), viz.,

L. -4-The Sturm-Liouville representation will then be of the form

wy,t) = f ^ w M - ^ / 7 ) , (n.2.1.16)

where the APiCrare again the eigenvalues of the Sturm-Liouville equation, Eq. (11.2.1.8) with LF given by Eq. (11.2.1.15) rather than by Eq. (11.2.1.7). It is evident that, since LF is again effectively the same as for the corresponding normal diffusion problem, that the Ap of the normal diffusion problem will again scale according to Eq. (11.2.1.9). The scaling effect is particularly important in the context of escape of a particle over a potential barrier (the Kramers problem treated in Chapter 1). As has been repeatedly emphasised, the smallest nonvanishing

1 dV) + •

d</>2 (11.2.1.15)

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Chapter 11. Anomalous Diffusion 615

eigenvalue of the Fokker-Planck equation, when written for the Brownian motion in a potential, yields in the high barrier limit, the Kramers escape rate. In the present context, since we consider the anomalous diffusion analogue of the overdamped Brownian motion, the scaling Eq. (11.2.1.9) suggests that the overdamped Kramers escape rate TK (here K denoting the Kramers value), namely [Eq. (1.13.1.20) of Chapter 1]

^o^co^ AV/(kT) (n.2.1.17)

is slowed by the factor T1_a so that

Ala~Tl-CT\=Tl-arK. (11.2.1.18)

The corresponding decay mode is governed by

Ea\-T^tlr)a'\ (11.2.1.19)

rather than by the e~^' of the normal diffusion. In order to understand how the anomalous relaxation behaviour

influences the dielectric properties, we first recall that according to linear response theory [35] (see Chapter 2, Section 2.8), the complex dielectric susceptibility Xi0*) is defined as

oo

Xco) = x'(G))-ix\G)) = -xXQ)\e-imCt)dt , (11.2.1.20) o

where X'(0) is t n e static susceptibility and the normalised relaxation function C(t) is, according to Eq. (11.2.1.16), given by

c(0 = X / f £ , K / ) ' (E„C,=D. 01-2.1.21) so that

Xi.G>) c p (11.2.1.22) Z'(0) ^pl + (iGKf/(TAp)

Here we have noted Eq. (11.2.1.9) and we have used the Laplace

transform of the Mittag-Leffler function Ea \-Xpi:flr)a~\, viz.,

^ [ - ^ , / f l . ] - _ J _ _ . (H.2, .23)

At low frequencies, ignoring the contributions of all decay modes in Eq. (11.2.1.22), save the slowest one, and noting Eq. (11.2.1.18), we will have for the complex dielectric susceptibility Xi®)

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616 The Langevin Equation

Z(G)) 1 1 *±p- = - = - , (11.2.1.24) Z(P) 1 + (iorrf l(x\) 1 + (ial coc f

where the characteristic frequency coc (where the imaginary part of %(a>) has a maximum) may thus be defined as

G > c = r - 1 ( ^ ) 1 / f f » r - 1 ( r r J f ) , / ' T . (11.2.1.25)

The characteristic time xc of the relaxation process may be defined as

Tc=a>;1 = x(x\)'lla. (11.2.1.26)

Equation (11.2.1.25) illustrates how anomalous relaxation influences the complex susceptibility arising from the slowest relaxation mode in the noninertial limit. In anomalous relaxation therefore, the Kramers escape rate can be best understood as playing the role of a decay parameter in the Mittag-Leffler function governing the relaxation behaviour of the system. We remark that since 0< a<\ and x is of the order 10~10 seconds, the characteristic frequency (Oc of the normal diffusion will be reduced by a factor of the order 105 for a- 1/2. Moreover, the slowest decay mode, the parameters of which are determined by Eq. (11.2.1.18), will be highly non-exponential in character as its behaviour will be dictated by the Mittag-Leffler function, Eq. (11.2.1.19), even though the Arrhenius character (albeit stretched according to Eq. (11.2.1.18)) of the activation process is preserved.

Equation (11.2.1.24) may be used to evaluate the complex susceptibility in the low-frequency range (d)< 0)c) only. In order to describe the dielectric spectra at all frequencies, we can generalise the method of Chapter 2, Section 2.13. In order to accomplish this generalisation, let us now suppose that the relaxation function C(t) from Eq. (11.2.1.21), which in general comprises an infinite number of Mittag-Leffler functions, may be approximated by two Mittag-Leffler functions only:

Ct)^^Ea[-tlx)axA,] + l-^)Ea[-tlx)axlxw\ (11.2.1.27)

The parameters A| and %are defined by Eqs. (2.13.11) and (2.13.12) of Chapter 2, viz.,

W V - 1 V,>u-1 _ ( 1 ,.2.1.28) ^ - 2 + l / ( ^ ) W \-\ltef

Here \IX\, % and xm are the characteristic times of normal diffusion (o= 1), viz., \IX\ is the inverse of the smallest nonvanishing eigenvalue of the Fokker-Planck operator, which is usually associated with the long time behaviour of C(t) (slowest relaxation mode), the integral relaxation

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Chapter 11. Anomalous Diffusion 617

(or, in linear response, correlation) time Tint defined as the area under C(t):

oo

TiM=iC(t)dt = ,Zpcp/Ap, (11.2.1.29) o

and the effective relaxation time % defined by

r e / = - l / C ( 0 ) = l / X / A (H.2.1.30) (which gives precise information on the initial decay of Ct) in time domain). We shall show now that the above three time constants completely characterise the dynamic susceptibility in anomalous relaxation. Indeed, according to Eqs. (11.2.1.20) and (11.2.1.27) the spectrum of ;jf (fi>) is a sum of two Cole-Cole spectra

Z^= *! + ^ , (11.2.1.31) Z'(0) \ + iQ)IO)cf 1 + dCO/O^f

where the characteristic frequencies ct)c and C0w are given by

(OC=Z-T\)V,J, (Ow^T^it/Tw)1"7. (11.2.1.32)

Here, we implicitly suppose that the contribution of the high-frequency modes to %(co) may be approximated as a single Cole-Cole spectrum with characteristic frequency and half-width given by c%. Having evaluated Tin„ %, and A\ for normal diffusion, we may predict zi®) f° r

anomalous diffusion in all frequency ranges of interest (as we have seen for normal diffusion in Chapters 4, 6, 8, and 9). We remark that the parameters At and % can be estimated by the method described in Chapter 1, Section 1.20.

The results of this section apply to all the models of noninertial rotational Brownian motion, which we have developed in the earlier chapters. The scaling relation Eq. (11.2.1.10) is also relevant in free diffusion in the context of the application of the theory to the calculation of higher order relaxation functions such as those pertaining to Kerr-effect relaxation etc. We shall return to the Sturm-Liouville method later in the context of diffusion in phase space. In the following subsection, we shall briefly indicate how the continued fraction method of solution of the fixed axis rotator problem of Chapter 4 may be extended to anomalous diffusion.

11.2.2 Anomalous diffusion of a planar rotator in a mean field potential

In Chapter 4, we have described how the complex susceptibility Zi03) °f a fixed axis rotator moving in a potential

V(0) = Usm20, (11.2.2.1)

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618 The Langevin Equation

may be calculated by converting the problem of solving the Smoluchowski equation into the calculation of successive convergents of a differential-recurrence relation. Here, we indicate how a similar approach may be applied in anomalous relaxation. Referring to the potential, Eq. (11.2.2.1), and to the operator Eq. (11.2.32), we have, by expanding the distribution function W(j),t) in Fourier series

WW,t)= f; eip*cp(t), (11.2.2.2) p=-oo

a differential-recurrence equation

fp(t) = T-\DJ-'7Bp[fp_2(t)-fp+2(t)]-p2fp(t), (11.2.2.3)

where fp(t) = Re[cp(0]/(2tf) = cosP(/>)(t), (11.2.2.4)

B = U l2kT) is the barrier height parameter (we are using B here in order to avoid confusion with the anomalous exponent a). Applying the generalised theorem of Laplace transformation,

r i „ i f*1-ff/(*)-oA~ff/(Ol,-n. 0<<7<1, L oA / ( 0 = J^'\t-o> (11.2.2.5)

[s^fis), 1<<X<2,

we have from Eq. (11.2.2.3)

srfP (s)~fp(0) = (ST)1-* Bp [fp_2 (s) - fp+2 (5)] - p2fp (s),(11.2.2.6)

where the initial values ,(0) are given by Eq. (4.4.2.3) of Chapter 4. The solution of the three-term recurrence Eq. (11.2.2.6) can be obtained, as in normal diffusion, in continued fraction form (see Chapter 2, Section 2.7.3). We have (cf. Chapter 4, Eqs. (4.4.3.4) and (4.4.3.6)]

/i(0) + Z^- / 2 p + i (0)r t / , ( . ) - r ( s r )

(ST)° +l-B + BpS3(s)

SPW = T ^ 2 - - . >• (H.2.2.8)

(11.2.2.7) with successive convergents being calculated from the continued fraction

Bp

(STr + p2+BpSp+2(s)

Thus, just as normal diffusion, we may calculate the longitudinal complex susceptibility ^(<y) using linear response theory to any desired degree of accuracy by setting s = i6), viz.

M = l-toM, (11.2.2.9) z'(0) ^ MO)

where the static susceptibility ;f'(0) is (see Chapter 4, Section 4.4.3)

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Chapter 11. Anomalous Diffusion 619

10°'

io-2-

l(f-

1 2 3

• : r * i ^ " ' " 7 r T »

3 \

8 = 0.01 8 = 2.5 8 = 5.0

2 S. >

<r=0.5

4

Fig. 11.2.2.1. x'io) and x'i03) evaluated from the exact continued fraction solution [Eqs. (11.2.2.7) and (11.2.2.9): solid lines] for o - 0.5 and various values of B and compared with those calculated from the approximate Eq. (11.2.2.11) (stars). The low-(dotted lines) and high-frequency (dashed lines) asymptotes are calculated from Eqs. (11.2.2.15) and (11.2.2.16), respectively.

• ?

* 1 - (7=1.00

2-<r=0.75 3 - <r= 0.50

3 \

,1

A

8 = 5.0

V \

><

10 "

vr io" an

Fig. 11.2.2.2. The same as in Fig. 11.2.2.1 for B = 5 and various values of <T.

Z'(0) = M2N0Il(B) + I0(B)

(11.2.2.10) kT 2I0(B)

The ease of calculation of Xi03) represents the chief advantage over the Sturm-Liouville method of the continued fraction method as applied to anomalous diffusion.

The results of the numerical calculation of the normalised (//NQI (kT)= 1) susceptibility %(G>) from the exact continued fraction solution [Eqs. (11.2.2.7) and (11.2.2.9)] are shown in Figs. 11.2.2.1 and 11.2.2.2. In these figures, the results of calculations from the approximate Eq.(l 1.2.1.31), viz.

X(G>) -+-

1-A, (11.2.2.11) Z'(0) l + (ico/o)c)

a l + iicolo^f

are presented. Here, the characteristic frequencies CDC and C0w are defined

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620 The Langevin Equation

by Eq. (11.2.1.32) and are expressed in terms of the integral relaxation time Tim, the effective relaxation time %, and the smallest nonvanishing eigenvalue X\ for the normal diffusion. We have already calculated Tim, Tef, and X\ for the normal diffusion in a two-fold cosine potential in Sections 2.10, 2.11, and 2.12 of Chapter 2. These are given by Eqs. (2.10.27), (2.11.1.16), and (2.12.8), respectively. In the notation of this section, these equations are

^int ~ Te

IB

4B[I1(B) + I0(B)] J e-Bcos2«*erf2(V2Bsin^)^, (11.2.2.12)

AlT = n (-1)' -i-i

-IB 'p+1/2 (B) (11.2.2.13)

V (11.2.2.14)

l - e - ° ^ 0 2 p + l

_J0(B) + Il(B)

IQB)-IYB)

Thus, having determined lMi, T, and %, one may evaluate the complex dynamic susceptibility %co) for the anomalous relaxation from Eqs. (11.2.2.11) and (11.2.1.32). The results are given in Figs. 11.2.2.1 and 11.2.2.2. In these figures, the low- and high-frequency asymptotes, viz.

*'(0) ~\-Z-^i(OT)a +..

for <w—»0, and Z(<»)

- + ..

(11.2.2.15)

(11.2.2.16) X'(0) (i(DT)aTeS

for 6>—>°o, are also shown. Equations (11.2.2.15) and (11.2.2.16) can be obtained from Eqs. (11.2.1.22), (11.2.1.29), and (11.2.1.30). Apparently, both the low- and high-frequency behaviours of the imaginary part of X(G>) are completely determined by the normal diffusion characteristic times Tim and Tef.

Thus, Eq. (11.2.2.11) correctly describes the behaviour of ^(<») at all frequencies. The characteristic frequency coc = T~ (TJ^) a given by Eqs. (11.2.2.13) and (11.2.2.17), is shown in Fig. 11.2.2.3 as a function of a and B. We recall that in the high barrier limit (B » 1), the dependence of X\ on B is determined by the asymptotic equation (see Chapter 4, Section 4.4.3)

\T ~ 8Be~2B J'71 so that we have a simple asymptotic equation for 0)c, viz.

cocT~mi7T)Vae-1BICJ (B»\). (11.2.2.17)

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Chapter 11. Anomalous Diffusion 621

0 ' ' / ' * N * V ' . - I

- 5 - V ~ * "•• ^ '

Fig. 11.2.2.3. Characteristic frequency coc as a function of trand B.

11.3 Divergence of Global Characteristic Times in Anomalous Diffusion

We shall now consider how the global characteristic times associated with normal diffusion, such as correlation time, mean first passage time etc, may be treated in the context of anomalous diffusion. We shall demonstrate that, in general, global characteristic times defined in this sense do not exist in anomalous diffusion. This behaviour is to be expected in the fractal time random walk model due to the absence of a characteristic waiting time. We illustrate the divergence of global characteristic times in anomalous diffusion by considering the integral relaxation time rim. We remark that, as far as the integral relaxation time is concerned, the formal definition of that time as the area under the decay curve of the electric polarisation or the magnetisation, still holds good in anomalous as well as in normal diffusion. If we consider the simplest possible case of relaxation, namely, the normalised after-effect

function f(t) = e~tlT for the normal diffusion, for which the Laplace transform is

oo

f(s) = Le-'/T=\f(t)dt = - ^ - (11.3.1) 1 J * ST + l

(in the notation of Section 1.22.3) and set s = 0, we have the integral relaxation time Tin, = / (0) = T. In anomalous relaxation governed by the Cole-Cole equation (see Section 1.22.3)

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622 The Langevin Equation

£$- = , (11.3.2) Z(0) l + isrf

the after-effect function/(f) is the Mittag-Leffler function, Ea\ -tiff 1,

with Laplace transform

fs) = LEa[-it,rr] = - - ^ . (11.3.3)

The function f(s), when evaluated as 5 —> 0, tends to infinity, except in

the special case of normal diffusion, where <J= 1. This behaviour, in retrospect, is obvious because of the long tailed character of the Mittag-Leffler function [see Eq. (11.2.1.14)], and emphasises the nonlocal temporal character of the fractal time relaxation process, which may arise from [4] obstacles or traps which delay the motion of the rotator and introduce memory effects into the motion.

We shall now turn our attention to the calculation of the mean first passage time. We shall commence by outlining the calculation of this quantity for normal diffusion.

11.3.1 First passage time for normal diffusion

We commence this subsection by defining the mean first passage time for a stochastic process. This is defined as the time when the process, starting from a given point, reaches a predetermined level for the first time and is a random variable [5]. Thus, one may introduce the concept of a mean first passage time (MFPT). Following Risken [6], we may in general consider first passage time as the time at which a random variable first leaves a given domain. For simplicity, we shall confine ourselves (following the exposition of Risken [6]) to the one-dimensional case of a random variable £(f) starting with the realisations at t = 0 with <f(0) - x say, the first passage time 7\ is the time when g(t) reaches a boundary for the first time. Let us suppose that the boundaries (xu x2) are absorbing, then £(T) =x2 or £(7\) =xi [6]. 7\ varies from realisation to realisation of the process, so, Tx is a random variable. The probability density function underlying g(t) is then W(x,t\x',0) so that Wdx is the probability for the random variable £(?), starting at t - 0 from the fixed point £(0) = x, to reach x at time t. Furthermore, if %i) reaches either x

or x2 for the first time, we no longer count its realisations. Hence, W must satisfy the Fokker-Planck equation

dW ^L = LFPW, (Xl<x<x2), (11.3.1.1) at

with the initial conditions

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Chapter 11. Anomalous Diffusion 623

W(x,t\x',0) = S(x-x) (11.3.1.2)

and the boundary conditions W(x,t\x',0) = 0, forx = XiOrx = x2. (11.3.1.3)

The probability Q(x,t) of realisations that have started at x = x and which have not yet reached either boundary in a given time t is

x2

<2(x',0=J W(x,t\x',0)dx. (11.3.1.4)

The probability dQ of a realisation reaching one of the boundaries in a time interval [t,t + dt] is thus

-dQ = -\ W(x,t\x',0)dxdt. (11.3.1.5) x\

Therefore, the distribution function q( x ,T\ ) for the first passage time T

is

q(x,,Tl)=--^rQ(x',Tl)=—^-tw(x,T1\x',0)dx. (11.3.1.6) dTx dTx x\

The moments of the first passage time distribution are [6]

(?;"(*')) = j7iB$(jc',7i)<*7'1 = \pnx,x)dx , (11.3.1.7) 0 Xi

where 00

Pox,x') = -\ Wx,Tx \x',0)dTl =5x-x) (11.3.1.8) 0

since the particle was definitely at x = x at time t = 0. Furthermore, by integrating by parts

00

pn(x,x') = n\ Tf^Wix,^ \x',0)dTv (n > 1). (11.3.1.9) 0

Hence, on applying the Fokker-Planck operator LFP to Eq. (11.3.1.9), we have the following set of coupled differential equations [6]

LFpPn(x<x')= ~npn_i(x,x'). (11.3.1.10)

Thus, we can obtain pn(x,x') by solving this set of equations successively starting with the first one, the boundary conditions for pn

being the same as those for W: pn (x, x') = 0 for x = Xi or x = x2.

The first equation of the set of Eq. (11.3.1.10), viz.,

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624 The Langevin Equation

LFPplx,x') = -8(x-x) (11.3.1.11) describes [6] the stationary probability density. If at x' a unit rate of probability is injected into the system, the integration of the first equation from x' - e to x' + e yields

J(x'+£,t)-J(x'-e,t) = l, (11.3.1.12) where

J(x,t)-WdV dW^

+ D g dx dx j

(11.3.1.13)

is the probability current given by the continuity equation

^ 4 = 0. (11.3.1.14) at ox

Coffey [7] and Coffey et al [8] have shown how for normal diffusion (cr= 1), the MFPT for a fixed axis rotator moving in the potential Us'm20 and for space rotation in the potential Kvsin & (i?is the polar angle) may be calculated using the differential-recurrence relations Eqs. (4.4.3.1) and (7.4.2.1) which are generated by averaging of the Langevin equation. The calculation in Ref. [7] for a fixed axis rotator is accomplished by_ noting that from Eq. (11.2.38), we have for the Laplace transform W(0,s)

sW(0,s)-W(<p,O) = i - Cd<i> \d2W

T d<p2 (11.3.1.15)

which, for becomes

a delta function initial distribution of orientations at (p'',

sW(/),s)-S<l>-o$

ld2W

r d(/)2 (11.3.1.16)

yielding Green's function (the transition probability) for the system. The use of the differential recurrence relation of Eq. (4.4.3.1) to calculate the MFPT is based on the observation that if, in Eq. (11.3.1.16), one ignores the term sW(0,s) which is tantamount to assuming that the process is quasi-stationary, i.e., all characteristic frequencies associated with it are very small, one then has

-S<!>-</>') = d(/> £d<p

\d2W

T dtp2 (11.3.1.17)

which is Eq. (11.3.1.11). The MFPT, i.e., the time at which the random variable £(t) specifying the angular rotation of the rotator first leaves a domain D defined by the absorbing boundaries <j>\, <jh, where W(0,t\0',O) and consequently W(j),s = 0|^') say, vanishes (because as we have seen

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Chapter 11. Anomalous Diffusion 625

the boundary conditions for both W and W are the same) may now be calculated. We may do this because Eq. (11.3.1.17) is now identical to Eq. (11.3.1.11) for px(x,x), so that by Eq. (11.3.1.7)

<h (W))=J W(M)dfi. (11.3.1.18)

<h Equation (11.3.1.18) is useful because it clearly demonstrates that (Ti) may be calculated from the zero frequency limit of the solution of Eq. (4.4.3.8) et seq. of the recurrence Eq. (4.4.3.1) for delta function initial conditions. For example, here we are interested [7] in escape out of the domain D specified by a single cycle of the potential, that is, a domain of arc length K (the domain of a well for the potential given by Eq. (11.2.2.1)). Since the potential of Eq. (11.2.2.1) has a maximum at <j>- Jtl 2 and minima at 0 = 0 and (/)= K, it is convenient to take our domain as the interval -Ttl 2 < (p<7cl2. Thus, we will impose absorbing boundaries at 0= -nl 2 and (/>= nl 2. Next, we shall impose a secondary condition that all the rotators are initially located at the bottom of the potential well so that </>'= 0. Hence, we have in the zero frequency limit

(Ti) = I Z/2p+i(0)cos(2p + i ) ^ , (11.3.1.19) -nil P=0

where /2/>+1(0) must be calculated from Eqs. (4.4.3.1) and (4.4.3.8).

Thus, the MFPT is

*e=2(Tl) = 4fj^-f2p+l(0). (11.3.1.20) P=o lP +1

In the high barrier limit, B » 1 this reduces by the properties of the Riemann-Zeta function /3(\), Eq. (23.2.30) of [10] (details in [9]) to

r * — e 2 B . (11.3.1.21) e AB

On the other hand, for small barriers, B « 1, we have

Te=^- (11.3.1.22)

by the properties of the Riemann-Zeta function $3 ) , Eq. (23.2.32) of [10], viz.

00 (—]\P IT3

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626 The Langevin Equation

In like manner, for 3D rotation in the potential #vsin2 tf, one may show [8] that for the escape time out of the domain 0 < t? < K 12

r.-*W-2f%fr + y ' / M < « > . 01.3.1*, „=o 2 (n + l)\n\

where /2n+1(0) is the zero frequency limit of the solution of Eq. (7.4.2.1) and (Ti) is the MFPT from 0 to nl 2. The high barrier limit B » 1 again yields Brown's result [3]

Te=T—B-V2eB, (11.3.1.25)

while for vanishing small barrier heights, we retrieve the result of Klein [11], namely

Te=2t\n2. (11.3.1.26) An approach to the calculation of the escape time which is essentially equivalent to the one outlined above has been given by Malakhov and Pankratov [12] in their exact solution of the Kramers escape problem for piecewise parabolic potentials, which they obtain by posing the solution as a Sturm-Liouville problem [12]. We shall now demonstrate how the method of calculating the first passage time distribution, which we have just described, may be generalised to anomalous diffusion.

11.3.2 First passage time distribution for anomalous diffusion

In order to illustrate the calculation of the first passage time distribution distribution for anomalous diffusion in configuration space, we shall use an example given by Rangarajan and Ding [5] as this provides an excellent exposition of the principles involved. The example is, the calculation of the first passage time distribution for the translational anomalous diffusion of a free particle, where two absorbing barriers are situated at finite distances from the origin, where the source is situated. The appropriate fractional diffusion equation is Eq. (11.2.37), viz.

^Wx,t) = Ka0D]-a^-TWx,t), (11.3.2.1) at dx

The boundary conditions assuming that the absorbing barriers are situated at x = -b and x = a are

W(-b,t) = OandW(a,t) = 0. (11.3.2.2) The initial condition is

W(x,0) = S(x). (11.3.2.3)

This W is the transition probability or Green function of the system.

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Chapter 11. Anomalous Diffusion 627

The distribution function for the first passage time Tx is then

d q(t) = f W(x,t)dx,

dt (11.3.2.4)

-b where we have used Eq. (11.3.1.6) with t = Tu x = 0 and the notation T\ for the first passage time random variable. The mean first passage time may now be calculated from the transition probability or fundamental solution Why noting that from Eq. (11.3.1.7)

Tl) = \tq(t)dt. o

Now, noting Eq. (11.2.6), Eq. (11.3.2.5) may be rewritten as

&>-£*•> .v=0

(11.3.2.5)

(11.3.2.6)

which is a convenient way of evaluating the MFPT from the Laplace transform of the transition probability. We solve the fractional diffusion Eq. (11.3.2.1) by the method of separation of variables. We have writing as in Section 11.2.1

W(x,t) = X(x)F(t) (11.3.2.7)

and choosing A, as a negative value to ensure oscillatory solutions KX"(x) = -AX(x), (11.3.2.8)

F = -U/K)D,aF, (11.3.2.9)

where X denotes the separation constant. The solution of Eq. (11.3.2.9) is again the Mittag-Leffler function as in Eq. (11.2.1.16), while Eq. (11.3.2.8) has solutions of the form

X(x) = Ccos(jA/K x) + C2 sm(y/A/K x), (11.3.2.10)

where C\ and C2 are two constants of integration. In view of the boundary conditions embodied in Eq. (11.3.2.2), the condition that a solution should exist is

*L = _nW = 2 (H.3.2.11) K (a + b)2 n

say, n > 1. The solution of Eq. (11.3.2.1), noting the boundary conditions, Eq. (11.3.2.2), may now be expressed in the form

W(x,0 = XA.sin

n=\

n7t(b+x)

a+b (a + b)2 Kt° (11.3.2.12)

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628 The Langevin Equation

The Fourier coefficients A„ are determined by imposing the initial condition, Eq. (11.3.2.3), viz.,

nnb + x) W(x,0) = <?(*) = £ 4 , sin

«=i a + b (11.3.2.13)

We have by orthogonality, on multiplying both sides ofEq. (11.3.2.13) by sin AH'(b + x) and integrating between the limits x = -b,a,

2 . (Knb An = sin

a + b a + b The complete solution for the transition probability is then

W(x,t) = sin -(Knb

a + bn=x ya+b sin

nK(b+x)

a + b

n V

(a+b)2 Kf

(11.3.2.14)

.(11.3.2.15)

Thus, we have from Eq. (11.3.2.4) and (11.3.2.15) the first passage time distribution function q(t), viz.

*>=4\*i-. ' dt [ ^2n + \)7t sin

(2n + \)7tb

a+b

(2n + \)n

a+b

Here, we have used the fact that

nTtib + x)

Kf

(11.3.2.16)

sin a + b

dx = 2(a + b)

nn

for odd n and zero for even n. In order to evaluate the derivative of the Mittag-Leffler function in Eq. (11.3.2.16), we introduce the generalised Mittag-Leffler function [14]

V ^ I w T ^ f ' («,/*><>), (11.3.2.17)

where, obviously, the Mittag-Leffler function Ea(z) is EaX(z). If we

combine Eq. (11.3.2.17) with the rule for differentiating a function of a function, we obtain

— Ea-Xf) - -Xaf-xE'a(-ha) = -Af-lEaa(-Ata), (11.3.2.18)

dt where the prime denotes the derivatives

kz k-\

• = z (k + \)zk

•=z a Z ~Jz ff Z ~ t t o m + afc) kti T[l + ok + T)]~fe aT(jk + o)

because T[l + a(k +1)] = a(k +1) T\a(k +1)], so that

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Chapter 11. Anomalous Diffusion 629

dz a££ F(ak + a) a (11.3.2.19)

(Eq. (11.3.2.19) may also be obtained [5] using the Fox functions [13]). Equation (11.3.2.16), on combining with Eq. (11.3.2.18), yields the first passage time distribution as

q(t) = AxKt a-\

(a + by £(2n + l)sin n=0

(2n + \)xb

a + b o,o a + b Kf

(11.3.2.20) which is the result of Rangarajan and Ding [5].

Having explicitly determined the distribution q(t), we may now calculate the MFPT from Eq. (11.3.2.6). Essentially, the problem is to evaluate

g(s) = ] e-xtf-xEaa-lf)dt. (11.3.2.21)

We have

0

t'0r((7 + ak)0

e-«t°-lZ t^Ta+ak)

(-At°)k dt

(11.3.2.22) 1 °° b

k=0

where we have used a well known result in Laplace transformation [10]. Equation (11.3.2.22) has the explicit form

g(s) = — = — - — . (11.3.2.23) l + As-ff A + sa

The MFPT is entirely determined by the derivative of this function w.r.t.s at S = 0 as dictated by Eq. (11.3.2.6). We have

- * ' ( ' ) = •

as A2s-ff+l+sff+l+2As

(11.3.2.24) (Jl + sa)2

which becomes infinite when evaluated at 5 = 0 and a< 1 unless a- 1, to which case we shall return below. Thus, in anomalous diffusion, we may conclude that the MFPT for two absorbing barriers, at finite distances from the origin, is infinite. This is quite unlike normal diffusion as we now demonstrate. Eq. (11.3.2.24) for <r= 1 has the value when evaluated at 5 = 0

-g'Q) = X~2. (11.3.2.25) Thus, Eqs. (11.3.2.6) and (11.3.2.20) yield

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630 The Langevin Equation

) _ j(g^££_i_^r(2»+i)«»i N ' A ntS(2« + i)3 [ a+ft J

For symmetric barriers a = A, b = A), Eq. (11.3.2.26), may be written as

) = i^.jziyL (113227) T3K„=o (2n + l)3

The sum of the series in Eq. (11.3.2.27) is the Riemann-Zeta function PO) = tf3 /32, Eq. (11.3.1.23). Thus, we have

(T1) = A2/(2K) (11.3.2.28) which is, as to be expected, the transposed Einstein result for the displacement of a free Brownian particle, if we identify A2 as ((A xf) and (Ti) as t. Thus, only in the exceptional case of normal diffusion does the MFPT for two absorbing barriers, at finite distances from the origin, exist. The fact that the MFPT does not in general exist, is entirely in accordance with the CTRW picture, in which the random walker may be trapped in a given configuration for an arbitrary time before making a jump.

An important conclusion that may be drawn from the solution of the simple two absorbing barriers problem, Eq. (11.3.2.16) et seq. is that the MFPT will also not in general exist (unlike normal diffusion), for anomalous diffusion in a potential. This follows from Eq. (11.2.1.16) where the calculation of the probability density W<j>,t) is posed as a Sturm-Liouville problem. In which case, the variation of W is always like

whence it follows from Eq. (11.3.2.24) that the MFPT is in general infinite, which is entirely consistent with the assumption of a waiting time distribution of the form of Eq. (11.2.26).

The divergence of all the global characteristic times for anomalous diffusion, as defined in their conventional sense (which we have amply illustrated in this section, and which is a natural consequence of the underlying Levy distribution) rendering them useless as a measure of the relaxation behaviour, signifies the importance of seeking characteristic times for such processes in terms of their frequency domain representation. This has been accomplished in Sections 11.2.1 and 11.2.2. There we have shown, in particular, that the anomalous diffusion process in the periodic potential will have two associated characteristic times, namely the inverse of the over barrier frequency coc and the inverse of the well frequency 0)w. Thus the global relaxation times of the normal diffusion process (tin, and %) appear as parameters of the anomalous

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Chapter 11. Anomalous Diffusion 631

diffusion process. Moreover, we have the novel and surprising result that the anomalous dielectric relaxation at any frequency is governed by the characteristic times A^1, Tint, and % of the normal diffusion processes. Thus, the difficulties associated with divergent global characteristic times in anomalous diffusion may be avoided. In conclusion, we remark that throughout this discussion, we have confined ourselves to an interpretation of Cole-Cole behaviour. Phenomenological fractional ordinary differential equations in the time domain describing other types of anomalous relaxation such as Cole-Davidson behaviour have been given in Ref. [34].

We shall now discuss the role played by inertial effects in anomalous relaxation on the basis of various generalisations of the Klein-Kramers equation to anomalous diffusion. Moreover, we shall demonstrate that the exponential decays, typical of the relaxation functions calculated from the normal Klein-Kramers equation, do not in general simply map onto Mittag-Leffler decays, unlike the limiting noninertial diffusion.

11.4 Inertial Effects in Anomalous Relaxation

We have seen throughout this book that the omission of inertial effects in the relaxation process gives rise, in the context of dielectric relaxation, to the phenomenon of infinite dielectric absorption at high frequencies. Restoration of the inertial terms in the Langevin equation, however, yields a return to transparency at high frequencies. The simplest example of this behaviour yields the Rocard equation, Eq. (1.15.1.9) of Chapter 1. This equation pertains to dielectric relaxation of both fixed and moving axes free rotators. It arises from the second convergent of the infinite continued fraction solution for the complex susceptibility described in Chapter 3 and exhaustively described in Chapter 10. Moreover, it is an accurate approximation to the exact solution for small inertial effects, and exhibits the desired correction (i.e., return to transparency) to the Debye equation at high frequencies. In the context of Brownian motion in the presence of an external potential, we have also seen that inertial effects may be included to any desired degree of accuracy by either averaging the inertial Langevin equation or by constructing the Klein-Kramers equation for the evolution of the probability density function in phase space. The resulting differential recurrence relation may then be solved by matrix continued fraction methods to yield the complex susceptibility.

Exactly the same considerations regarding the failure to return to transparency at high frequencies apply to the Cole-Cole equation for the complex susceptibility [Eq. (11.3.2) or Eq. (1.22.3.22)] as to the Debye

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632 The Langevin Equation

equation. Thus, once again in order to give a physically meaningful description of the high-frequency behaviour, inertial effects must be included [16] in anomalous relaxation just as in normal relaxation. At the time of writing, an approach based on the Langevin equation is still under active development [17,18,19] and such solutions as exist [16,20,21] have been mainly based on attempts to extend the Klein-Kramers equation to fractional dynamics. Since the latter approach, although by no means complete, has been more fully developed than the Langevin one, we shall describe it here. The various generalisations of the Klein-Kramers equation which have been proposed have been fully described by Metzler [22] and we shall merely list them here. Moreover, we shall select the particular generalisation of the Klein-Kramers equation which provides a physically acceptable description of the far-infrared absorption in dipolar fluids. The generalised Klein-Kramers equation which achieves this appears to be that proposed by Barkai and Silbey [23] rather than that given by Metzler and Klafter [1,24] which we first describe.

11.4.1 Slow transport process governed by trapping

Metzler and Klafter [24] and Metzler [22] have proposed a fractional Klein-Kramers equation, which, according to them, corresponds to a multiple trapping picture where the tagged particle executes Brownian motion in accordance with the Langevin equation

mx(t) = -mj3x(t)-—V[x(t),t] + F(t), (11.4.1.1) dx

where F(f) is the usual white noise driving force. The particle then gets successively immobilised in traps whose mean distance apart is

A = (kT/m)T*, (11.4.1.2) where T* is the mean time between successive trapping events. The time spans spent in the traps are governed by the waiting time probability density function

w(t) ~ Aa rl-a, ( 0 < « < 1 ) , (11.4.1.3) which corresponds, as before, to a divergence of the characteristic waiting time due to [4] "the relatively frequent occurrence of long waiting times". The process is thus [22] a separable fractal time random walk with finite jump length variance and so describes subdiffusion. The divergence of the average waiting time combined with the separability of the process is very significant if the walk is used to model dielectric relaxation as it leads to infinite integral absorption. We illustrate this by referring to the motion of a fixed axis rotator the normal Brownian rotation of which is described by the Langevin equation (see Chapters 4 and 10)

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Chapter 11. Anomalous Diffusion 633

I$(t) = -lMt)~V[ftt),t] + Mt). (11.4.1.4) 30

The procedure based on a generalised Chapman-Kolmogorov equation in phase space proposed by Metzler and Klafter [1,22,24] then leads assuming the diffusion limit to the following generalisation of the Klein-Kramers equation

~d7~°Dt T •,dw idvdw i a -„A krd2w)

—(</>W)+-, (11 .4 .1 .5 ) - d ) — + ^ + J3 .vr..,.

d</> Id<j> d(f> \d(j> I d</>2 ) where W = W(<f>,0,t) and the Debye relaxation timer is here identified with T*. According to Metzler [22], the entire Klein-Kramers operator in the square brackets acts non-locally in time, i.e., drift friction and diffusion terms are under the time convolution and are thus affected by the memory. In the present application, the ensuing non-local behaviour causes divergence of the absorption coefficient as we shall presently illustrate. Thus, a model based on a fractional Klein-Kramers equation of the form of Eq. (11.4.1.5) cannot produce the desired return to transparency at high frequencies. Before illustrating this behaviour however, we remark that according to Metzler and Klafter [1,24] and Metzler [22], Eq. (11.4.1.5) reduces in a manner analogous to the reduction of the Klein-Kramers equation, to the Smoluchowski equation in the high friction limit. In other words, Eq. (11.4.1.5) simplifies, in the high friction limit, to the fractional Fokker-Planck Eq. (11.2.38) for the evolution of W in configuration space </> only, if the replacement cr —> or is made. Equation (11.4.1.5) may also be separated, just as the normal Klein-Kramers equation, in the time and space variables in the form

W = &ty,f)F(t). (11.4.1.6) This procedure yields the time behaviour corresponding to a given phase space eigenmode <£>pn of the normal diffusion solution as a Mittag-Leffler function instead of the exponential decay characteristic of the Klein-Kramers equation just as in Eq. (11.2.1.12). Moreover, the scaling laws for the eigenvalues of the Fokker-Planck operator described in Section 11.2 will also apply. In addition, one stationary solution will be the Maxwell-Boltzmann distribution, while the other, just as in Chapter 1 will pertain to a steady diffusion current over a barrier [1]. According to Metzler and Klafter [1,24], the second solution leads to a generalisation of the Kramers theory of escape of particles over potential barriers to anomalous diffusion including inertial, and so, energy controlled diffusion effects.

Assuming a solution of Eq. (11.4.1.5) in the form of Eq. (11.4.1.6) corresponds to posing the solution as a Sturm-Liouville

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634 The Langevin Equation

problem. However, the problem may be also be cast into the form of a solution to a differential recurrence relation, and so as a calculation of successive convergents of a continued fraction in the frequency domain. This is accomplished, just as in normal diffusion, by means of a Fourier expansion in the Hermite polynomials in the angular velocity and the circular functions to yield, inter alia, the Laplace transform of the complex susceptibility as an infinite continued fraction in the frequency domain, as we now describe.

11.4.2 Calculation of the complex susceptibility

In Eq. (11.4.1.5), we assume, in the present context, that a weak uniform electric field E applied along the initial line is suddenly switched off at t = 0, so that linear response theory may be used to describe the ensuing response. Now, the fractional derivative in (11.4.1.5) acts only on functions of the time. Hence, we may seek [16] a solution of Eq. (11.4.1.5), just as in normal Brownian motion, in the form of the Fourier-Hermite series

in n=oq=-°°2 n\

where the Hn are the Hermite polynomials and J] = ^1 l2kT). The linearised initial distribution is

W(M0)=-^-e-^2 1 ME 1 + —-cos<z> + ...

kT

(11.4.2.2)

Straightforward manipulation [16] of the recurrence relations of the Hn, essentially as in Chapter 10, then leads by the orthogonality of the H„, to a differential-recurrence relation for the Fourier coefficients cn (t), viz.,

cn,?(0 + r-%D^jj=[cn+1,?(0^^

where y = \l(2rf 01) is a dimensionless inertial (Sack's) parameter [25]. It is obvious from Eq. (11.4.2.3) how all recurrence relations associated with the Brownian motion may be generalised to fractional dynamics governed by Eq. (11.4.1.5). Since we consider the linear dielectric response, this may be written for the only case of interest, namely, q = 1 (q = - 1 follows by symmetry) as a recurrence relation for cn>1 (t). The

Laplace transform of this set, noting the generalised integration theorem of Laplace transformation, Eq. (11.2.2.5), is

yTs)a c01 (s) + ijyi2cu s) = coa (0)yr (rsf'1, (11.4.2.4)

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Chapter 11. Anomalous Diffusion 635

r(Ts)a+njcnl(s) + i4ri2[cn+ul(s) + 2ncn_u(s)]^0, (11.4.2.5)

(n > 1). In writing Eqs. (11.4.2.4) and (11.4.2.5), we have noted that on account of the initial condition, Eq. (11.4.2.2), and the orthogonality property of the //„,all the cnl (0) will vanish with the exception n = 0.

Equation (11.4.2.4) is a three-term algebraic recurrence relation for the cnl (s) which may be solved exactly for c0l (s) using standard

continued fraction methods as in the Brownian motion (see Chapter 2, Section 2.7.3):

. rr(STf-lc0l(0) T(sT)a-lc0l(0) 0 ,1 V< V<r\a 4- / . / v /9 .?_ (l\ . , ry 1 ysT)a + i4rl2Sxs) sT)a +

1 + y(sz)a + ^ 2 + r(ST)a+....

(11.4.2.6) where successive convergents are calculated from the continued fraction

Sn(s) = ^L=^ . (11.4.2.7) y(sT)a+n + ijyl2Sn+ls)

The normalised complex susceptibility %(a>) may be given by linear response theory

%(0) = ^d = l - i C 0 ^ ^ - - (H.4.2.8) Z(0) c01(0)

We remark in passing that c0tl (s) will also yield the Laplace transform of the characteristic function of the configuration space probability density function. Equations (11.4.2.6-8) then lead [16] to the generalisation of the Gross-Sack result [25,49] for a fixed axis rotator (Chapter 3, Section 3.4) to fractal time relaxation governed by Eq. (11.4.1.5), namely,

Xi.co) = \ y(mf_ (11.4.2.9)

yion)a + r- — 1 + y(icor)a +

2 + y(io)T)a +.... Equation (11.4.2.9), in turn, can be expressed in terms of the confluent hypergeometric (Kummer) function M (a,b,z):

2(0)) = 1—(lT0) nM(\,\ + y\\ + (izco)a\y). (11.4.2.10)

1 + 0'tttf)

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636 The Langevin Equation

Equation (11.4.2.10) can be readily derived by comparing Eq. (11.4.2.9) with the continued fraction

M(a,b,z) 1

(b-l)M(a-l,b-l,z) b j , , az

b-z + - (a + l)Z

(11.4.2.11)

b + l-z + ... where

a=l,z= %and b = 1 + y\\ + (iTO))a 1,

by noting that M(0,b-l,z) = l [10]. The continued fraction (11.4.2.11) can be obtained from the known recurrence relation [10]

b(l-b + z)M(a,b,z) + b(b-l)M(a-l,b-l,z) = azM(a + l,b + l,z).

We remark that M(\,l + b,z) = bz~bezy(b,z), where

r(b,z)=je-'tb-1dt o

is the incomplete gamma function [10]. For a- 1, Eq. (11.4.2.10) can be reduced to Sack's result [Eq. (3.19c) of Ref. [25]; see also Eq. (3.4.1.10) of Chapter 3].

Successive convergents of Eq. (11.4.2.9) may be calculated as in Chapter 10. The first convergent is the generalisation to fractional dynamics of the Debye result given in Chapter 1, Section 1.22.3, namely

XiC0)= 1 (11.4.2.12)

Z(o>) = „ „ , g , (11.4.2.13)

The next convergent, namely, J_

1 + io)T)a + yicor)2

represents a generalisation of the Rocard result [Eqs. (1.15.1.9) or (3.4.1.11)]. Moreover the effect of the Brownian (intertrapping) dynamics appears explicitly through the inertial parameter y. Thus, one expects intertrapping effects to manifest themselves at high frequencies. We also remark that Eq. (11.4.2.11), unlike the Rocard equation, does not obey the Gordon sum rules [26] [see Eq. (11.4.2.17) below] which will have an important bearing in our subsequent discussion.

Equation (11.4.2.3) may also be used to calculate the angular velocity correlation function (AVCF) in the fractional dynamics. We have from Eq. (11.4.2.3) with n = 1 and q = 0,

clfi(t) + (r-a I y) 0DJ-aclfi(t) = 0. (11.4.2.14)

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Chapter 11. Anomalous Diffusion 637

0.0 H — ~ • ' i •• "i " i ' 0.0 0.1 0.2 0.3 0.4 0.5

t

Figure 11.4.2.1 Normalised AVCF cJO/cJO) for a= 0.7 and y= 0.1 (1 and 1'), 0.5 (2 and 2'), 0.9 (3 and 3'). Note the pronounced increase in the long time tail as yis increased.

On noting Eq. (11.2.30) and choosing a sharp set of initial values of 0(0), say $,, Eq. (11.4.2.14) has a solution in the time domain

clo(t) = 0oEa[-(t/T)a/r, (11.4.2.15)

where, as usual, Ea(z) is the Mittag-Leffler function. The equilibrium AVCF cat) = (^(0)^(0) then follows by multiplying Eq. (11.4.2.15) by $, = 4 ( 0 ) and averaging over a Maxwell distribution of $,, thus

c0)t) = kTII)Ea[-(tlT)aly\. (11.4.2.16)

Equation (11.4.2.16) represents the generalisation of the conventional result based on the Ornstein-Uhlenbeck (inertia-corrected Einstein) theory of the Brownian motion to fractional dynamics (see Chapter 3). By way of illustration, we show in Fig. 11.4.2.1 the variation of the AVCF with inertial parameter y, for a fractional index a= 0.7. The long time tail (t» t) due to the asymptotic t~a like dependence of the Mittag-Leffler function is apparent as is the stretched exponential behaviour at short times (t« f).

Despite the success in predicting reasonable behaviour of the AVCF, and the recovery of the Debye result in the limit of small inertial effects, the theory based on Eq. (11.4.1.5) and the exact continued fraction solution, Eq. (11.4.2.9), or any of its convergents, utterly fails to explain the return to transparency at high frequencies as is illustrated in Fig. 11.4.2.2. Here, we show the absorption coefficient co^Xco) for a- 1 (normal diffusion) and ar=0.5. It is apparent that just as in the Debye theory (Chapter 1, Section 1.15.1), Eq. (11.4.2.10) predicts nonphysical behaviour of the absorption coefficient as it does not predict the return to transparency at high frequencies. The effect of including inertia based on

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638 The Langevin Equation

1.5

I" 0.0

.

/' /

/ /

/ /

ff=0.5, ^=10

cc — \.J, y—v o=1.0, y=0 a=1.0, y=10

i i •

log.Jan)

Figure 11.4.2.2. (Ox\<o) vs. log10(fi«) for a= 0.5 and y= 10 (solid line), ar= 1 and 7=10 (dot-dashed line with the Debye plateau), a=0.5 and y=0 (dotted line), and a= 1.0 and / = 0 (dashed line).

the model of Eq. (11.4.1.5) is simply to produce an enhanced Debye plateau for all a < 1, thus the Gordon sum rule for the dipole integral absorption of one-dimensional rotators [26,35]:

\coxX<o)dco = xNn2

4/ (11.4.2.17)

is violated [20]. (It is significant that the right-hand side of Eq. (11.4.2.17) is determined by dipole parameters only and does not depend on the model parameters a and £). This unphysical behaviour in the context of the present model appears to be a consequence of allowing the fractional operator, or memory function, to operate on the Liouville terms in the fractional Klein-Kramers equation, so that, the form of the underlying Boltzmann equation is not preserved.

Another way of regarding the divergence of the integrated absorption is to recall that in Brownian motion, the characteristic time associated with the far-infrared absorption for a fixed Debye time is essentially the angular velocity correlation time as expressed through Sack's parameter y. In the separable fractal time random walk, however, in Eq. (11.4.1.5) unlike in Brownian motion, the angular velocity correlation time no longer dictates any particular time scale for the process. Such behaviour results from the fact that both angular velocity and configuration variables are subject to the same separable fractal time random walk, since a fractional derivative appears in the Liouville term. As is usual in separable fractal time random walks, no global characteristic time scales exist. Hence, the problem must be considered in the frequency domain. Thus, the only meaningful time scale is that provided

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Chapter 11. Anomalous Diffusion 639

(at low frequencies) by the inverse of the characteristic frequency at which xi®1) has a maximum. No very short time scale exists because the far-infrared absorption is effectively infinite. Thus, it is impossible to use the inverse of the maximum far-infrared absorption frequency to define a very short time scale. The lack of such a short time scale is synonymous with infinite integrated absorption.

Faced with these difficulties, we shall presently illustrate that if a generalisation of the Klein-Kramers equation, first proposed by Barkai and Silbey [23], where the fractional derivatives do not act on the configuration variables, is used, then the desired return to transparency at high frequencies is achieved. Moreover, the Gordon sum rule, Eq. (11.4.2.17), is satisfied. Before illustrating this, it is however instructive to comment on the telegraph equation [27] which has been used by several authors in order to write down a partial differential equation in configuration space from a fractional, or indeed, a conventional Klein-Kramers equation, despite the refutation of the use of such an equation for that purpose by Hemmer [28] and Risken[6].

In conclusion of this subsection, we remark that the divergence of the integral absorption is not unusual in models which incorporate inertial effects. For example, in the well-known Van Vleck-Weisskopf model [31,35,36], the divergence results from the stosszahlansatz used by them, just as in the problem under discussion.

11.4.3 Comment on the use of the telegraph equation as an approximate description of the configuration space distribution function including inertial effects

The telegraph equation "approximation" [27] for the evolution of the space distribution, including inertial effects (which effectively replaces the parabolic Klein-Kramers equation with a damped wave equation in configuration space, which is of course of hyperbolic type), has been criticised by Hemmer [28] and Risken [6]. They assert that the telegraph equation provides a poorer approximation to the configuration space distribution function than the (noninertial) Smoluchowski equation. The error in using the telegraph equation to describe the configuration space distribution, stems from the singular perturbation procedure (exhaustively described in Risken's book [6]) used to derive a perturbation expansion for the configuration space distribution from the phase space distribution function. Indeed, it is impossible to derive any simple partial differential equation in configuration space, which includes inertial effects from the phase space Klein-Kramers equation. Thus, in general, it is impossible, either in fractional or normal dynamics, to draw any conclusions about

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640 The Langevin Equation

the inertial behaviour of the system from a telegraph equation. The sole exception is the Rocard equation, which may be obtained from the telegraph equation.

The telegraph equation for this problem is

/ a2 1 a2

^f0,t) + -—f(t>,t) = -~f(^>,t), (t>0). (11.4.3.1) at (, dt T d<j>

Assuming an after-effect solution of the form

f(0,t) = ^-2K

u.E 1 + g(t) —-COS(Z>

kT Eq. (11.4.3.1) leads to the Rocard equation

Z(G)) = l-ia)g(ico) = -1

(11.4.3.2)

(11.4.3.3) \ + im-coLltkT)

which is the second convergent of Eq. (11.4.2.9) for a = 1, and which is, of course, valid for small inertial effects only, i.e.,

y = kTlC « l . (11.4.3.4) Exactly the same considerations apply to fractional relaxation based on Eq. (11.4.1.5) where Davies' procedure is used to write down [22] a fractional analogue of the telegraph equation which is retaining an external potential

/ = - It a-\

c v, t f + 0Dt T d2/ ( a r / dv d(/)2 + d(/)\kT d(/>

(11.4.3.5)

Now, for dielectric relaxation, we are interested in the ac response, where

V(0) = -MEeiwcos0. On assuming a solution of the form

/ = J_ In

l + B(a))^eio"cos<p kT

(11.4.3.6)

(11.4.3.7)

and ignoring nonlinear terms

iO)B(o)) + IT"'1 (ia)1+aB(a))/C = T'a(iO))l-a [l -B(co)], (11.4.3.8)

we have

Z(co) = B(a»= l (11.4.3.9) 1 + (im)a + y(io)T)2a

This is the complex susceptibility as rendered by the fractional telegraph equation, Eq. (11.4.3.5). It is the second convergent of the exact continued fraction solution Eq. (11.4.2.9), and as such, is only valid for very small inertial effects, i.e., in the high friction limit. We remark that although the telegraph equation is able to correctly reproduce the second

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Chapter 11. Anomalous Diffusion 641

convergent of the continued fraction for the complex susceptibility (in effect the one-sided Fourier transform of the characteristic function of the system) the agreement must be regarded as purely fortuitous. In all other cases, the telegraph equation provides a poorer approximation to the high order susceptibilities (such as those occurring in the Kerr effect relaxation [29]), than the Debye approximation or its fractional equivalent. For a practical demonstration of this, based on calculation of successive convergents of the continued fraction solution, see Coffey et al. [29]. This conclusion is reinforced by the discussion in pages 257-265 of Risken's book [6]. The fact that Eqs. (11.4.3.3) and (11.4.3.9) (which incidentally, by inverse Fourier transformation from K, S) to (0,t) variables, may be used to generate the telegraph equation) are only valid in the overdamped limit, y«1, means that to take the limit of small damping in the telegraph equation, in order to obtain the undamped wave equation, is erroneous.

We shall now describe how the fractional Klein-Kramers equation, given by Barkai and Silbey [23], provides a more acceptable description of the inertia-corrected relaxation process.

11.5 Barkai and Silbey's Form of the Fractional Klein-Kramers Equation

We have seen from the particular form of the fractional Klein-Kramers Eq. (11.4.1.5) that the effect of allowing the entire Klein-Kramers operator to act nonlocally in time is to cause infinite dielectric absorption just as the noninertial case. Thus, Eq. (11.4.1.5) provides a physically unacceptable picture of the high-frequency dielectric behaviour. The root of this difficulty apparently being that in writing Eq. (11.4.1.5), the convective derivative or Liouville term, in the underlying Klein-Kramers equation, is operated upon by the fractional derivative. Hence, the convective derivative is under the time convolution and so is affected by the memory, thus exhibiting nonlocal behaviour or "Levy sneaking", a term coined by Metzler [22]. Thus, the form of the Boltzmann equation for the single particle distribution function is not preserved meaning in the present context that both angular velocity and configuration variables are governed by the same separable fractal time random walk. Such a walk has finite jump length variance and fractal waiting time distribution. Thus, the angular velocity correlation time can no longer dictate the time scale of the fast high-frequency process giving rise to the far-infrared absorption, ultimately resulting in divergence of the integrated absorption. The diverging integrated absorption resulting from a separable fractal time walk and to a lesser extent the infinite variance

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642 The Langevin Equation

associated with Levy flights suggests that, in general, separable CTRWs may not be useful as a description of inertial effects in dielectric relaxation.

In the fractional Klein-Kramers equation proposed by Barkai and Silbey [23] in contrast, the memory or fractional derivative term is supposed to act only on the dissipative part of the normal Klein-Kramers operator, so that Barkai and Silbey's equation is of the form, again referring to fixed axis rotators [cf. Eq. (11.4.1.5)]

at 6(f) I dip \d</> I d<jr ,

This equation according to Barkai and Silbey is given for 0 < a < 1, which pertains to subdiffusion in velocity space. We shall demonstrate by solving this equation using the continued fraction method that the subdiffusion in velocity space gives rise to enhanced diffusion in configuration space. Although Eq. (11.5.1) has hitherto been regarded as valid for subdiffusion only, we shall demonstrate that if the equation is also regarded as describing enhanced diffusion in velocity space, then the enhanced diffusion in velocity space gives rise to subdiffusion in configuration space. We remark that if we set a-1 in Eq. (11.5.1) then the velocity 0 has the form of the Ornstein-Uhlenbeck process (Chapter 2). Thus, in anomalous relaxation described by Eq. (11.5.1), the velocities acquire a fractional character and we are dealing with the fractional Ornstein-Uhlenbeck process in the velocity variables. In order to justify a diffusion equation of the form of Eq. (11.5.1), Barkai and Silbey [23] consider a "Brownian" test particle that moves freely in one dimension and that collides elastically at random times with particles of the heat bath which are assumed to move much more rapidly than the test particle. The times between collision events are assumed to be independent, identically distributed, random variables, implying that the number of collisions in a time interval (0 ,0 is a renewal process. This is reasonable, according to Barkai and Silbey, when the bath particles thermalise rapidly and when the motion of the test particle is slow. Thus, the time intervals between collision events rt are described by a probability density w(t) which is independent of the mechanical state of the test particle. Hence, the process is characterised [23] by free motion with constant velocity for a time Tu then an elastic collision with a bath particle, then a free motion evolution for a time fy, then again a collision. The most important assumption of the model is that w(f) decays as a power law for long times, namely,

w(t)~t-l'a, ( 0<a r< l ) . (11.5.2)

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Chapter 11. Anomalous Diffusion 643

The above stochastic collision model then leads to the generalisation, Eq. (11.5.1), of the Fokker-Planck equation for the evolution of the phase distribution function for mechanical particles, known as the fractional Klein-Kramers equation, where the velocities acquire a fractional character [23], rather than both the displacements and the velocities as in Eq. (11.4.1.5). In the present context, all these comments apply, of course, to rotational Brownian motion.

An equation which resembles Eq. (11.5.1), and indeed, coincides with it when the external potential is zero, has been proposed by Metzler and Sokolov [30], in order to extend the equation [of which Eq. (11.4.1.5) is an example] originally proposed by Metzler and Klafter, to include enhanced diffusion in configuration space. We should also remark that according to Metzler, the Barkai and Silbey equation arises from a Levy random walk, that is, a walk governed by a jump probability density function in which the step lengths are coupled to the waiting times, so that, a long jump is penalised by a large time cost. In the particular Levy walk corresponding to Eq. (11.5.1), according to Metzler (who analysed the problem using a generalised Chapman-Kolmogorov equation), the length of the step associated with a jump is curtailed by utilising a jump probability density function of the form (v0 is the sharp velocity in the free motion of the test particle)

p(x,T) = p(x\u) = S(x- v0t)w(r)/2 . (11.5.3)

Thus, the explicit spatio-temporal coupling characteristic of the Levy walk is retained. The coupling ensures finite jump length variance in contrast to a Levy flight, which arises from a separable random walk with finite mean jump time. The behaviour is thus quite different from the simple one-dimensional CTRW model considered in Section 11.1, where, in order to derive the one-dimensional fractional Fokker-Planck equation, Eq. (11.2.32), for diffusion in configuration space, the jump length and waiting time random variables are assumed independent [see discussion following Eq. (11.1.8)]. However, we shall presently see from the exact continued fraction solution of the Barkai and Silbey equation, termed by Metzler, the Levy rambling model, that in the high friction limit the result for the complex susceptibility is the same, namely, Eq. (11.4.2.12), as predicted by the fractal time random walk model (see Chapter 1, Section 1.22.3 for the extension of the Debye theory to anomalous relaxation). This is of course true of the normal Klein-Kramers equation. However, Eq. (11.5.1) is not separable. Indeed, the stationary solution, with W = 0 can only be obtained by supposing that both convective and dissipative terms each vanish separately. Nevertheless, Eq. (11.5.1) can also be solved by continued fraction methods as we shall demonstrate.

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644 The Langevin Equation

The fact that Eq. (11.5.1) does not separate in the space and time variables appears to be entirely consistent with the Levy walk picture where a jump length involves a time cost and vice versa. This concludes our abbreviated discussion of the advantages and drawbacks of the various fractional diffusion equations, which have been proposed in order to generalise the Klein-Kramers equation. We shall now justify our assertion that the Barkai-Silbey equation leads to a physically acceptable description of the absorption coefficient at high frequencies.

11.5.1 Complex susceptibility

Just as in Section 11.4.2, we may seek a solution of Eq. (11.5.1) in the form of the Fourier series, Eq. (11.4.2.1). The linearised initial (at t = 0) distribution function is given by Eq. (11.4.2.2). Straightforward manipulation of the recurrence relations of the Hermite polynomials Hn, as in Section 11.4.2, leads to a differential-recurrence relation for the Fourier coefficients c (t) as [cf. Eq. (11.4.2.3)]

^ ( 0 + - ^ [ c n + 1 , q ( 0 + 2nCn_1,9(0] + r , - % D , , - ^ ( 0 = 0.(11.5.1.1)

Here, as above, y is the Sack inertial parameter. Noting the initial condition, Eq. (11.4.2.2), all the c„,(0) in Eq. (11.5.1.1) will vanish with

the exception n = 0. On using the integration theorem of Laplace transformation, Eq. (11.2.2.5), as generalised to fractional calculus, we have from Eq. (11.5.1.1) the three-term recurrence relation [cf. Eqs. (11.4.2.4) and (11.4.2.5)] for the only case of interest q= 1 (since the linear dielectric response is all that is considered)

TS+—(TS) V

\-a ^ W + ^ [ c » + 1 , i W + H - . . i W ] = % ( 0 ^ . ( 1 1 . 5 . 1 . 2 )

(n > 1). Equation (11.5.1.2) is yet another example of how using the Laplace integration theorem above, all recurrence relations associated with the Brownian motion may be generalised to fractional dynamics. Moreover, Eq. (11.5.1.2) is a three-term algebraic recurrence relation for the cnl(s), which may be solved exactly for c0l(s) as in the previous section, viz., - , , rc0il(0) rc0jl(0)

sr + i^yl2Sx(s) 1

sr + (sr)l~a 1 y + r sr + 2(sT)1-a/y + ....

(11.5.1.3)

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Chapter 11. Anomalous Diffusion 645

where successive convergents are calculated from the continued fraction -inJly

Sn(s) = V ' _ . (11.5.1.4) sr + nsrta I y + ijyl 2Sn+x (s)

The normalised complex susceptibility xi®)= Zi®)~ix'i®) is> a s

usual, given by linear response theory Eq. (11.4.2.8). On combining Eqs. (11.4.2.8) and (11.5.1.3), we have after simple algebra the generalisation of the Gross-Sack result [25,49] for a fixed axis rotator to fractional relaxation governed by the Barkai and Silbey Eq. (11.5.1), namely [20] [cf. Eq. (11.4.2.9)]

* « ) = ! ^ ^ • <".5.1.5) Biany +

B 9R

\ + B(jiartf + — 2 + B(icorf+-

3 + ...

where cr = 2-a and B = y(ia)T)2<'a~l). Just as in Section 11.4.2, Eq. (11.5.1.5) can be expressed in terms of the confluent hypergeometric function M(a,b,z) [cf. Eq. (11.4.2.10)], viz.,

Z(0)) = l-l™r M(\,\ + B\\ + izcQ)a\B). (11.5.1.6)

Equation (11.5.1.6) can be readily derived, by comparing Eq. (11.5.1.5) with the continued fraction Eq. (11.4.2.11), where now

a = 1, z = B, and b = 1 + fi[l + iTO))a~\.

In the high damping limit (y« 1), Eq. (11.5.1.6) can be simplified, yielding the generalisation to fractional dynamics, governed by the Barkai and Silbey Eq. (11.5.1) of the Rocard equation, namely

X0)) = j - . (11.5.1.7) 1 + (icor) - y(a)T)

We remark that Eq. (11.5.1.7), unlike the form of the Rocard equation of the Levy sneaking model, Eq. (11.4.1.5), has an inertial term similar to the Rocard equation for normal diffusion. This has an important bearing on the high frequency behaviour because return to transparency can now be achieved, as we shall demonstrate presently. The exact solution, Eq. (11.5.1.5), also has satisfactory high frequency behaviour. We further remark that, on neglecting inertial effects y^> 0), Eq. (11.5.1.6) yields the Cole Cole formula, viz.,

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646 The Langevin Equation

10'

10"'

10'

1,1': a=0.25 2,2':a=0.5 3,3':«=1.0 4,4': a= 1.5

y >

y s :

/A/, yf...../\ ••••«! ,f,

f. .*

.*r

•V^"

/

C-^. T A 2 \ *'*'

\W3* • T A \

r=oai

'"*<'

\ 3'

*..

10" 10 10 T)(0 10" 10

Figure 11.5.1.1. Dielectric loss spectra %"(co) for y= 0.02 and various values of a. 0.25 (curves 1 and 1'), 0.5 (2, 2'), 1 (3, 3'), and 1.5 (4, 4'). Solid lines (1, 2, 3, and 4): Eq. (11.5.1.6); crosses (1', 2', 3', and4'): Eq. (11.5.1.8); filled circles: Eq. (11.5.1.9).

Figure 11.5.1.2. z"((o) for a= 0.5 and various values of y. 210 8 (curves 1 and 1'), 2-10-* (2, 2'), 2-10"4 (3, 3'), 2-10"2 (4, 4'), and 2 (5, 5'). Solid lines (1, 2, 3, 4, and 5): Eq. (11.5.1.6); crosses (1', 2', 3', 4', and 5'): Eq. (11.5.1.8); filled circles: Eq. (11.5.1.9).

Figure 11.5.1.3. The same as in Figure 11.5.1.2 for a= 1.5.

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Chapter 11. Anomalous Diffusion 647

Z(a» = —-1—, (11.5.1.8) \ + (l(OT)

that is, the result predicted by the noninertial fractional Fokker-Planck equation, of Section 1.22.

As in Section 11.4.2, dielectric loss spectra z"(co) vs. corj, for various values of or and /are shown in Figs. 11.5.1.1-11.5.1.3. It is apparent that the spectral parameters (the characteristic frequency, the half-width, the shape) strongly depend on both the anomalous exponent a (which pertains to the velocity space), and the inertial parameter y. Moreover, the high-frequency behaviour of %"(co) is entirely determined by the inertia of system. The nonphysical a dependence of %"G>) from Eq. (11.4.1.5) at high frequencies is absent (see Section 11.4.2). It is apparent, just as in Brownian dynamics, that inertial effects produce a much more rapid fall off of %"(a>) at high frequencies. Furthermore, the Gordon sum rule for the integral absorption, Eq. (11.4.2.17), is satisfied. Such behaviour is quite unlike that resulting from the hypothesis that the fractional derivative acts on the convective term [cf. Eq. (11.4.1.5)]. Moreover, it is apparent from Figs. 11.5.1.1 and 11.5.1.3, assuming that the Barkai-Silbey Eq. (11.5.1) also describes subdiffusion in configuration space (that is, a< 1 or 1< a<2), that Eq. (11.5.1.6) also provides a physically acceptable description of the dielectric loss spectrum. In the high damping limit (/« 1), the low-frequency part of Z"(co) may be approximated by the modified Debye Eq. (11.5.1.8). When <7> 1 enhanced diffusion in configuration space), the low-frequency behaviour of z'i®) is similar (see Figs. 11.5.1.1 and 11.5.1.2) to that of the dielectric loss ZFRW

m ^ free rotation limit (£ = 0), which is given by [20]

Z"FR(0)) = ^TTjcoe-riW. (11.5.1.9) In order to calculate the equilibrium AVCF cmt) = (^(0)^(0),

one can simply use the same method as in Section 11.4.2. Since the dynamics of the Barkai-Silbey model are also governed by Eq. (11.4.2.14), we have the same result, Eq. (11.4.2.16), as the Metzler-Klafter model, viz.,

ca)(t) = (kT/I)Ea[-(t/T)a/r]. (11.5.1.10)

This is simply the result of Barkai and Silbey [23] for the translational velocity correlation function (x(0)x(t)), where the translational quantities are replaced by rotational ones. For a> 1, the AVCF exhibits oscillations (see Fig. 11.5.1.4), which is consistent with the large excess

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648 The Langevin Equation

Figure 11.5.1.4. Normalised angular velocity correlation function ca(t)/ca(0) for y= 1

and various values of a. 0.5 (curve 1), 1.0 (curve 2), and 1.5 (curve 3).

absorption occurring at high frequencies. We remark that both the Barkai-Silbey and Metzler-Klafter generalisations of the Klein-Kramers equation yield identical results for the AVCF in the absence of the external potential due to the decoupling of the velocity and phase space.

It thus appears, unlike the fractional kinetic equation of Section 11.4.1, namely Eq. (11.4.1.5), that the Barkai-Silbey [23] kinetic equation, Eq. (11.5.1), can provide a physically acceptable description of the high frequency dielectric absorption behaviour of an assembly of fixed axis rotators. The explanation of this appears to be the fact that in the equation proposed by Barkai and Silbey, the form of the Boltzmann equation, for the single particle distribution function, is preserved, that is, the memory function of which the fractional derivative is an example, does not affect the Liouville terms in the kinetic equation. Exactly the same conclusions apply to an assembly of rotators, which may rotate in space, that is, the needle model considered for normal diffusion in Chapter 10. We very briefly summarise the result.

11.5.2 Fractional kinetic equation for the needle model

Although the fixed-axis rotator model considered above reproduces the principal features of dielectric relaxation of an ensemble of dipolar molecules and allows considerable mathematical simplification of the problem, this model may be used for the qualitative evaluation of dielectric susceptibility only. The quantitative theory of dielectric relaxation requires an analysis of molecular reorientations in three dimensions (cf. Chapter 10). Here, we shall generalise the results given above and demonstrate how the analogous fractional Klein-Kramers equation pertaining to rotation in space may also be solved [21] to yield the complex dielectric susceptibility in terms of continued fractions, thus extending the results of Sack [25] (originally given for normal rotational

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Chapter 11. Anomalous Diffusion 649

diffusion in space including inertial effects) to fractional dynamics. We shall confine ourselves to the simplest model of rotation in space, namely an assembly of noninteracting needle-like dipolar molecules (see Chapter 10, Section 10.4). The results for the more complicated sphere model then follow, if desired, by analogy with the work of Sack [25] for normal diffusion.

We consider the rotational motion of a thin rod, or rotator, representing the polar molecule, which is subjected to an external electric field E. We assume that the field E is parallel to the Z axis of the laboratory coordinate system OXYZ. In the molecular coordinate system oxyz rigidly connected to the rotator, the components of the angular velocity <o of the rotator and of the torques K produced by the field E are (Chapter 10, Section 10.4)

<o = (a)x,(Oy,c0z) = (i!,<psmd,<pcosi!), (11.5.2.1)

K =(-juE sin tf,0,0), (11.5.2.2) where &t) and <p(t) are the polar and azimuthal angles, and \i is the dipole moment of the rotator. Here, the internal field effects are ignored meaning that the effects of the long-range torques due to the interaction between the average moments and the Maxwell fields are not taken into account. Such effects may be discounted for dilute systems in first approximation. Thus, the results obtained here are relevant to situations where dipole-dipole interactions have been eliminated by extrapolation of data to infinite dilution.

In order to describe the fractional Brownian rotational motion, we use the fractional Klein-Kramers equation for the evolution of the probability density function, W, in configuration angular-velocity space for linear molecules, in the same form as we have used above, for fixed axis rotators. For rotators in space, the fractional Klein-Kramers equation becomes (see Chapter 10, Section 10.4)

dW

dt + 0),

dW + 0)vCOtl?

dd y

( a>„

dW

dd)r

0)r

dW_

da>„

<Pt \-a V~a/3

(

d(Or

0)XW+ kTdW

I da.

juE . adW

/ dcor

+-'x J d(0„

0)yW+ kTdW

I dco. y J

(11.5.2.3)

where / is the moment of inertia of the rotator about the axis of rotation, r is the intertrapping time scale which we identify with the Debye relaxation time £l2kT) for linear molecules, and a is the exponent characterising the anomalous diffusion process. Again, the fractional derivative is a type of memory function or stosszahlansatz for the

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650 The Langevin Equation

Boltzmann equation underlying the fractional Klein-Kramers equation. For a- 1, Eq. (11.5.2.3) reduces to that corresponding to normal inertia corrected rotational diffusion considered by Sack [Eq. (10.4.4.8)]. Just as for a= 1, Eq. (11.5.2.3) is independent of the azimuthal angle <p and the z-component of the angular velocity &., so that for the problem in question, one may ignore the dependence of Won ^and o\.

Our starting point, as before, is the Fourier series method of solution of the fractional Klein-Kramers equation. We expand of the distribution function in Fourier series (in the notation of Section 10.4)

WAa>x,<oy,t) = rfe'n>1^± £ Z ^^(t)smn (a>x,<oy)/f (costf)

7=0 m=0 n=0

a^m are normalising coefficients given in [21]). We then use the generalised theorem of Laplace transformation, Eq. (11.2.2.5), and after considerable algebra which is similar to that used for the normal diffusion of the needle model described in Chapter 10 (see [21] for details), we have a system of algebraic recurrence relations for the Laplace transform

of c]imt) (m = 0, 1) governing the dielectric response:

[T?S + 2nx]cln

0(s)-2cln

l_l(s)-c1n\s)/2 = Sn0T]c^(0), (11.5.2.4)

[Tjs + (2n + l)x]cln

l(s) + 2(n + l)[cl;0(s) + cZ(s)/4]^0. (11.5.2.5)

Here x = PXicor)1'01, /?'' = J3TJ = C, I\]2IkT is the inertial parameter (large /T corresponds to small inertial effects and vice versa, unlike the

Sack parameter y = \l2/3'2), cf. Section 11.5.1). Moreover, we have

noted that all the c '°(0) vanish with the exception of n = 0. Equations

(11.5.2.4) and (11.5.2.5) may be solved exactly for c0l(s) as in Chapter

10, Section 10.4.2. This procedure then allows us to obtain the following continued fraction solution for the normalised complex susceptibility

X(o)) = x((0)lxX<d) = \-io$?i6))lc]?Q)) [cf. Eq. (10.4.2.7), Chapter 10]

Z(co) = l i-^L .(11.5.2.6) j]ia>+

x+t]ico+ -z 2x + 77*'<y+

3x + rno)+ 4x+rjico+...

Here ^'(0) = M2N0 /(3kT) is the static susceptibility.

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Chapter 11. Anomalous Diffusion 651

The infinite continued fraction Eq. (11.5.2.6) is useful for calculations because the complex dielectric susceptibility can be readily evaluated from Eq. (11.5.2.6) for all values of the model parameters 7, P, and a. For a= 1, the anomalous rotational diffusion solution Eq. (11.5.2.6) coincides with that of Sack (Chapter 10, Section 10.4) for normal rotational diffusion. Moreover, in a few particular cases, just as rotation about a fixed axis, Eq. (11.5.2.6) can be considerably simplified. In the free rotation limit £= 0), which corresponds to the continued fraction evaluated at x = 0, that fraction can be expressed (just as normal rotational diffusion) in terms of the first order exponential integral function E\z) [10], so that the normalised complex susceptibility is [21]

Z(o)) = 1 + ?72<y V 7 ^ 2 E x (-TJ2Q)2). (11.5.2.7) Furthermore, again as in the one degree of freedom fixed axis rotation model, Eq. (11.5.2.6) in the high damping limit (J3'»l) can be simplified yielding the generalisation to fractional dynamics of the Rocard equation (for an assembly of needles), namely

Xico)= I (11.5.2.8)

where a = 2-a. On neglecting inertial effects (77—>0), Eq. (11.5.2.8) again reduces to the Cole-Cole equation:

XiC0)= l (11.5.2.9) 1 + (iCDT)

Dielectric loss / *(<w) and absorption co%"(a>) spectra for various values of or and /?' are shown in Figs. 11.5.2.1 and 11.5.2.2. The Cole-Cole plot [x"(0) vs. z'i®) ] is presented in Fig. 11.5.2.3. It is apparent [21] that the half-width and the shape of the dielectric spectra strongly depend on both a (which pertains to anomalous diffusion in velocity space) and /?'(which characterises the effects of molecular inertia). In the high damping limit (fi'» 1) and for a> 1 corresponding to a< 1 (subdiffusion in configuration space), the low-frequency part of ;£"(©) may again be approximated by the modified Debye equation (11.5.2.9). On the other hand, the high-frequency behaviour of %"CQ) is entirely determined by the inertia of system. For a given value of fl', the inertial effects become more pronounced when a—> 2 (see Fig. 11.5.2.3). Just as in Brownian dynamics, it is apparent that inertial effects produce a much more rapid falloff of %"(co) at high frequencies. As before the fractional needle model satisfies the Gordon sum rule for the dipole integral absorption of rotators in space [21,26]:

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652 The Langevin Equation

Figure 11.5.2.1. Dielectric loss spectra ,£"(&>) for /?'= 5 and various values of a. a= 0.5 (curves 1 and 1'), 1.0 (curves 2 and 2'), and 1.5 (curves 3 and 3'). Solid lines (1,2, and 3), Eq. (11.5.2.6); crosses (1', 2', and 3'), Eq. (11.5.2.9).

Figure 11.5.2.2. Dielectric absorption spectra cox"co) for a = 0.5 and a= 1.5 and various values of /? ' 5000 (curves 1 and 1'), 500 (curves 2 and 2'), 50 (curves 3 and 3'), 5 (curves 4 and 4'), and 0.5 (curves 5 and 5'). Solid lines (1, 2, 3, 4, and 5), Eq. (11.5.2.6); crosses (1', 2', 3', 4', and 5'), Eq. (11.5.2.9).

0.5-

0 0 -

1 - a= 1.00

2 -a=1 .25 3 - B = 1 . 5 0

4 - a = 1 . 6 0 ^ '

2

£ = 50

Re[£(<u)]

Figure 11.5.2.3. Cole-Cole plots for /?'= 50 and various values of a: a=\ (curve 1), 1.25 (curve 2), 1.5 (curve 3), and 1.6 (curve 4). Solid lines (1, 2, 3, and 4), Eq. (11.5.2.6); symbols, Eq. (11.5.2.9).

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Chapter 11. Anomalous Diffusion 653

Table 11.5.2.1. Comparison of the results for fixed-axis rotators and rotators in space

Fixed axis rotators Rotators in space Characteristic relaxation time

T = C/(kT) T = p(2kT)

Static susceptibility

Z'(0) = jU2N/2kT) Z\0) = jU2N/(3kT)

Generalised Rocard equation

i(«)= I

l+(ia*)ff-2(flizr Z(<0) =

1 \+(iax)°'-(corf)1

Gordon's sum rule \cox (co)dco = —^-i 4 /

JQ)z"(o))d CD = xNoJu

3/

Dielectric loss at (= 0 (free rotation limit) X"FR(0) = 4nr)coe * rFR(0)) = mjWe-^2

\cox\co)dco = xNoJu

37 (11.5.2.10)

Again, it is significant that the right-hand side of Eq. (11.5.2.10) is determined by molecular parameters only, and is independent of the model parameters a and £ In contrast, the fractional Debye model does not predict the correct value of the integral absorption: e.g., for a> 1, it predicts infinite integral absorption (see Fig. 11.5.2.2) and our earlier discussion, Section 11.4.2.

The behaviour of the dielectric spectra for the two rotational degree of freedom (needle) model is however, similar, but not identical to that for fixed axis rotators considered above. Here, the one and two rotational degree of freedom models (fractional or normal) predict dielectric parameters that may considerably differ from each other. The differences in the results predicted by these two models are summarised in Table 11.5.2.1. It is apparent that the model of rotational Brownian motion of a fixed-axis rotator treated in Section 11.5.1 merely qualitatively reproduces the principal features (return to optical transparency, etc.) of dielectric relaxation of dipolar molecules in space, for example, the dielectric relaxation time obtained in the context of these models differs by a factor of 2.

The result we have obtained for the complex susceptibility is again of particular interest in the theory of dielectrics as it demonstrates how the unphysical high-frequency divergence of the absorption coefficient in the far-infrared region due to the neglect of inertia may be removed in fractional relaxation just as in inertia-corrected Debye relaxation (see Figs. 11.5.2.1-3). We remark that the continued fraction

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654 The Langevin Equation

solutions that we have given, with a few elementary modifications, also yield the Laplace transform of the characteristic function of the configuration space distribution function including inertial effects. Thus, all desired statistical averages such as the mean square angular displacement, etc., may be simply calculated by differentiation. This problem is the fractional dynamics equivalent of the Ornstein-Uhlenbeck process. We further remark that the present calculation constitutes a good example of the solution of the fractional Klein-Kramers equation for a multi-degree of freedom system and is, to our knowledge, the first example of such a solution.

11.6 Anomalous Diffusion in a Periodic Potential

It is the purpose of this section to include [19] the effect of an internal field potential (and so dielectric relaxation due to barrier crossing by dipoles) in the fractional Brownian dynamics model considered above. This will be accomplished using the matrix continued fraction method and the generalised integration theorem of Laplace transformation. These methods will also allow us to consider the mechanism underlying the high-frequency (far-infrared) absorption peak in fractional dynamics. Moreover, it will facilitate the extension of the cage model of polar fluids to fractional dynamics. In order to simplify our presentation, we shall confine ourselves to the linear response to a small ac applied field.

We again illustrate the solution by using the simplest microscopic model of dielectric relaxation discussed above, namely: an assembly of rigid dipoles, each of moment p , each rotating about a fixed axis through its centre. A dipole has moment of inertia / and is specified by the angular coordinate </> so that it constitutes a system of one (rotational) degree of freedom. However, we now introduce a model of the internal field due to molecular interactions which is represented by an TV-fold cosine potential.

V (0) = -Vo cos N<f>. (11.6.1) As usual, we suppose that a uniform electric field E (having been applied to the assembly of dipoles at a time t = -°° so that equilibrium conditions prevail by the time t = 0) is switched off at t = 0. In addition, we suppose that the field is weak (i.e., fjE«kT). For t<0 and f-»«>, the distribution functions are linearised Maxwell-Boltzmann distributions:

e-it*?Hv**N* ( 1 + ^ C 0 S ( ^ _ Q ) ) Wt<0 ~ IK

J g-OH>>+#v<»s*#(1 + £ c o s ( 0_ 0 ) ) r f 0 ( H 6 2 )

0

= W o ( ^ ) [ l + £cos(0-0)-<f(cos(0-0))o]

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Chapter 11. Anomalous Diffusion 655

and

WU. = wo M) = Z-le-™)2+^cosN* , (11.6.3)

respectively. Here Z is the partition function, 0 is the angle between E and the z axis in the plane zx, the field and barrier height parameters are, respectively,

$ = fiEHkT), £v=V/(lcT), (11.6.4)

and ( )0 means the equilibrium statistical averages over W0((/),<j)).

Our goal is, as usual, to evaluate the transient relaxation of the electric polarisation, defined as

P£(O = / / iV o [ (cos(^-0)) (O-(cos(^-0)) o ]

= cos ©/J, (0 + sin 0P± (0,

where

Jf,(f) = //Wo[(cos0)(O-(cos0)o] and P±(t) = iNo[(sm0)(t)-(sm0)o]

are the longitudinal and transverse components of the polarisation, respectively, iVo is the concentration of dipoles, and the angular brackets ()t) denote the statistical averages over the assembly of rotators. We have seen in Chapters 4 and 10 that according to linear response theory, the decay of the longitudinal and transverse components of the polarisation of a system of noninteracting planar dipoles, when a small uniform external field E is switched off at time t = 0, is

/?,(*) = cos 0E ^ - ^ 2 . C„(f) and P,(O = s i n 0 £ ^ - ^ - C , (?), 11 kT " kT where

C„(0 = <cos^(O)cos0(O>o -<cos<z>(0))o (11.6.6) and

Cx (0 = (sin 0(0) sin </>(t))0 - (sin <f>(0))20 (11.6.7)

are the longitudinal and transverse correlation functions. The longitudinal and transverse components of the complex susceptibility tensor are defined as

,,2>

0

(11.6.8) By supposing that the local configuration potential is uniformly distributed in a plane [cf. Eq. (10.7.35)], we may also define the averaged susceptibility %(co) as

M2N0 CrQ>)-iG)\e-iaxCr(t)dt (rHU).

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656 The Langevin Equation

Xa)) = [x^(D) + xLG))\l2, (11.6.9)

which becomes, after elementary manipulation of Eqs. (11.6.6) and (11.6.7)

X(co) = M2N0

2kBT 1 -icoj (cos A0(t))oe~ioxdt (11.6.10)

where A0(t) = <f>t) - 0(0).

The starting point in our calculation of Zi®) fr°m Eq. (11.6.10) is the fractional Klein-Kramers equation for the probability density function W(0,0,t) in the phase space (0,0) [10,11], which we shall take in the Barkai-Silbey form

- + 0— (A/VosinA^ + / /£s in0) r 60 Id0

/ . . „ . , „ . N (11-6.11)

dt

= oA 1-a-l-a ,

60 I 60 J

where the various quantities have the same meaning as in Section 11.5, except where otherwise stated. As in the preceding sections, we seek a solution of Eq. (11.6.11), for the case E = 0 at t > 0, by using the method of separation of variables in the form of the Fourier-Hermite series

oo oo

^ 0 = T 5 7 T ^ TZzhMq(t)Hn(rj0)e In- n=0q= 2nn\

Thus, using the recurrence and orthogonal properties of Hn, we have the recurrence relation for

fn^t) = (Hn(jj0)e-^)(t) (11.6.12)

which is given by

Vfn,q(t) + (iq/2)[fn+Uq(t) + 2nfn_lq(t)]

+(inNv /2)[fn„Uq+N(t) - fn_lq_N (t)] = -Tl~a 0Dl-anrjfifn,q(t).

On using the integration theorem of Laplace transformation generalised to fractional calculus, Eq. (11.2.2.5), we then have

2rjsf^q(s) + iqflq(s) = 2Tjf^(0), (11.6.14)

~2rjs + ny'2-a r]sfa 1 /B>, (s) + iq [fn+hq (s) + 2nfn_Uq (s)]

(11.6.13)

+inN£v [fn_hq+N(s) - fn_lq_N(s)] = 0, (n > 1). (11.6.15)

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Chapter 11. Anomalous Diffusion 657

Here / = £^2l(IkT) is again the inertial effects parameter (so that large

Y characterises small inertial effects and vice versa) and / (0) = 0 for

n > 1 because

( H „ c - ^ ) o = 0 , (11.6.16)

for the equilibrium Maxwell-Boltzmann distribution. We remark that the calculation of the longitudinal and transverse components of the complex susceptibility tensor differs only in the term /o,g(0), which must be evaluated at © = 0 and Q-TC/2, respectively. The calculation of the averaged susceptibility from Eqs. (11.6.9) and (11.6.10) can be carried out formally by solving Eqs. (11.6.14) and (11.6.15) for the functions

^ ( o = ( f f , H ^ , N ( B , ) 0 <n-6-17) which obey the same recurrence Eq. (11.6.13) as the fn^(t). We then

have

Xco) = ^-[\-icdcwco)\, (11.6.18) 2kT

where

cn,qm = \cn^t)e'iMdt. (11.6.19) o

The initial conditions are In

\ e^-^e^ cosN*d</>

^(0) = (e-'(*-w)o=JL^ = V . ^ 7 # v > (n-6-20) je*"*»N<dt Wv)

o where the /„ are the modified Bessel functions of the first kind of order n and we have noted that

e-Kl-Me$, COSN* = £ j^yW-q+l)* _ (H.6.21)

m=—oo

We choose as an example of an internal field potential a double-well potential (N = 2) which will allow us to treat overbarrier relaxation (for N = 1 corresponding to a uniform electric field this process does not exist). In order to solve Eqs. (11.6.14) and (11.6.15), we shall again use matrix continued fractions. This is accomplished as follows. We can rearrange as in Chapter 10, Section 10.3, the recurrence Eqs. (11.6.14) and (11.6.15) as the matrix three-term recurrence equation

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658 The Langevin Equation

(2^-Qn(w))Cn(«)-Q:C„+ 1(«)-Qn-C„_1(^) = 2^n lC1(0). (11.6.22)

The column vectors C„ (co) are now given by r :

f do,-

%

: \

-2O)

- i(«)

cQl(a)

c02co)

V : /

Ln-l,-2

Ln-1

_2(«)

C!(ffl)= T 1 ) . ? and C„(fli)= cB_li0(fl9

cB-i,i(®)

c„-i .2(®)

with

Ct(0) =

(n>2) , (11.6.23)

0

/ 2 ( & ) 0

h (&) 7 0 (&) 7 0 (&)

0

0

/2(<?v)

and the matrices Q„(ft>), Q* and Q„ are defined by

Q>~i

•• -2

•• 0

•• 0

•• 0

•• 0

0

- 1

0

0

0

0 0

0 0

0 0

0 1

0 0

0 ••

0 ••

0 ••

0 ••

2 ••

(11.6.24)

(11.6.25)

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Chapter 11. Anomalous Diffusion

l-a, Qn(a)) = -y'2-ai7](oy-"(n-l)l,

659

(11.6.26)

Q;=-2(/I-I)

- -4 o -4,

... o •••

••• 0 0

••• 0 0

-2 0

0 -1

-$, o - - & 0 •••

4 0

0

0

S,

... o 6 -0 £ 1 0

0 2

0 0 •••

0 0 •••

... o ••• &

0 £v -

(11.6.27)

where I is the unit matrix of infinite dimension. The exceptions are the

matrices Q* and Q2 , which are given by

Qi" = -«"

•• -2

•• 0

•• 0

•• 0

0

-1

0

0

0

0

0

0

0

0

1

0

0 ••

0 ••

0 ••

2 ••

(11.6.28)

-2/

- -Zv 0 ... 0 - &

••• 0 0

••• 0 0

••• 0 0

-2

0

-& 0

0

0

-1

0

~<fv 0

0 0

tv 0

0 tv

1 0

0 2

0

0

0

tv 0

0 •••

0 •••

0 •••

0 •••

#v -

(11.6.29)

By invoking the general method for solving the matrix recurrence Eq. (11.6.22) given in Chapter 2, we have the exact solution for the spectrum Cj(fiJ) in terms of a matrix continued fraction, viz.,

Cl(co) = 2rjAl(co)Cl(0), (11.6.30) where the matrix continued fraction is defined by

An(o)) = [2irjcol-Qn - Q X + i O ^ Q ^ i j ' - (11.6.31)

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660 The Langevin Equation

11.6.1 Calculation of spectra

The exact matrix continued fraction solution [Eq. (11.6.30)] we have obtained is again very useful for computation. As far as practical calculation of the infinite matrix continued fraction Eq. (11.6.31) is concerned, we again approximate that by some matrix continued fraction

of finite order (by setting Q~,Q^ =0 at some n - N). Simultaneously, we restrict the dimensions of the matrices Q~,Q^ and Qn to some finite number M. Both of the numbers N and M depend on the barrier height £v

and damping ^parameters and must be chosen by stipulating the degree of accuracy of the calculation (with decreasing ^and increasing <fv both TV and M must be increased). Having calculated C^co) from Eqs. (11.6.30) and (11.6.31), we may evaluate the complex dielectric susceptibility %co) from Eq. (11.6.18) for all values of the model parameters 77, •/, £v and a.

The real, xXo>) > a nd imaginary, x'i®), parts of the complex susceptibility for various values of oc, £v, and Y a r e shown in Figs 11.6.1.1-7 [the calculations were carried out for jU2N0/(2kT)-l]. Clearly, the shape of the dielectric spectra again depend strongly on the anomalous exponent a (here pertaining to anomalous diffusion in velocity space), %v (which is the barrier height parameter), and YiY~> 0 characterises large inertial effects and Y~> °° characterises small effects). In general, three bands may appear in the dielectric loss x'i®) spectra, the corresponding dispersion regions are visible in the spectra of x'i®) • One anomalous relaxation band dominates the low frequency part of the spectra and is due to the slow overbarrier relaxation of the dipoles in the double-well cosine potential as identified by Frohlich [36]. The characteristic frequency (OR of this low-frequency band strongly depends on the barrier height t;v and the friction parameter Yas well as on the anomalous exponent a. Regarding the barrier height dependence, the frequency C0R decreases exponentially as the barrier height £V is raised [cf. Eq. (11.2.2.17) which describes the behaviour of (OR in the noninertial limit]. Such behaviour occurs because the probability of escape of a dipole from one well, to another, over the potential barrier, exponentially decreases with £y(cf. Figs. 11.6.1.1-3).

As far as the dependence of the low-frequency part of the spectrum for large inertial effects / « 1 is concerned, the overbarrier frequency a>R decreases with decreasing / , for given values of E,y and a [cf. curves 1-3 in Fig. 11.6.1.6 a- 1, normal diffusion) and curves 1-4 in Fig. 11.6.1.7 a- 0.5, enhanced diffusion in configuration space)]; for

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Chapter 11. Anomalous Diffusion 661

>< 1- a = 0.5 2 - <*=1.0

3- a=1.5

r '=10 £,= 3

10 '• > '

><

io- 10 10 T]0)

Figure 11.6.1.1. Real and imaginary parts of the complex susceptibility xi®) vs-normalised frequency TJCO for Y- 10 and gv = 3 and various values of the fractional parameter a.

-H

0.5-

0.0'

" \ ~ ~ ~ v .

4 \ 3 \

1 -£,=0.1 \ \ 2 - £= 1.0 \ \ 3 -£,= 3.0 \ . 4 - £ = 5 . 0 \ ~ _ _ _ _ _ _

y' = 0.4 a=1.5

2\ i

\ \

\ i \

10 10 10 10 t\co

Figure 11.6.1.2. Real and imaginary parts of the complex susceptibility z(co) vs. jja> for a = 1.5 and Y= 0.4 and various values of the barrier height parameter £v .

>< 0.5-

0.0-

^ \

1 - £=0.1 2 - £,=1.0 3- £=3.0 4- £=5.0

\ 3 \ 2 \ 1

v V H a=1.0 f = 0A It 1 ,0" V<o 10~

Figure 11.6.1.3. The same as in Fig. 11.6.1.1 for a= 1.0.

the subdiffusion (a> 1), however, this frequency does not show such a behavior (Fig. 11.6.1.5). In like manner, for small inertial effects, y> 10, the frequency (OR decreases as /'increases as is apparent by inspection of curves 3-6 in Fig. 11.6.1.5, curves 4-6 in Figs. 11.6.1.6 and 11.6.1.7. Thus, on decreasing /from high values ('/» 1), that is allowing inertial

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662 The Langevin Equation

0.5

0.0-

-OS-

" ^ ~ s

V

1-£F0.1 3 -^3 .0

N^2

\ 3

4 -^5 .0 <"0i

i i

,•• = 0.4

f

'

10 10 10 10 10 10 rjco

a=05 (-' = 0.4

Figure 11.6.1.4. The same as in Fig. 11.6.1.1 for a= 0.5.

><

1.0-

0.5-

0.0-

6 \ '•

l - r '=o.oi \ 2- r ' = 0.1

3- r ' = i 4- y'=10 5- r '=100 6- r'=1000

\ \ \ ^ \ \ \

^^^^^d^fes r^

o=1.5 £, = 3 1

10 ' 10 '

Figure 11.6.1.5. Real and imaginary parts of the complex susceptibility vs. normalized frequency 77ft) for a = 1.5 and gv = 3 and various values of the friction parameter y'.

0=1 £,=3

\

-H

/

m //// ' •'/' • • /

?K

// 4 ,-

2 -

3 -

4-5-6-

v ' V y ' ''• ' v \ \ N-

6 \

r'=o,oi r'=o.\

y'=10 C'=100 r'-\ooo

• ^ >

n-

V 5

1 1

fv

-Xi 3 1

= 1

A Ti

\

1 0 „„, 10 10 10 7 0

Figure 11.6.1.6. The same as in Fig. 11.6.1.4 for cc= 1.0.

effects to come into play, the low-frequency part of the dielectric loss initially shifts to high frequencies. However at some critical value of Y~ 1-10 (which depends on a), the band shifts back to low frequencies as ^ - > 0 (that is, for very large inertial effects) (cf. Figs. 11.6.1.4-6). Such behaviour is a consequence of the frictional and anomalous exponent dependence of (OR. The frictional behaviour has its origin in the

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Chapter 11. Anomalous Diffusion

rjm rja>

Figure 11.6.1.7. The same as in Fig. 11.6.1.4 for a= 0.5.

various turnover regimes of the Kramers' escape rate (very high damping, low damping, etc.). Equation (11.2.2.17) is a simple explicit example showing the behaviour of (OR in the very high damping limit. For a>\ (subdiffusion in configuration space), the low-frequency behaviour of Z"((D) may be approximated by the Cole-Cole equation:

Z'(0) l + (ico/o)R)2-a

We remark that a very high-frequency band is visible in all the figures. This band is due to the fast inertial librations of the dipoles in the potential wells. This band corresponds to the THz (far-infrared) range of frequencies and is usually associated with the Poley absorption [38]. For £ v » 1, the characteristic frequency of librations fit increases as ~ (gv /I)112 . As far as the behaviour as a function of / i s concerned, the amplitude of the high-frequency band decreases progressively with increasing / ' for small inertial effects Y> 1» a s o n e would intuitively expect. On the other hand, for large inertial effects y « 1. a fine

structure appears in the high-frequency part of the spectra (due to resonances at high harmonic frequencies of the almost free motion in the cosine potential) again in accordance with intuition. We further remark that the high frequency (d)» tffc) behaviour of x'i®) is again entirely determined by the inertia of system. Moreover, just as in the normal Brownian dynamics, the inertial effects produce a rapid falloff of %"(0) at high frequencies. Moreover, it can be demonstrated that the fractional model under consideration again satisfies the Gordon sum rule for the dipole integral absorption of rotators in a plane, Eq. (11.4.2.17). For a= 1, the anomalous rotational diffusion solution coincides with that for normal rotational diffusion. Finally, it is apparent that between the low-frequency and very high-frequency bands, at some values of model parameters, a third band exists in the dielectric loss spectra (see, e.g., Fig.

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664 The Langevin Equation

10 10 /[Hz]

Figure 11.6.1.8. Broad-band (0-THz) dielectric loss spectrum of 10% v/v solution of probe molecule CH2CI2 in glassy decalin at 110° K. Symbols are the experimental data [37]. Curve 1 is the best fit for the anomalous diffusion in the double-well cosine potential (or =1.5, gv = 8, and Y= 0.003); curve 2 is the best fit for the normal diffusion (a= 1, & = 7, and / = 0.001). Dashed line is the Cole-Cole Eq. (11.6.1.1)

11.6.1.4). This band is due to the high-frequency relaxation modes of the dipoles in the potential wells (without crossing the potential barrier) which will always exist in the spectra even in the noninertial limit. Such relaxation modes, as we have seen in Chapters 4,6, and 7, are generally termed the intrawell modes. The characteristic frequency of this band depends on the barrier height £v and the anomalous exponent a.

In Fig. 11.6.1.8, a comparison of experimental data for 10 % v/v solution of a probe molecule CH2C12 in glassy decalin at 110° K [34] with

the theoretical dielectric loss spectrum £"(G))~(£o-£oo)z"((0)/z'(Q) calculated from Eqs. (11.6.18) and (11.6.30) is shown. The reduced moment of inertia Ir used in the calculation is defined by [38]

lr 'b ^ l c

where Ib and Ic are the principal moments of inertia about molecular axes perpendicular to the principal axis a along which the dipole moment vector is directed. For the CH2C12 molecule, Ir= 0.24 10~38 gem2. The use of the reduced moment Ir (cf. Chapter 10) allows one to obtain the correct value for the dipolar integral absorption for two-dimensional models. The phenomenological model parameters gv, f, and a were adjusted by using the best fit of experimental data. Fig. 11.6.1.8 indicates that our generalised Frohlich model explains qualitatively the main features of the broad-band dielectric loss spectrum of the CH2Cl2/decalin solution.

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Chapter 11. Anomalous Diffusion 665

The model we have outlined incorporates both resonance and relaxation behaviour and so may simultaneously explain both the anomalous relaxation (low-frequency) and far-infrared absorption spectra of complex dipolar systems. Moreover a third mid-frequency relaxation band may appear in the dielectric loss spectra at low temperatures due to intrawell relaxation modes. The present calculation also constitutes an example of the solution of the fractional Klein-Kramers equation for anomalous diffusion in a periodic potential and is to our knowledge the first example of such a solution. The approach which is grounded in a theorem of operational calculus generalised to fractional exponents and continued fraction methods clearly indicates how many existing results of the classical theory of the Brownian motion may be extended to fractional dynamics.

11.7 Fractional Langevin Equation

In previous sections, we have treated anomalous relaxation in the context of the fractional Fokker-Planck equation. As far as the Langevin equation treatment of anomalous relaxation is concerned, we proceed first by noting that Lutz [18] has introduced the following fractional Langevin equation for the translational Brownian motion in a potential V:

m^v(t) + mra0D]-av(t) + dxV[x(t),t] = A(t), (11.7.1) at

where v(t) = x(t) is the velocity of the particle, m is the mass of the particle, ya is the friction coefficient, mya0DJ~av(t) and A(t) are respectively the generalised frictional and random forces with the properties

1(0 = 0, Xt')Xt)= mkT7a \t-tf~2 (11.7.2) r ( t f - l ) 1 '

(the parameter or corresponds to 2-a as used in Ref. [18]). The overbar means the statistical average over an ensemble of particles starting at the instant t with the same sharp values of the velocity and the position. The

fractional derivative QD)~a in Eq. (11.7.1) has the form of a memory function so that Eq. (11.7.1) may be regarded as a generalised Langevin equation (a didactic account of the generalised Langevin equation is given by Mazo [39])

m—v(t)+\ Ka(t-t')v(t')dt' + dxV[x(t),t] = Mt). (11.7.3) dt J

Q

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666 The Langevin Equation

The memory function KJj) is given (in accordance with the fluctuation dissipation theorem) by

1 Ka(t) = —M0)Mt), (H.7.4)

kT and the parameter a corresponds to 2 - a as used in Ref. [18]. Lutz [18] also supposed that the random force Xt) is Gaussian. Equation (11.7.3) may also describe non-Gaussian processes. However, in that case, the moments A,(tl)Mt2)—Mtn) may not be expressed in terms of A(t) and A(t')A(t).

In passing, we remark that Oppenheim and co-workers [52] have demonstrated how long-time asymptote behaviour of a memory function of the form t~m appears naturally in the theory of the diffusion coefficient of dense liquids. This result is achieved by means of a Tauberian theorem [50] which tells one [39] how the asymptotic (in this case the long time) behaviour of a function is determined by the small s behaviour of its Laplace transform. Such Tauberian theorems have been used on many occasions throughout the book.

The formal exact solution of Eq. (11.7.3) for a free Brownian particle (V=0) may readily be obtained using Laplace transforms [17,18]. We have

v-(s) = _-^_+l^L („.7.5) S + S Ya m S + S Ya

so that, noting Eq. (11.3.3),

v(t) = v(0)Eatl(-taYa) + m-lf EaA(-(t-tYYa)W)dt' (11.7.6)

o and

t

x(t) = x(0) + J v(t')dt' = x(0) + v(0)tEaa (-YJa) o

< t" (11.7.7)

+m-lj J Ea<l(-(t-trYa)Mt')dt'dt" o o

Here we have noted that

\ Ea,(-Yat'a)dt' = tEa2(-Yat

a), o

where E^iz) is the generalised Mittag-Leffler function defined by Eq. (11.3.2.17). In particular, one can readily obtain the first order statistical moments. We have for the mean displacement [18]

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Chapter 11. Anomalous Diffusion 667

x(t) = x(0) + v(0)tEaa (-yata) (11.7.8)

and for the first moment of the velocity

v(t) = v(0)EaA(-yata), (11.7.9)

since for the Maxwell-Boltzmann distribution (v2(0))0=A:r/m. Moreover, we have from Eq. (11.7.9) the equilibrium velocity correlation function cv(t) [cf. Eq. (11.5.1.10) for the angular velocity correlation function]

cv(r) = (v(0)v(0>0 =kTlm)EaA(-yata). (11.7.10)

Noting that x\i) is given by

* 2 (0 = *2(0) + 2j x(t')vt')dt', (11.7.11) o

we may obtain from Eqs. (11.7.6) and (11.7.7)

x2' ( 0 ) o - ( x 2 ( 0 ) ) o ~ f I Ea(-ya(t'-ty)*'*' ' ° m 0 0

= 2*^j t-t')Ea(-yat-tT)dt'=—t2E^(-yata).

(11.7.12)

m o m Lutz also compared his results with those predicted by the

fractional Klein-Kramers equation for the probability density function f(x,v,t) in phase space for the inertia corrected one-dimensional translational Brownian motion in a potential V of Barkai and Silbey [23] which in the present context is

( 3 UT 32 r \ df , 9 / i dv df _ ._ d ,^ , *r 3 2 / dt dx m dx dv dv m dv 2 (11.7.13)

Here * and v = x are the position and the velocity of the particle, respectively. Lutz showed that Eqs. (11.7.10) and (11.7.12) can be obtained in the context of both (Langevin and Fokker-Planck) methods. However, the two methods apparently predict different equations for the second order velocity moment, viz.,

v2(t) = (v2(0)-kT/m)Ell(-yata) + kT/m, (11.7.14)

by the Langevin method and

v2(t) = (v2(0)-kT/m)Eal [-2yata) + kT Im, (11.7.15)

by the Fokker-Planck method. Likewise, each approach apparently

predicts different results for all higher order moments (e.g., vnt) for n >

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668 The Langevin Equation

2). Thus, it has been concluded [18] that the fractional Eqs. (11.7.13) and (11.7.1) describe fundamentally different stochastic processes although they share striking common features.

The above results are indicative of a wider problem, which will be encountered in all attempts to calculate higher-order statistical moments from the generalised Langevin equation. Namely, the fact that a knowledge of the first two moments of the random force is insufficient to calculate higher moments. In other words, the advantage conferred by Isserlis's theorem (see Chapter 1, Section 1.3) in the calculation of statistical moments for Markovian Gaussian processes is entirely lost when memory effects are taken into account. This is particularly important in the context of the averaging procedure for the construction of differential-recurrence equations from the Langevin equation, which we have used throughout the book, as it is no longer apparent how the general term of the hierarchy of the averages may be calculated. Thus, at the present time, it is not clear how our procedure may be extended to Langevin equations with a memory term and so to fractional Langevin equations. Similar arguments will of course apply to fractional Fokker-Planck equations such as the Barkai-Silbey or Metzler-Klafter equations, since an analogue of the Isserlis theorem is needed in order to justify truncation of Kramers-Moyal expansion (see Chapter 1). The general nature of the problems that are encountered in identifying a generalised Langevin equation with a Fokker-Planck equation possessing a memory kernel have been succinctly discussed by Mazo in Chapters 10 and 11 of Ref. [39]. For example, taking as dynamical variable the momentum/? of a particle and retaining his notation, that is, we have the generalised Langevin equation

^- = -[K(t-t')p(t')dt' + F(t), (11.7.16) dt J

0

he demonstrates that this equation cannot, in general, be identified with the Fokker-Planck equation [ / = f(x, p, t) ]

f + pf-JjK(t-t')j-(^- + -)f(x,p,t')dt'. (11.7.17) at ox • opyap mkT)

In view of these difficulties, it appears then that the best course to adopt at the present time is nearly to regard the right-hand side of a fractional Klein-Kramers equation as a stosszahlansatz for the Boltzmann equation of which, e.g., the Barkai-Silbey equation is simply a special case. We remark that there is nothing unusual about this hypothesis, as there are many examples of collision operators (kinetic models) in statistical mechanics, where it is possible to have a well behaved right-hand side of

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Chapter 11. Anomalous Diffusion 669

the Boltzmann equation, where no corresponding Langevin equation exists, because one cannot separate the stochastic forces into systematic and random parts. Indeed, the classical theory of the Brownian motion is rather particular as in so far as a Newtonian-like equation of motion, e.g., the Langevin equation underlies the dynamical process. There exist many examples of collision operators, where a dynamical (Langevin) equation is not defined, e.g., the Van Vleck-Weisskopf model [31], the Bhatnagar-Gross-Krook model [40,48,49], etc. These kinetic models yield physically acceptable results for the observed variables (such as the complex susceptibility). Therefore, at the present stage of development, it appears to us that the best way forward is to regard kinetic equations such as the Barkai-Silbey equation, as belonging to that particular class of kinetic (collision) models which takes into account long-memory effects. Collision models are in general described by a Boltzmann equation such as [25,40,48,49]

df df dvdf Sf m , , m

r f + v ^ - + — -J- = -^-, (11.7.18) at ox ox dv ot

where the right-hand side represents the disturbance of the streaming motion of the distribution function due to collisions. In particular, Sack [25] and Gross [49] (considering rotation about a fixed axis) by means of an expansion of the distribution function in Fourier series in the angular variable (f> have shown how differential-recurrence relations for four distinct collision mechanisms may be obtained. These, in turn, may be solved [25,49] using continued fraction methods to yield the complex polarisability. We may summarise by stating that as far as progress using the generalised Langevin equation is concerned, the main problem is the lack of a stochastic integral formalism (analogous to the Wiener integral) which would allow one to calculate the statistical averages needed for the construction of the hierarchy of differential-recurrence equations from the Langevin equation.

This brings to an end our long discussion of the theory of the Brownian motion. Throughout the book, we have emphasised how the Langevin (dynamic) approach for the Brownian motion may be extended far beyond his original treatment of the one-dimensional translational Brownian motion of a free particle. We have demonstrated this by showing how his method may be used to solve a variety of problems involving both linear and nonlinear relaxation in a potential. Moreover, we have shown how the Langevin approach yields a comprehensive account of rotational Brownian motion regardless of the geometry, which is considered, and how the dynamical approach offers considerable simplification of any given problem over that yielded by the Fokker-

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670 The Langevin Equation

Planck equation. We remark that the Langevin and the Fokker-Planck methods are entirely equivalent and one method is complementary to the other. However, if a knowledge of the distribution function is not required (which is true in the majority of physical applications) the Langevin equation is much simpler to use as well as providing more insight into the physics of problem than the corresponding Fokker-Planck equation. An important consequence of the work described in the book is that in many cases, complex relaxation processes in a potential involving many eigenmodes, may in practice be approximated by simple analytical formulae for the quantities of interest. Progress in this respect is due to the matrix continued fraction method as it allows one to solve the problem exactly, and thus to gauge the accuracy of the various simple approximate solutions which we have mentioned above. The merit of the simple approximate results, which we have derived in the book is that they may readily be used for comparison with experiment.

The classical theory of the Brownian motion based on the Langevin equation may now be considered as well understood. However, the question of the equivalence of the generalised Langevin equations and probability density diffusion equations must still be regarded as a subject for active development. This is obvious from the treatment of anomalous diffusion, which is a particular case of a Langevin equation with memory, which we have described in Chapter 11. Further development of this subject is of the upmost importance in view of the large number of physical systems exhibiting anomalous relaxation behaviour.

Appendix: Fractal Dimension, Anomalous Exponents and Random Walks

The path of a random walker is a random object, e.g., Fig. (1.1.1). It is self-similar and is a fractal (Latin fractus - broken) or dilation invariant object, which exhibits irregularities on all scales. One of the most important characteristics of a fractal is the fractal dimension [44,45], a useful working definition of which has been given by ben-Avraham and Havlin [43]. If we consider a regular object such as a line segment, square or cube of Euclidian dimension d magnified by a factor R, the original object would fit FF1 times into the magnified one. The foregoing sentence may serve as a working definition of the fractal dimension df. As an example, let us consider the well-known Koch curve, which is a fractal [44,45]. The Koch curve is constructed (see Fig. ll .A.l) from an initiator, which is a unit segment (a) according to the generator (b) which replaces the middle third of the initiator by two other segments of length

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Chapter 11. Anomalous Diffusion 671

(a) (b) (c)

jpJ'W ' L A - .

(d) Figure l l .A.l Construction of the Koch curve.

1/3 making the tent shape shown in (b). The same procedure is repeated and results in (c) (after 2 iterations) and in (d) (after 5 iterations) and is carried on indefinitely. The limiting curve is of infinite length (after n iterations, it has increased (4/3)"-fold over the initial triangle [45]) but confined to a plane. The Koch curve, however, is not one-dimensional neither, on the other hand, is it two-dimensional - its area is zero [44]! Thus, its fractal dimension df should lie between one and two [44] and can be estimated by counting the number TV (= 4) of self-similar line segments created by the generator inside the magnified segment R(R = 3, in the figure). The scaling of the steps or segments N with magnification R defines the fractal dimension

so that Rdf ~ N

df =lnfl/lnN = ln4/ln3=: 1.262.

Dilation invariance or self-similarity is manifested in a deterministic fractal such as the Koch curve [44] by the fact that if we look at the object at any given magnification R, we will see the same pattern occurring over and over again. This is the self-similarity property of a fractal. We have so far dealt with deterministic fractals. For our purposes, we must treat random fractals. Examples of natural random fractals are tree bark, coastlines, and lightning flashes. The trajectory of a Brownian particle is also a random fractal (the generator of which is ultimately the molecular impacts or stosszahlansatz), where the path of the particle at any given resolution on average appears the same. The trajectories are not exactly self-similar. They are self-similar only in a statistical sense. In normal Brownian motion, a typical displacement made by the Brownian particle scales with the number of steps N as

if N is large. Here, the exponent 2 may be characterised as the fractal dimension of the random walk by replacing XN by R [32]. Thus, the

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672 The Langevin Equation

fractal dimension in this case is two. In the Weierstrass random walk, the large distance behaviour of a typical displacement XN scales with a large number of steps N made by the walker as [32]

XaN~N.

Thus the anomalous exponent a occurring in Weierstrass random walks is the fractal dimension of the walk [32]. In like manner, in the fractal time random walk, the anomalous exponent crmay be interpreted as the fractal dimension of the set of waiting times between jumps. (Methods for measuring the fractal dimension of random fractals are described in [44]).

References

1. R. Metzler and J. Klafter, Phys. Reports 339, 1 (2000). 2. P. Debye, Polar Molecules, Chemical Catalog, New York, 1929, Reprinted by

Dover Publications, New York, 1954. 3. W. F. Brown, Jr, Phys. Rev. 130, 1667 (1963). 4. R. Metzler and J. Klafter, Adv. Chem. Phys. 116, 223 (2001). 5. G. Rangarajan and M. Ding, Phys. Rev. E 62, 120 (2000). 6. H. Risken, The Fokker-Planck Equation, 2nd Edition, Springer, Berlin, 1989. 7. W. T. Coffey, J. Chem. Phys. I l l , 8530, (1999). 8. W. T. Coffey, D. S. F. Crothers, and S. V. Titov, Physica A 298, 330 (2001). 9. W. T. Coffey, Yu. P. Kalmykov, E. S. Massawe, and J. T. Waldron, J. Chem. Phys.

99,4011(1993). 10. Handbook of Mathematical Functions, edited by M. Abramowitz and I. Stegun,

Dover, New York, 1964. 11. G. Klein, Proc. R. Soc. Land. A 211, 431 (1952). 12. A. N. Malakhov and A. Pankratov, Adv. Chem. Phys. 121, 357 (2002). 13. C. Fox, Trans. Am. Math. Soc. 98, 395 (1961). 14. Higher Transcendental Functions, H. Bateman and A. Erdelyi, Eds., McGraw-Hill,

New York, 1953. 15. D. S. F. Crothers, Personal Communication (2002). 16. W. T. Coffey, unpublished (2002). 17. K. G. Wang, Phys. Rev. A 45, 883 (1992). 18. E. Lutz, Phys. Rev. E 64, 051106 (2001). 19. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. E in press (2003). 20. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. E 65, 032102 (2002). 21. W. T. Coffey, Yu. P. Kalmykov, and S. V. Titov, Phys. Rev. E 65, 051105 (2002). 22. R. Metzler, Phys. Rev. E 62, 6233 (2000). 23. E. Barkai and R. S. Silbey, /. Phys. Chem. B 104, 3866 (2000). 24. R. Metzler and J. Klafter, J. Phys. Chem. B 104, 3851 (2000). 25. R. A. Sack, Proc. Phys. Soc. Lond. B 70, 402, 414 (1957). 26. R. G. Gordon, /. Chem. Phys. 38, 1724 (1963). 27. R. W. Davies, Phys. Rev. 62, 1169 (1954).

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28. P. C. Hemmer, Physica 27, 79 (1961). 29. W. T. Coffey, S. G. McGoldrick, and K. P. Quinn, Chem. Phys. 125, 99 (1988). 30. R. Metzler and I. M. Sokolov, Europhys. Lett. 58, 482 (2001). 31. J. H. Van Vleck and V. F. Weisskopf, Rev. Mod. Phys. 17, 227 (1945). 32. W. Paul and J. Baschnagel, Stochastic Processes from Physics to Finance, Springer

Verlag, Berlin, 1999. 33. G. Temple, 100 Years of Mathematics, Duckworth, London 1981. 34. R. R. Nigmatullin and Ya. A. Ryabov, Fiz. Tverd. Tela (St.Petersburg) 39, 101

(1997) [Phys. Solid State 39, 87 (1997)]. 35. R. Kubo, M. Toda, and N. Nashitsume, Statistical Physics II. Nonequilibrium

Statistical Mechanics, Springer Verlag, Berlin, 1991. 36. H. Frohlich, Theory of Dielectrics, 2nd Edition, Oxford University Press, London,

1958. 37. C. J. Reid and M. W. Evans, J. Chem. Soc, Faraday Trans. 1175, 1369 (1979). 38. M. W. Evans, G. J. Evans, W. T. Coffey, and P. Grigolini, Molecular Dynamics and

Theory of Broadband Spectroscopy, Wiley, New York, 1982. 39. R. Mazo, Brownian Motion: Fluctuations, Dynamics and Applications, Oxford

University Press, Oxford, 2002. 40. P. L. Bhatnagar, E. P. Gross, and M. Krook, Phys. Rev. 94, 511 (1954). 41. E. W. Montroll and M. F. Shlesinger, On the Wonderful World of Random Walks,

Eds. J. L. Lebowitz and E. W. Montroll, in Non Equilibrium Phenomena II from Stochastics to Hydrodynamics, Elsevier Science Publishers, BV, Amsterdam, 1984.

42. G. H. Vineyard, Phys. Rev. 110, 999 (1958). 43. W. T. Coffey, J. Phys. D: Appl. Phys. 11, 1377 (1978). 44. D. ben-Avraham and S. Havlin, Diffusion and Reactions in Fractals and Disordered

Systems, Cambridge University Press, London, 2000. 45. M. Schroder, Fractals, Chaos and Power Laws, W. H. Freeman, New York, 1991. 46. J. Klafter, A. Blumen, and M. F. Shlesinger, Phys. Rev. A 35, 3081 (1987). 47. E. W. Montroll and G. H. Weiss, / Math. Phys. 6, 167 (1965). 48. E. P. Gross, Phys. Rev. 97, 395 (1955). 49. E. P. Gross, J. Chem. Phys. 23, 1415 (1955). 50. N. Wiener, The Fourier Integral, Cambridge University Press, Cambridge, 1933. 51. W. Feller, An Introduction to Probability Theory and its Applications, 3rd Edition,

vol.1, Wiley, New York, 1967; vol. 2, 2nd Edition, Wiley, New York, 1966. 52. I. Oppenheim, K. Shuler, and G. Weiss, Stochastic Processes in Chemical Physics:

The Master Equation, MIT Press, Cambridge, MA, 1977.

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Page 700: The Langevin Equation Coffey_Kalmykov_Waldron

Index

A action-angle variables, 524 additive noise, 76 after-effect function, 163, 207 anisotropy constant, 122,407,490 angular moment um operator, 341, 379 anomalous dielectric relaxation, 160 anomalous diffusion, 152, 617, 620,

654 anomalous relaxation, 156, 601, 631 Arrhenius law, 86 associated Legendre functions, 381,

531 asymptotic expansion, 268, 317, 255 asymmetric top, 378, 388 autocorrelation function, 13, 59, 206,

224, 245, 270, 303 Avogadro's Number, 16 axially symmetric potential, 397,582

B birefringence, 437, 443 blocking temperature, 125 Brinkman equations, 517, 518 Brown's asymptotic formula, 136, 357,

580 Brown's Fokker-Planck equation, 131 Brownian fluctuation of a suspended

mirror, 16 Brownian motion, 3, 236 Brownian motion in a tilted periodic

potential, 282 Brownian motion of a free particle,

236 Brownian particles, 3, 4

C cage model, 558 central limit theorem, 56 centred random variable, 47, 245 Chapman-Kolmogorov equation, 69,

605 characteristic function, 52 Clebsch-Gordan coefficients, 331, 332,

384, 470

Cole-Cole equation, 156, 651, 663 Cole-Cole plots, 362, 394,652 complex polarisability, 203 confluent hypergeometric (Kummer)

function, 246, 351,529,635 continued fractions, 201, 266, 268,

291, 313, 350, 351, 358, 368, 372, 537, 596

continuity equation, 8, 31,116, 130, 624

continuous time random walk (CTRW), 156, 158, 600, 602, 608, 609

correlation coefficient, 48, 51 correlation function, 59, 312, 333, 346,

454, 536, 563 correlation time, 207, 264, 268, 316,

349 covariance, 47 covariance matrix, 51 crosscorrelation function, 59 cubic anisotropy, 132, 490 Curie point, 126 current-voltage characteristics, 112,

290

D Debye relaxation time, 116, 160, 331,

399,524,581,611 Debye theory of dielectric relaxation,

117 depletion effect, 143 deterministic drift, 180, 330, 381 dielectric relaxation, 113, 230, 243,

436, 443 differential-recurrence relations, 177 diffusion coefficient, 19, 72, 75, 601 diffusion equation, 19, 92 dipole moment, 121,327, 367, 437,

513 discrete orientation model, 127, 500 discrete time random walk, 600 distribution function, 46 Doob's interpretation, 171 drift coefficient, 72, 76

675

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676 The Langevin Equation

dynamic Kerr-effect, 112,230

E effective eigenvalue, 221, 258, 293,

320, 430 effective eigenvalue method, 221,371,

429, 458 effective relaxation time, 222, 256,

354, 360, 373,416, 502, 617, 620 egg model, 577 Einstein's formula, 14 Einstein's method, 17 Einstein's summation convention, 180 electric polarisation, 388, 439, 655 energy control diffusion, 44 ensemble average, 25, 26 equipartition theorem, 25, 285 ergodic hypothesis, 26 error function, 214, 317, 353 error function of imaginary argument,

231,276,353 escape rate, 95 Euler angles, 378 Euler-Langevin equation, 184, 327,

378,531,545

F far-infrared absorption, 558 Fast Fourier Transform, 528 ferrofluids, 141, 576, 587 ferromagnetic resonance (FMR), 131,

481,487 fluctuation-dissipation theorem, 15,

205, 665 Fokker-Planck equation, 6, 68, 79, 80,

116,210,214,229,311,336,343, 388,518,544,556,622

Fokker-Planck operator, 210, 616, 623 forward Kolmogorov equation, 9 Fourier-Hermite series, 518, 634 fractal, 5, 670 fractal dimension, 602, 670 fractional derivative, 158 fractional diffusion equation, 603 fractional Klein-Kramers equation,

633, 641, 649, 656 fractional Langevin equation, 665 fundamental solution, 85, 601, 611

G gamma function, 158

Gaussian distribution, 48, 50, 608 Gaussian random variable(s), 54, 55,

245, 593 Gilbert's equation, 124, 335, 337, 578 generalised Langevin equation, 665,

668 generalised Mittag-Lefler function,

628, 666 Gordon sum rule, 638, 651

H harmonic oscillator, 241 Hermite polynomials, 514, 518, 531 high damping limit, 37 hysteresis, 128 hypergeometric functions, 246, 270,

351

I impedance of the Josephson junction,

299 incoherent scattering of slow neutrons,

112 indicator function, 67 inertial effects, 21, 244, 507, 631 integral exponential function, 542, 651 integral relaxation time, 207, 417, 616,

620 intermediate to high damping (IHD),

37,88,91,99 intertrapping time, 160 Isserlis's theorem, 13, 55, 72, 591 itinerant oscillator, 557 Ito and Stratonovich rules, 171 Ito calculus, 175 Ito equation, 175

J joint probability distribution, 49 Josephson junction, 177, 282 Josephson radiation, 302 jump length variance, 603

K kinetic theory, 4 Kerr-effect relaxation, 112,436 Klein-Kramers equation, 9, 35, 41, 79,

633 Koch curve, 670 Kramers escape rate theory, 85 Kramers transition state theory, 86

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Index 677

Kramers-Kronig relations, 203 Kramers-Moyal expansion, 72, 74 Kubo relation, 206 Kummer transformation, 352, 355

L Landau-Lifshiftz equation, 578 Langer formula, 109 Langer's treatment, 104 Langevin equation, 10, 11, 60, 114,

183, 186, 236, 244, 253, 286, 337, 509,561,590

Langevin function, 121, 369 Laplace operator, 116,131, 330 Legendre polynomials, 334, 387 Levy flight, 603, 608 Levy distribution, 607, 611 linear impedance, 292, 298 linear response theory, 201, 210 Liouville equation, 29, 31, 34 locked region, 306 longitudinal correlation time, 256, 271,

349,352,401,410 longitudinal relaxation time, longitudinal susceptibility, 352, 369,

401, 472, 493, 582 longitudinal response, 368, 407 low damping (LD), 37, 90, 99

M macroscopic quantum tunnelling, magnetic after-effect, 121 magnetic domains, 122 magnetic viscosity, 124 marginal distribution function, 50 Markov process, 58, 63 Mathematica program, 290, 330, 332,

417,470 matrix continued fractions, 196, 397,

401, 412, 429, 439, 446, 457, 476, 497,521,541,553,570

matrix diagonalisation method, 191 matrix recurrence relations, 196, 397 Maxwell-Boltzmann law, 27 Maxwell-Boltzmann distribution, 25,

105, 229 mean square displacements, 13, 20, 36,

236 mean first passage time (MFPT), 622,

624, 627 mean waiting time, 603

method of steepest descents, 350 Mittag-Lefler function, 161, 610, 615,

622, 628, 637 mobility of superionic conductors,

112,282 modified Bessel functions, 195, 264,

289,355,369,403,521,657 moment generating function, 52 Mori theory, 566 multiplicative noise, 76, 172 multiplicative noise term, 76, 287, 311,

328, 381

N N-fold cosine potential, 252 Neel relaxation, 124, 128 Neel relaxation time, 125,133 nematic liquid crystals, 333, 347,450 non-axially symmetric potentials, 468 noise-induced drift, 78, 180, 186, 329,

381 nonlinear Langevin equation, 172, 179 nonlinear response, 226, 229, 436,

443, 527 noninertial limit, 285 Nyquist's theorem, 23, 62

O Onsager model, 558, 561 orientational space gradient operator,

339, 379 Ornstein-Furth formula, 238 Ornstein-Uhlenbeck process, 82 Ornstein-Uhlenbeck theory, 236

P paleomagnetism, 126 parabolic cylinder function, 314, 518 partition function, 28, 146 phase space, 25 Poley absorption, 119, 557, 558, 663 probability density function, 47 probability theory, 44

Q quantum noise in ring laser

gyroscopes, 112,

R random fractal, 5, 671 random process, 57

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678 The Langevin Equation

random variable, 46 random walk, 152, 600 realisations, 8 recurrence relations, 196 relaxation, 2 representative point, 26 Rice's method, 241 Riemann-Liouville definition, 160, 610 Riemann-Zeta function, 269, 625, 630 ring laser gyroscope, 282, 307 Robert Brown, 1 Rocard equation, 246, 538, 636, 645,

651 rotational Brownian motion, 20, 113,

252, 397, 530, 544 rotational diffusion, 326, 378

S saddle point, 97, 483 self-similarity, 5, 608, 671 sharp values, 76, 173, 311, 331, 382 single domain particle, 121, 141 smallest novanishing eigenvalue, 214,

224,269,317,354,480,485 Smoluchowski equation, 9, 83, 116,

262,441 Smoluchowski integral equation, 69,

152 spectral density, 60 spectrum of the Josephson radiation, spherical harmonics, 331 standard deviation, 47,41 standard normal distribution, 48 stationary process, 58, 59 stationary solution, 9 Stieltjes integral, 67 stochastic differential equation, 13,

328 stochastic process, 57 stochastic resonance, 149, 421 Stokes's law, 20 Stoner-Wohlfahrt calculation, stosszahlansatz, 7, 19,33, 153 Stratonovich definition, 174, 328, 338

Stratonovich equation, 175 Stratonovich calculus, 183, 533, 547 Sturm-Liouville equation, 262, 613,

614 subdiffusion, 158, 602, 611, 632 superdiffusion, 158, 602, 608 superparamagnetism, 121 susceptibility, 223, 352, 423, 429, 472,

493,615,618 symmetric top, 383, 544

T telegraph equation, 639 theory of angular momentum, 341 time average, 27 torsional oscillator model, 247 trajectory of a Brownian particle, 5 transition probability, 69 transition state theory, 85, 86 transverse polarisability, 274 transverse susceptibility, 359, 373,

429, 582 transverse relaxation time, 256, 274,

360 transverse response, 372

U unlocked region, 306 uniaxial potential, 136, 334, 347, 407,

457

V variance, 47 variance of the Wiener process, 64

W Weierstrass random walks, 608, 672 white noise, 12 white noise torque, 114, 243, 327, 378,

531 Wiener integral, 66, 247 Wiener process, 63 Wiener-Khinchine theorem, 59, 238 Wigner D functions, 59, 238

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The Langevin Equation Second Edition

This volume is the second edition of the first-ever elementary book on the Langevin equation method for the solution of problems involving the Brownian motion in a potential, with emphasis on modern applications in the natural sciences, electrical engineering and so on. It has been substantially enlarged to cover in a succinct manner a number of new topics, such as anomalous diffusion, continuous time random walks, stochastic resonance etc, which are of major current interest in view of the large number of disparate physical systems exhibiting these phenomena. The book has been written in such a way that all the material should be accessible to an advanced undergraduate or beginning graduate student. It draws together, in a coherent fashion, a variety of results which have hitherto been available only in the form of research papers or scattered review articles.

"I found this book a valuable addition to my library. It will be of interest to researchers and advanced students and the material could be used as the text for a course for advanced undergraduates and graduate students."

Irwin Oppenheim Editor of Physical Review E Journal of Statistical Physics

"This enlarged and updated second edition of the book: 'The Langevin equation' presents an extremely useful source for the practitioners of stochastic processes and its applications to physics, chemistry, engineering and biological physics, both for the experts and the beginners. It gives a valuable survey of solvable paradigms that rule many diverse stochastic phenomena. As such, it belongs onto the desk of all engaged in doing research and teaching in this area."

Peter Hanggi University of Augsburg

"... this enlarged and updated second edition constitutes an excellent and quite useful contribution to this important subject of relevance in physics, chemistry and several branches of engineering."

Mathematical Reviews

"This is a timely update of the theory and applications of the Langevin equal ion, which skillfully combines the elementary approaches with most recent developments such as anomalous diffusion and fractional kinetics. Both experts and beginners will benefit from this well-written textbook."

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R I \viv I fniversity

ISBN 981-238-462-6

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