The Jarzynski Equation the Fluctuation Theorem · The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Scope the Jarzynski equality the Fluctuation theorem … there are

Kirill Glavatskiy

The Jarzynski Equation and

the Fluctuation Theorem

Trial lecture for PhD degree

24 September, NTNU, Trondheim

2The Jarzynski equation and the Fluctuation theorem K. Glavatskiy

The Jarzynski equation and the fluctuation theorem

Fundamentalconcepts

Practical applications

Recents developments

Statiscical physics

Fluctuations


Scope

the Jarzynski equality

the Fluctuation theorem

… there are a few relations that describe

the statistical dynamics of driven systems

which are valid even if the system is

driven far from equilibrium ...

Gavin E. Crooks, Physical Review E, 61(3), p.2361, 2000

«

»


Outline

The Jarzynski equality and the Fluctuation theorem

The contents of the theorems

Applications and experimental verification

Discussions and critique

General introduction

«Characters in play»

Crash course in statistical mechanics

Thermodynamics and its range of validity


Part A.I

«Characters in play»

Scope of the theorems

Main authors

A.I


Jarzinsky equality

Equilibrium (reversible process):

Work = Δ Energy

Non-Equilibrium (irreversible process):

Work = Δ Energy + Lost Work

Work ≥ Δ Energy

depends on the process path

f(Work) = f(Δ Energy)

Christopher Jarzynski

«Nonequilibrium Equality for Free Energy Differences»

University of Maryland, Assoc. Prof. Chemistry and Biochemistry

Physical Review Letters, 78(14), p.2690, 1997A.I


Fluctuation theorem

grows exponentially with time

Probability ( Δ S )

Probability ( -Δ S )

«Probability of Second Law Violations in Shearing Steady States»

The Australian National University, Prof., Research School of Chemistry

Evans et al, Physical Review Letters, 71(15), p.2401, 1993

Denis J. Evans

Δ S ≥ 0 The Second Law of Thermodynamics:

Macroscopic processes — irreversible

Motion of molecules:

Newton's equations — reversible in time F = m a

A.I


Fluctuation theorem

Giovanni GallavottiE. G. D. Cohen

Garry P. Morriss Debra J. Searles

Gavin E. Crooks

The Rockefeller University, USA

Universita di Roma La Sapienza, Italy

Lawrence Berkeley National Lab., USA

The University of New South Wales, Australia

Griffith University, Australia

and others...A.I


Part A.II

Crash course in

statistical mechanics

Distribution function

Lyapunov exponent

A.II


0 20 k 0 k10

k Ek


k T =⟨E kin⟩32

Thermodynamic variables are averages of microscopic properties

{x1, v1 ; x2, v2 ; xN , vN }

N particles: microscopic configuration

E i

K energy intervals for N particles : distribution function

E tot = ∑i=1

N

E i = ∑k=1

K

k Ek Ek

⟨ E ⟩ = 1K ∑

k=1

K

k Ek E k

Ensemble averages:

Extensive properties:

A.II


Lyapunov exponent

t ≈e t 0

p r o b a b i l i s t i c d e s c r i p t i o n

t≈0 t≠0

Molecular motion reveals the similar behavior: dynamical systems

Divirgense of particle's trajectory :

Lyapunov exponent

A.II


Statistical mechanics


Lyapunov exponent

Detailed balance

Link between microscopic and macroscopic properties:

i ,⟨ E kin⟩

PA B=PB A

0

A.II


Part A.III

Thermodynamics

and its range of validity

Equilibrium systems

Fluctuations

Non-equilibrium prcesses

A.III


Equilibrium system

i=1Z

exp− H i

kT — Gibbs canonical distribution

Meaningfull only for systems

⋯1 2 3 M-1 M

M configurations with the same distribution function:

There are configurations with the same distribution function

There are configurations with different distribution functions

In equilibrium, the same distribution function belongs to the most of configurations Equilibtium state is described by this distribution function:

the most probable distribution

With large number of molecules, N

With no external perturbations

A.III


Fluctuations

Large number of molecules:

Small number of molecules:

All the distributions are incarnated equally often:

there is no most probable distribution No way to introduce

the state functionsA.III


Non-equilibrium processes

Relaxation

Steady statesTime-dependent conservative

Non-conservative

Transition between steady states

Aging state

Microscopic configuration evolves in time: non-equilibrium fluctuations

A.III


Newton's dynamics

Thermodynamics

process rate

number of

particles

Non-equilibriumthermodynamics

Local equilibrium

T r , pr

Global equilibrium

Equilibrium thermodynamics

T , pFluctuations

Thermodynamics ?

Fluctuations

A.III


Part B.I

Contents

of the theorems

Transient Fluctuation theorems

Jarzynski equality

Crooks Fluctuation theorem

B.I


Transient Fluctuation theoremsD. J. Evans, E. G. D. Cohen, G. P. Morris, Phys Rev Lett, 71(15), p.2401, 1993D. J. Evans, D. Searles, Phys Rev E, 50(2), p.1645, 1994G. Gallavotti and E. G. D. Cohen, Phys Rev Lett, 74(14), p.2694, 1995G. Gallavotti and E. G. D. Cohen, J. of Stat Phys, 80, p.931, 1995

Dynamical systems

Second law vs microscopic reversibility

There are two kinds of microscopic trajectories:

0

Panti

Pordinary

Anti-trajectories are less mechanically stable, then their corresponding trajectories

ordinary trajectoriesΔ S ≥ 0

anti-trajectoriesΔ S ≤ 0

P P −

= e t

~e− t

dissipation

B.I


Jarzynski equalityC. Jarzynski, Phys Rev Lett, 78(14), p.2690, 1997C. Jarzynski, Phys Rev E, 56(5), p.5018, 1997

P

VW

W rev= F

W irr=∫ f xdx ≥ Ff

〚W 〛≡ 1K ∑

1

K

W j

〚e− W

kT 〛=e− F

kT

Process average:

10x

2

⋮

W 1

W 2

W K

⋮

1t

2t

K t

⋮

1t =2t =⋯=K t same schedule:

W 1 ≠W 2 ≠⋯≠W K diffrent work:〚W 〛≥ FB.I


W

F

R

F

Crooks Fluctuation theoremG. E. Crooks, J. of Stat Phys, 90, p.1481, 1998G. E. Crooks, Phys Rev E, 60(3), p.2721, 1999G. E. Crooks, Phys Rev E, 61(3), p.2361, 2000

1 2F t

2 1R −t

Path ensemble: Initial thermal equilibrium (canonial distribution)

The process, perturbing from equilibrium

Direction of the process: Forward (F)

Reverse (R)

[F t ]=[R−t ] exp− Qk T Detailed balance:

〚A t e−

W − F H

kT 〛F= A−t

〚〛

〚〛R

〚e−

WkT〛=e

− F H

kT

Jarzynski equality

PF PR−

=

Fluctuation theorem

e

B.I

Path function: At A−t Any function defined for the process: and


Contents of the theorems

Transient Fluctuation theorems

Jarzynski equality

Crooks Fluctuation theorem

Family of

theorems

P P −

= e t

〚e− W

kT 〛=e−

F H

kT

B.I〚A t e

−W − F H

kT 〛F= A−t〚〛R

Reduce to the common expressions

in linear regime


Part B.II

Applicationsand

experimental verification

B.II


Applications

Physical processes

Biological machines

Colloids Turbulent flow

Energy conversion in ATP

Chemical reactions

B.II


MD Simulations

Relaxation

ln[P t =P t =− ] =

B.II


Experiments

pulling biomolecules: a bead in an optical trap

B.II

W =∫ f xdx


Experiments

B.II

WJE1 ↔ WJE2

WJE1 ↔ ΔF

Expectations:

Conditions: 40 folding-unfolding cycles

7 datasets with different molecules

Reversible work: slow rate

folding and unfolding curves coincide

Prerequisites: small number of molecules

both, Eq and Neq regimes

J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr., C. Bustamante, Science 296, p.1832, 2002


Part B.III

Discussions

and critique

B.III


Relevance

Aiming for a new understanding of Nature

Does the macroscopic description contain all the necessary information?

Mechanism of Life

Arrow of Time

A family of the relations must be treated together

It is the consistency of different approaches, which matters a lot

A complex verification is needed

Is it a coincidence for special processes or a general result?

Do experiments correspond to the required conditions?

The physical meaning of the used quantitiesB.III


Debates

E. G. D. Cohen and D. Mauzerall, J. of Stat Mech: T&E, P07006, 2004Received: 23 June 2004Accepted: 29 June 2004Published: 13 July 2004

C. Jarzynski, J. of Stat Mech: T&E, P09005, 2004Received: 6 August 2004Accepted: 30 August 2004Published: 21 September 2004

… The communities accepting the Jarzynski equality

consists overwhelmingly of chemists and biophysicists,

while the physicists have divided opinions ...

«»

E. G. D. Cohen and D. Mauzerall, J. of Stat Mech: T&E, P07006, 2004

B.III


Cohen arguments

B.III

E. G. D. Cohen and D. Mauzerall, J. of Stat Mech: T&E, P07006, 2004C. Jarzynski, J. of Stat Mech: T&E, P09005, 2004

The Jarzynski equality (JE) is not an equality in any mathematical sense, but can be a useful approximate equality in certain important fields, e.g. study of single molecules in solution

Essentially reversible isothermal experiments were performed True irreversible processes, have so far not been considered experimentally

Correct accounting for the heat exchange The system is subjected not only to the mechanical work,

but also to the simultaneous energy exhange with the surroundings

A rigorous derivation is possible only for «linear regime», which is already known

〚e−

WkT〛=e

− F H

kT Temperature of the initial equilibrium state

Usage of the temperature of the surroundings for every irreversible path

makes no physical sense


Vilar arguments

B.III

J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 100, 020601, 2008

L. Peliti, J. of Stat Mech: T&E, P05002, 2008J. Horowitz and C. Jarzynski, Phys. Rev. Lett. 101, 098901, 2008 (Comment)J. M. G. Vilar and J. M. Rubi, Phys. Rev. Lett. 101, 098902, 2008 (Reply)

JE is not general: there are systems, where it does not hold

Harmonic oscilator ⟨e−W ⟩=1 ⟨e−W ⟩=e− F Jarzynski:

The Jarzynski definition of the work is not general:Parameter א is not necessarily the (generalized) coordinate

Hamiltonian is defined up to an arbitrary time-dependent function

W =∫ dt ∂∂ t

∂ H∂

JE holds, but not between the work and free energy

⟨exp [−∫ f x dx ]⟩≠⟨exp [−∫ dH x , t ]⟩= Z t Z 0

≠exp [− FZ t ]

The experiments confirm JE beacuse of specialy chosen conditions Yet, the agreement is good, maily close to relatively slow perturbations


My arguments

B.III

Why does an irreversible process average

depends on an equilibrium state

( work vs free energy ) ?

… The microscopic history of the system and environment

will differ from one realization to the next, simply because

the initial microstate differs from one realization to the next ...

«»

〚〛 ≡ ⟨ ⟩process average

〚 A〛= 1N ∑

i=1

N

Ai ⟨ A⟩= 1N ∑

i=1

N

Ai Ai

canonical average

does ρ = 1 for an irreversible process? diversity is not only due to initial configuration


Kulinskii arguments

The definition of the work is misleading

JE does not hold for a simple process

W =∫ dt ∂∂ t

∂ H∂

H micro{x1, v1 ; x2, v2 ; x N , vN }

W micro=∫ dH = Hmicroscopic energy: microscopic work: NB! equality for any process:

molecules do not know about heat!

macroscopic configuration: H macro{T ,}

W macro=∫ force ∗ dxmacroscopic work:

Because of averaging over microscopic degrees of freedom we loose information: Q

〚e−W macro〛=e− F

W micro≠W macro

Jarzynski: and

Irreversible adiabatic expansion of ideal gas into vacuum

W =0

F ~lnV 2

V 1

〚e−W 〛≠e− F

no work on the system

increase of entropy and free energy

recent communications


The End

The story just begins, doesn't it?


Bibliography Books & Reviews:

Articles:

Kvasnikov. Thermodynamics and Statistical Physics. Editorial, 2002Landau, Lifshitz. Statistical Physics. Pergamon Press, 1980Rumer, Ryvkin. Thermodynamics, Statistical Physics and Kinetics. Nauka, 1972

Bustamante, Liphardt, Ritort. The Nonequilibrium Thermodynamics of Small Systems, Physics Today, p43, 2005Evans, Searles. The Fluctuation Theorem, Advances in Physics, 51(7), p1529, 2002Jarzynski. Nonequilibrium Fluctuations of a Single Biomolecule. Lecture Notes in Physics, 711, p201, 2007.Ritort. Nonequilibrium fluctuations in small systems: From physics to biology. arXiv, cond-mat:0705.0455

V. L. Kulinskii, Private communications, 2009J. M. G. Vilar and J. M. Rubi, PRL 100, 020601, 2008L. Peliti, J. of Stat Mech: T&E, P05002, 2008J. Horowitz and C. Jarzynski, PRL 101, 098901, 2008 J. M. G. Vilar and J. M. Rubi, PRL 101, 098902, 2008A. Imparato and L. Peliti. arXiv:cond-mat/0706.1134v1C. Jarzynski. PRE 73, 046105, 2006F. Douarche, S. Ciliberto, A. Petrosyan and I. Rabbiosi. EPL, 70(5), 2005, p593C. Jarzynski. J. Stat. Mech.: Theor. Exp. (2004) P09005E. G. D. Cohen and David Mauzerall. J. Stat. Mech.: Theor. Exp. (2004) P07006G. Gallavotti. arXiv:cond-mat/0301172v1J. Liphardt, S. Dumont, S. B. Smith, I. Tinoco Jr., C. Bustamante, Science 296, p.1832, 2002C. Jarzynski. PNAS, 98(7), 2001, p3636G. E. Crooks. PRE, 61(3), 2000, 2361G. E. Crooks. PRE, 60(3) 1999, p2721E. G. D. Cohen and G. Gallavotti. J of Stat Phys, Vol. 96, Nos. 5/6, 1999G. E. Crooks. J of Stat Phys, Vol. 90, Nos. 5/6, 1998C. Jarzynski. PRE, 56(5), p.5018, 1997C. Jarzynski. PRL, 78(14), p.2690, 1997D. J. Evans and D. J. Searles. PRE, 53(6), 1996, p52G. Gallavotti and E. G. D. Cohen. J. of Stat Phys, 80, p.931, 1995G. Gallavotti and E. G. D. Cohen. PRL, 74(14), p.2694, 1995D. J. Evans, D. Searles. PRE, 50(2), p.1645, 1994D. J. Evans, E. G. D. Cohen, G. P. Morris. PRL, 71(21), p.3616, 1993D. J. Evans, E. G. D. Cohen, G. P. Morris. PRL, 71(15), p.2401, 1993G. N. Bochkov and Yu.E. Kuzovlev. Physica 106A (1981) 443-479, Physica 106A (1981) 480-520


Detailed balance

P A xA , t A x A , t A ; xB , t B=xB , t B ; xA , t A P B xB , t B

exp− H A

k T xA , t A ; xB , tB= xB , tB ; xA , t A exp− H B

k T

Probability ( A → B ) = Probability ( B → A )

It is convenient to use propabilistic approach in stead of deterministic:

Newton's equations ma = F are time reversal: principle of detailed balance

xi t j Px t j=xi

state probability to be

in state A

transient probability to go

from state A to state B

transient probability to go

from state B to state A

state probability to be

in state B

A.II

The Jarzynski Equation the Fluctuation Theorem · The Jarzynski equation and the Fluctuation theorem K. Glavatskiy Scope the Jarzynski equality the Fluctuation theorem … there are

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