Dresden 2014 A tour of some fractional models and the physics behind them

A brief tour of some fractional models … and of the physics

they seek to embody Nick Watkins

([email protected])

MPIPKS, Dresden, 9th July, 2014

Addendum:

These slides were given as a seminar at MPIPKS Dresden on 9th July 2014. I have corrected errors/typos that I spotted or which were pointed out to me, and added the remaining references. These are meant to be indicative, not comprehensive, especially those written as “e.g”

I ran out of time and space in designing the talk, so SOC and turbulence ended up not being treated in any detail at all – I have added a couple of suggested book references to their slides in order to rectify this a bit.

I also expect that there are many other areas that could be improved. If using this material please consult a local expert, especially on topics like stochastic calculus and fractional derivatives.

Nick

Context:

The understanding of Brownian motion has been one of the great achievements of 20th century physics, mathematics, chemistry and economics. Its importance spans these fields, as well as biology and control engineering, at the very least. However, the increasing interest in processes which are not completely amenable to the now standard methods has led to new models for "anomalous" time series and diffusion, most notably the fractional stable family like Mandelbrot's fractional Brownian motion, and the family of fractional continuous time random walks. In addition fractals and fractal calculus play a key role in models for the known physical phenomenon of turbulence, and the postulated one of self-organised criticality.

Aim

• Informally review the “zoo” of fractional models and the physics each one brings with it …

• Driven in part by my own difficulties over the years in working out which fractional model corresponds to what physics… have become a fractal zookeeper in my spare time …

• Main aim is to promote insight via discussion, and build on the excellent talks at EXEV14; by Rainer & Alexei during their visits; etc.

• WARNING: Haven’t yet reconciled notation.

So if it looks wrong/puzzling please do ask.

Sorry !

“A brief tour of some fractional model … and of the physics

they seek to embody Nick Watkins

MPIPKS, July, 2014

… and never use “brief tour” in an abstract …B. Kliban

Origins:

"On the motion of small particles suspended inliquids at rest required by the molecularkinetic theory of heat.“--A. Einstein,

Ann. D. Physik, 4th ser. 17 549 (1905)

Standard diffusion picture due to Bachelier, Einstein, Smoluchowski, Langevin et al.

Three motivating questions

• Q1 stochastics: If we “break” that picture, what models then result for “anomalous” behaviour, and why ? Do we have more choices ? 2 key examples are fractional kinetics (e.g. CTRW), and fractional motions (e.g. FBM) …, … How differ-what’s common ? (*) [Main topic]

• Q2 statistics: How can we reliably distinguish models, measure exponents, etcwhen many observables show same or similar behaviour ? …[I will largely ignore, has been discussed by Barkai, Klages, Chechkin, Froemberg, Klafter, Sandev and others at PKS over the last year]

• Q3: stat mech: What physical scenarios do these models correspond to … can that help us design better tests … how do scenarios relate to other observed physical phenomena such as turbulence, and to postulated ones such as SOC ? …[I will actually very have little to say about SOC and turbulence]

(* May also briefly mention nonlinear Fokker-Planck and Levy walk cases).

Cornerstones

2

Wiener process

( ) ( ) (0,1)t dt

tdX tX t dt t N

2

Diffusion equation

( , )( , )

( , ) is Gaussian

P y tD P y t

t

P y t

1/

1/

Stability property: Pdf looks the same under

' , ' where =2

( ', ') ( , )

( , ) is Gaussian,

and is an attracting fixed

point for short tailed pdfs.

t yt y

P y t P y t

P y t

Ohmic Langevinequation and FDT

Kinetic description

Stability property, Wienerprocess & central limit theorem

( ) ( )Mq q V q f t

e.g. Lemons, 2002; Paul and Baschnagel, 2013

( ) 0 and

( ) ( ') 2 ( ')B

f t

f t f Tt k t t

Cornerstones of classical stochastics & equilibrium stat mech [e.g. Lemons, 2002]• Mathematical BM: Finite variance stable pdf; the central limit

theorem; and embodiment as a random walk. Einstein-BachelierBrownian motion, modelled as the Wiener process (WBM).

• Physical BM: embodiment by the Langevin equation (LE) of the fluctuation-dissipation theorem (and equipartition of energy). LE follows velocity v, and thus position x, of an individual particle on timescales comparable with the dissipation time scale.

• An approximation (Smoluchowski’s) that bridges LE physics and WBM maths. Leads to the Fokker-Planck and diffusion equations which follow the pdf p(x,t) for position, on timescales long compared to dissipation time scale.

Central Limit Theorem & Mathematical BM

• Adding r.v.’s = convolution of pdfs or multiplication of characteristic functions [e.g. Mantegna & Stanley, 2000; Bouchaud and Potters, 2003 ]. Can also be viewed as instance of renormalisation group process [e.g. Sornette, 2004]

Langevin equations and physical BM

• “We know that the complementary force … is indifferently positive and negative and that its magnitude is such as to maintain the agitation of the particle, which, given the viscous resistance, would stop without it ”-Langevin, 1908, in Lemons, 2002.

2Einstein: ( ) ( ) (0,1)t dt

tX t dt X t dtN

2

Langevin: V( ) ( )

(0,1) )( t dt

tV t dt

t dt V t

dtN

Fluctuation-Dissipation Relation

• “We know that the complementary force … is indifferently positive and negative and that its magnitude is such as to maintain the agitation of the particle, which, given the viscous resistance, would stop without it ”-Langevin, op. cit..

( ) ( ') 2 ( ')

( ) 0

BTf t f t k t t

f t

The Smoluchowski approximation

• Leads to a diffusing X:

2(dV = 0 = (0,1)) t dt

tV t dt d Nt

dX = Vdt2

2

2

0 0 2

dX = (0,1)

whose solution is

( ) (x , )

t dt

t

t

dt

tN

N

X t

So which leg(s) could we break ?

2

Wiener process

( ) ( ) (0,1)t dt

tdX tX t dt t N

2

Diffusion equation

( , )( , )

( , ) is Gaussian

P y tD P y t

t

P y t

1/

1/

Stability property: Pdf looks the same under

' , ' where =2

( ', ') ( , )

( , ) is Gaussian,

and is an attracting fixed

point for short tailed pdfs.

t yt y

P y t P y t

P y t

Ohmic Langevinequation and FDT

Kinetic description

Stability property, Wienerprocess & central limit theorem

( ) ( )Mq q V q f t

e.g. Lemons, 2002; Paul and Baschnagel, 2013

( ) 0 and

( ) ( ') 2 ( ')B

f t

f t f Tt k t t

Symptoms of complex transport: 1

10 July 2014 15

One symptom is existence of very long jumps

(“flights”) compared to the <jump>

Extending the CLT -> Lévy flights

• Goes beyond the CLT relatively unambiguously, by dropping assumption of finite variance.

• Result is Extended Central Limit Theorem and the family of α-stable distributions, defined by pdf’s characteristic function:

~ exp((k) )LP k~ ( )( ) e dkx ikx

LP kP

Lévy “flight”

10 July 2014 17

Terminology came from Mandelbrot’s “Fractal Geometry of Nature” [1977, p. 289] & his picture of a

rocket traveling between fractally distributed galaxies. Actually a random walk but with non

Gaussian, heavy-tailed jumps. Waiting times still short tailed---temporal memory is short ranged.

Use α-stable distribution,

has asymptotic

power law tail

for its pdf P(x)

with exponent α

(1 )P(x) x

0<α< 2

α-stable

Gaussian

Stability

• Stable distribution has property of keeping its shape under convolution [e.g. Mantegna & Stanley, 2000; Sornette, 2004; Bouchaud and Potters, 2003] but the parameters rescale.

Gaussian

Cauchy

A kinetic equation for Lévy flights I

• 1st edition of Paul and Baschnagel (it’s now in Sec 4.2) gives a heuristic derivation of the kinetic equation that such a pdf must obey. Note sometimes µ replaces α, in the physics literature. Also watch out for a µ that differs from α by 1 e.g. classic papers on Levy flights & walks and DFA.

| |ˆCF for N step walk: ( ) Na k

LP k e

| |ˆCF for one step walk: ( ) a k

LP k e

A kinetic equation for Lévy flights II

( / ) t| |ˆIf each step takes then ( , ) a t kt P k t e

t| |ˆIf each step takes then ( , ) D kt P k t e

0Effective diffusion coefficient lim

t

aD

t

ˆˆA solution of | |

PD k P

t

Paul & Baschnagel, op cit.

Space fractional kinetic equation

Defining fractional derivat

1| |

2

ive th

ikxdk e kx

fractional kinetic equationWe get

( , ) ( , )P x t D P x tt x

Paul & Baschnagel, op cit.

Next questions …

• When do the FDT and/or equipartition “break” ? [e.g Klages & Chechkin, various PKS talks]

• What happens if we use full (generalised) Langevin equation ? [e.g. Sandev, ditto]

• Can we then still make Smoluchowski approximation ? [e.g. Lutz, QMUL seminar, 2006]

• What happens if we break ergodicity – and when might we ? [e.g. Froemberg, EXEV14]

• Why are some processes semi-martingales [*], & others not? [Weron, EXEV14]

• [* and what is a semi-martingale ?]

At least two ways to “break” classical diffusion

• One is explicitly non-Markovian,

via generalised Langevin

equation … (fractional motions)

• … Another is semi-martingale route.

Factorising CTRW is of this type,

modifies Brownian diffusion

by using subordination in time

(fractional kinetics).

[Weron and Magdziarz, 2008]

Non-Markovian route: Fractional Motions (e.g. fractional Brownian motion)• Non-Markovian, and not a semi-martingale

• Keeps a specified (stable) pdf,

• & a (fractional) Langevin equation [e.g. Lutz, 2001;Kupferman, 2004],

• but sacrifice much of the intuition built up about diffusion equations -except as a formal solution that gives the pdf of fBm/LFSM-as process no longer has the semi-martingale property.

• Might seem less intuitive of the 2 routes, BUT to understand what we can say about fractional motions physically, first look back at the derivation of the Langevin equation …

Where does Langevin equation come from ?

• Fundamentally LE is the equation of a preferred degree of freedom (“system”) interacting with a reservoir made of a set of harmonic oscillators, …

• … which are usually taken to be a thermalised heat bath-requires us to impose conditions on the oscillators.

• We are most used to the “ohmic” (linearised resistance) limit of LE:

• But what’s the physical picture behind LE?

( ) ( )Mq V q fq t

Damped harmonic oscillator [Yurke, 1984]:

• As oscillator moves up and

down it launches waves along string. These

carry away oscillator energy & motion damped.

• Waves propagating along string towards

oscillator will deposit energy and excite it.

• If we give wave modes a thermal spectrum

oscillator is then connected to a heat bath

and the Langevin equation results

Lumped:

Microscopic model of Brownian motion …

• System interacting with oscillators, often known as Caldeira-Leggett model. Used in study of decoherence in QM.

• Combined Lagrangian for system, interaction, reservoir (and a counter term):

• System

• See e.g. Caldeira, 2010 from where following slides are taken, and also Paul & Baschagel, 2013, section 3.3.

I R TS CL L L L L

2 ( )1

2S MqL V q

… Caldeira-Leggett model

• Interaction

• Reservoir

• Counter term

I k k kqqL C

2 2 21 1

2 2R k k k k k km qL m q

22

2

1

2k

C k

k k

T

CqL

m

Solve via Euler-Lagrange equations …

• Force on system:

• Force on k-th oscillator of the reservoir:

2

2( ) k

k k k k

k k

CM Cq q

mV q q

2

k k k k k kq qm m q C

… and Laplace transforms

2 2

2 2 2

2

2 2 2 2

2( )

( )

q whic

1 ( )

2

h be

(0) (0)1

co e

2

m s

istk k

F kP D

istk

ik k k k

kk k

k k k k

ik k k

q sF

C s q se

q

Mq V q

F F V q C e dsi s s

CC q

m

dsm i s

Fluctuating force term

Dissipation term

The dissipation term and the spectral function

2

2 2 2

2

2 0

0 0

1 ( )

2

cos ( )

2 ( )cos ( )

istk

Di

k k k k

tk

kk k k

t

Cd sq sF e ds

dt m i s

Cdt t q t dt

dt m

d Jd t t q t dt

dt

Above we went from sum over oscillators to an integral by defining a “spectral function” J. Note J not spectral density, which is effectively J(omega)/omega):

2

( )2

kk

k k k

CJ

m

Simplify dissipation term: Ohmic ansatz for J(ω)

2

2 0

2cos cos 2 ( )k

kk k k

Ct t d t t t t

m

Choose a form for the bath’s spectral function and then take the limit of large cutoff frequency . Brick wall cutoff not necessary, can also use a smooth exponential cutoff as discussed in Watkins and Waxman [2004], and Caldeira & Leggett papers cited therein.

( ) if

and 0 if

J

022 ( ) ( ') ' ( ) (0)

t

DF t t q td

qdt t qdt

Linearity of damping in velocity is reason for name “Ohmic” c.f. Ohm’s law.

Simplify fluctuating force using equipartition

Assume each oscillator initially in equilibrium about

2

(0)(0) and (0) (0) (0) 0k

k k k k

k k

C qq q q q

m

(0) (0) Bk k kk

k

k Tq q

m

(0) (0) Bk k kk

k

k Tq q

m

2(0( )0) /k k k kq C q m

Can use these, and our expression forfluctuating force to show that :

(0) (0) (0)k k kq q q

( ) 0 and

( ) ( ') 2 ( ') 0B

f t

f t t k t tf T

Essentially a version of the FDT

Ohmic Langevin equation

• Note used hypothesis that environmental oscillators are in equilibrium to get rid of spurious term

( ) ( )q tV fMq q

2 ( ) (0)t q

Beyond the Ohmic caseMore generally we can consider other types of spectral function including but not limited to power laws :

( )

where s 1 is super-Ohmic

and s 1 is sub-Ohmic

sJ

And in the presence of a memory in the heat bath we have the generalised Langevin equation of the form:

.. .

0'(q) ( ) ( ) ( )

t

q V M dt t t t f tM q Where memory kernel replaces constant eta [e.g. Kupferman, 2004 and Caldeira, 2010]

Fractional Langevin equation

(1 2 )If memory kernel has slowest decay ( ) ~ d

.. .

0(( ) ( )then GLE: M ' ) )(q

t

M dt t t f tV tq q ..

(1 )

0

2

(1 2 ) 2becomes FLE: M (q)

( ) 1where frac. derivative is

(

( )( )(

)

)

(

F

)d

d d

t

fM

t

q t

d F

ttq

tt

V

Fractional Brownian motion

• Instead of being defined purely as a stochastic process, the development in terms of an FLE allows some physical insight into meaning of fractional Brownian motion.

• It is the noise term in the FLE we have just described [Kupferman, 2004; Lutz, 2001]

• If we want to allow non-Gaussian heavy tailed jumps we can replace Gaussian steps in fBm by stable ones, to get linear fractional stable motion (see e.g. refs in Watkins, 2013).

At least two ways to “break” classical diffusion

• One is explicitly non-Markovian,

via generalised Langevin

equation … (fractional motions)

• … Another is semi-martingale route.

Factorising CTRW is of this type,

modifies Brownian diffusion

by using subordination in time

(fractional kinetics).

[Weron and Magdziarz, 2008]

Semi-Martingale: Fractional Kinetics (FFCTRW)

• Keep a specified pdf (though no longer a stable one), and a (fractional) diffusion equation [e.g. Klafter and Sokolov, 2011; Brockmann et al, 2006].

• Lose the ergodicity---the reservoir is explicitly nonequilibrium.• Rather than one Langevin equation now have two coupled ones [e.g

Fogedby, 1994], -> CTRW. • Need to have a second LE because this is BM subordinated to fractal

time.• Can retain a lot of stochastic calculus methods & a fractional Taylor

expansion for the pdf.• Is above true for any CTRW, or just the factorising CTRW ?• Physical picture: “flights in sticky space and trapping time”.

10/07/2014

One motivation for CTRW is dynamics ofHamiltonian chaos where the environment isnot “just” hierarchical, like the power law bath spectral density of the FLE, but also spatially structured in the KAM sense:

“Chaotic dynamics can be considered as a physical phenomenon that bridges the regular evolution of systems with the random [case] … What kind of kinetics should [there] be for chaotic dynamics that is intermediate between completely regular (integrable) and completely random (noisy) cases ? … These are the subjects of this paper, where the new concept of fractional kinetics is reviewed for systems with Hamiltonian chaos.” – Zaslavsky, Physics Reports, 2002

Continuous Time Random Walk (CTRW)-simulated discretely

10 July 2014 42

1

( ) i

n

i

X t

Impose iid random jumps

/( )x

Impose iid random times

1i

n

ni

t

Scale factors included

Notation as Fulger et al, PRE (2008)

Factorising CTRW

10 July 2014 43

( , ) ( ) ( )i P

Can in principle study CTRW where the pdfs are

coupled, in practise a factorising ansatz is often made

for the pdf: again use Fulger et al’s notation

CTRW = renewal reward process

10 July 2014 44

Here jumps at {J}

become

rewards {W} and

waiting times

become holding

times {S}

CTRW useful as time series model provided one can define & measure events

at arbitrary times ?

Fixed sampling intervals t motivate different class e.g. fractional motions

Symptoms of complex transport: 2

10 July 2014 45... longer waiting times

10 July 2014 46

Fractional time process

10 July 2014 47

Jumps still Gaussian but

waiting times now come

from a heavy-tailed,

distribution.

Here use Mittag-Lefler pdf

with parameter β. When β=1

it becomes exponential.

taking β < 1 fattens the tail.

Mittag-Leffler waiting time ccdf, from Fulger et al, PRE (2008)

Reality is ambivalent

• Frequently one sees both heavy tailed jumps and waiting times-or at least non-Gaussian ones ... Brockmann et al (2006) coined apt phrase “ambivalence” for this property.

• Various models proposed. One is simply to have a decoupled CTRW with heavy tails in both waiting time and jump size-known as the Fully Fractional Continuous Time Random Walk (FFCTRW).

10 July 2014 48

FFCTRW traces

10 July 2014 49

0 0.5 1 1.5 2 2.5

x 108

-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

4

Time

x,y

com

ponents

Time series

x

y

α=1.5, β=0.7

-14000 -12000 -10000 -8000 -6000 -4000 -2000 0 2000 4000-3.5

-3

-2.5

-2

-1.5

-1

-0.5

0

0.5x 10

4 Spatial pattern

x

y

time series

space

Dollar bills [Brockmann, 2006]

10 July 2014 50

Data fitted to an FFCTRW with α, β both about 0.6

10/07/2014

Zaslavsky, Physics Reports, 2002 (and book, Chapters 14 & 16) gave

argument for how a fractional Fokker-Planck equation could be obtained for

ambivalent processes. First consider standard F-P route:

3 3 1 1 2 3 3 2 2 2 2 1 1( ; ; ; ) ( ; ; ; ) ( ; ; ; )W x t x t dx W x t x t W x t x t

( , ; ', ') ( , '; - ')W x t x t W x x t t

00 0

( , ; )( , ; ) ( , ; )

W x x tW x x t t W x x t t

t

00 0

0

( , ; )1lim { ( , ; ) ( , ; )}t

W x x tW x x t t W x x t

t t

0( , ) ( , ; )P x t W x x t

10/07/2014

Zaslavsky, op. cit. … then makes fractional modification:

00 0

0

( , ; )1 ( , )lim { ( , ; ) ( , ; )}

| |t

W x x t P x tW x x t t W x x t

t t t

0

( , ) 1lim { [ ( , ; ) ( )] ( , )}

( )t

P x tdy W x y t t x y P y t

t t

( )| |

PP

t x

10/07/2014

From supp. Info. of [Brockmann et al, 2006] comes very useful schematic-

NB they defined (,) opposite way to Zaslavsky 2002

Actually not fBm,but rather in fact the fractional time process (FTP).

Don’t believe everything youread in Nature ;-)

CTRW vs fixed t models

10 July 2014 54

Table from Watkins et al, PRE, 2009. Here2 12 1[ ] 2

HH

tt Ht

Why name “FF” CTRW

10 July 2014 55

FF CTRW is bottom right example, α, β are the

orders of fractional derivatives--- β= αH

Markovian: Nonlinear FP

• Markovian or at least local in time: Can no longer have stable pdf solution because D is changing in space or time. Keep duality of diffusion equation and Langevin - > nonlinear Fokker-Planck equations. Are these in semi-martingale class ?

• Examples:

Wheatcraft and Tyler, 1988; Bassler et al, 2006

Hnat et al, 2003

2

2( , ) ( ) ( , )P x t D t P x t

t x

2

2( , ) (x) ( , )P x t D P x t

t x

Levy walk: couples space & time

10 July 2014 57

Gives a finite velocity by introducing a jump duration τ’ & coupling the jump size to

it – idea known as Lévy walk [Shlesinger & Klafter, PRL (1985)]. Obviously can

be done in many ways, simplest is just to make size proportional to duration.

( , ') ( ' | ) ( )

(| | ') ( )

Lévy walk

( , ) ( ) ( ) Uncoupled CTRW

In above τ’ means flight duration in Levy walk, and τau waiting time in CTRW.

Deliberately changed notation for these quite different quantities. In

foraging & other literature, walks & flights, durations & waiting times often

treated in a very cavalier fashion – NOT SAFE TO DO SO !!!

Turbulent cascade processes

Many systems haveaggregation, but not by an additive route. Classic example is turbulence.

SOC

Data-inspired question:

• How much of any given complex system’s time series properties can be predicted from just a few parameters e.g. from its pdf and power spectral exponents ? Examples include wind power paper, solar wind, AE etc.

• One motivation for why this might be possible is the effective collapse onto a few degrees of freedom used not only in LD chaos, but also in the synergetics idea.

• In particular, how much does this hopefully simple parameterisation then govern the extremes ?

• Begs a “physics” question: are these parameters actually physical in origin ? Or more to do with the aggregation and measurement processes ?

Sources Used I:

• Bassler, K., et al, “Markov processes, Hurst exponents, and nonlinear diffusion equations”, Physica A 369, 343, 2006 [and subsequent exchanges of comment(s) with Frank].

• Bouchaud, J.-P., and M. Potters, “Theory of financial risk and derivative pricing”, 2nd edition,CUP, 2003.

• Brockmann, D., et al, “The scaling laws of human travel”, Nature, 439, 462, 2006.

• Caldeira, A. O., “Caldeira-Leggett model”, Scholarpedia, 2010.

• Fogedby, H. C., “Langevin equations for continuous time Levy flights”, PRE, 50, 1657, 1994.

• Fulger, D., et al, “Monte Carlo simulation of uncoupled continuous-time random walks yielding a stochastic solution of the space-time fractional diffusion equation”, PRE, 77, 021122, 2008

• Hnat, B., et al, “Intermittency, scaling, and the Fokker-Planck approach to fluctuations of the solar wind bulk plasma parameters as seen by the WIND spacecraft”, PRE, 67, 056404, 2003

• Lemons, D., “An introduction to stochastic processes in physics”, Johns Hopkins, 2002.

• Mantegna, R., and H. E. Stanley, ”Introduction to Econophysics”, CUP, 2000.

• Kupferman, R., “Fractional kinetics in Kac-Zwanzig heat bath models”, J. Stat. Phys., 114, 291, 2004.

• Lutz, E., “Fractional Langevin equation”, Phys. Rev. E, 64, 051106, 2001.

Sources Used II:

• Paul, W., and J. Baschnagel, “Stochastic Processes from Physics to Finance, 2nd Edition”, Springer , 2013.

• Klafter, J., and I. M. Sokolov, “First Steps in Random Walks”, OUP, 2011.

• Shlesinger, M. F. and J. Klafter, Phys. Rev. Lett., 54, 2551, 1985.

• Sornette, D., “Critical phenomena in natural sciences”, 2nd edition, Springer, 2004.

• Watkins, N. W. and Waxman, D., “Path integral derivation of Bloch-Redfield equations for a qubit weakly coupled to a heat bath: application to nonadiabatic transitions”, arXiv:cond-mat/0411443v1, 2004.

• Watkins, N. W., et al, “Kinetic equation of linear fractional stable motion and applications to modelling the scaling of intermittent bursts”, PRE, 79, 041124, 2009.

• Watkins, N. W., “Bunched black (and grouped grey) swans: dissipative and non-dissipative models of correlated extreme fluctuations in complex systems”, GRL , 40, 1-9, doi:10.1002/GRL.50103, 2013.

• Weron, A., and M. Magdziarz, “Anomalous diffusion and semi-martingales”, EPL, 86, 60010, 2009

• Wheatcraft, S. W. and S. W. Tyler, “An Explanation of Scale-Dependent Dispersivity in Heterogeneous Aquifers Using Concepts of Fractal Geometry”, Water Resour. Res., 24, 566, 1988

• Yurke, B., “Conservative model for the damped harmonic oscillator”, Am. J. Phys., 52(12), 1099, 1984

• Zaslavsky, G., “Chaos, fractional kinetics, and anomalous transport” , Physics Reports, 371, 461, 2002 (and his book, OUP, 2005)