Lévy Flights and

Fractional Kinetic Equations Lévy Flights

and Fractional Kinetic Equations

A. ChechkinInstitute for Theoretical Physics of National Science Center «Kharkov Institute

of Physics and Technology», Kharkov, UKRAINE

IEP Kosice, 9 May 2007 IEP Kosice, 9 May 2007

In collaboration with:

In collaboration with:

V. Gonchar, L. Tanatarov (ITP Kharkov) J. Klafter (TAU) I. Sokolov (HU Berlin)

R. Metzler (University of Ottawa) A. Sliusarenko (Kharkov University)T.Koren (TAU)

A. Chechkin, V. Gonchar, J. Klafter and R. Metzler, Fundamentals of Lévy Flight Processes. Adv. Chem. Phys. 133B, 439 (2006).

LÉVY motion or “LÉVY flights” LÉVY motion or “LÉVY flights” (B. Mandelbrot): paradigm of non-(B. Mandelbrot): paradigm of non-

brownian random motionbrownian random motion

Brownian Motion Lévy Motion

1. Central Limit Theorem 1. Generalized Central

Limit Theorem

2. Properties of Gaussian probability laws and processes

2. Properties of Lévy

stable laws and processes

Mathematical Foundations:

Paul Pierre Lévy (1886-1971)

Lévy stable distributions in Mathematical Monographs:

Lévy (1925, 1937)Хинчин (1938)Gnedenko & Kolmogorov (1949-53)Feller (1966-71)Lukacs (1960-70)Zolotarev (1983-86, 1997)Janiki & Weron (1994)Samorodnitski & Taqqu (1994)

B.V. Gnedenko, A.N. Kolmogorov, “Limit B.V. Gnedenko, A.N. Kolmogorov, “Limit Distributions for Sums of Independent Distributions for Sums of Independent

Random Variables“ (1949)Random Variables“ (1949)

The prophecy: “All these distribution laws, called stable, deserve the most serious attention. It is probable that the scope of applied problems in which they play an essential role will become in due course rather wide.”

The guidance for those who write books on statistical physics: “… “normal” convergence to abnormal (that is, different from Gaussian –A. Ch.) stable laws … , undoubtedly, has to be considered in every large textbook, which intends to equip well enough the scientist in the field of statistical physics.”

First remarkable propertyof stable distributions

1 21

. . .n d

n i ni

X X X X c X

2 2X X t

1 1

1

, 1/ 1/ 2n

ii

X X n t

1. Stability: distribution of sum of independent identically distributed stable random

variables = distribution of each variable (up to scaling factor)

ccnn = = nn1/a1/a , 0 < , 0 < 2, 2, is is Lèvy indexLèvy index

Gauss: Gauss: = 2, = 2, ccnn = = nn1/21/2

CorollaryCorollary 2. 2. Normal diffusion law Normal diffusion law, , Brownian motionBrownian motion1 2 1 2

1

n

ii

X X n t

or the rule of summingor the rule of summing variancesvariances

CorollaryCorollary 3. 3. SuperdiffusionSuperdiffusion, , LLéévyvy

CorollaryCorollary 1. 1. Statistical self-similarityStatistical self-similarity, , random fractalrandom fractal

MandelbrotMandelbrot: When does the whole look like its parts : When does the whole look like its parts description of random description of random fractal ptocessesfractal ptocesses

QuestionQuestion: : XXii are not stable r.v. are not stable r.v.

secondsecond remarkableremarkable propertypropertyof stableof stable distributionsdistributions

2 2 ( )x x f x dx

Property 2. Generalized Central Limit Theorem: stable distributions are the

limit ones for distributions of sums of random variables (have domain of

attraction) appear, when evolution of the system or the result of experi-

ment is determined by the sum of a large number of random quantities

Gauss: attracts Gauss: attracts f(x)f(x) with with finitefinite variancevariance

LLéévy: attract vy: attract f(x)f(x) with with infiniteinfinite variancevariance

2 2 ( )x x f x dx

2 , 0 2x

2 2( ,1, )

Dp x D

D x

2 2( ,1, )

Dp x D

D x

21( ,2, ) exp

44

xp x D

DD

21( ,2, ) exp

44

xp x D

DD

10 2

1( ; , ) exp( ) exp | | , 0 2, 0 ( , , )ˆ

| |X X

xp k D ikX D k D p x D

x

10 2

1( ; , ) exp( ) exp | | , 0 2, 0 ( , , )ˆ

| |X X

xp k D ikX D k D p x D

x

third remarkable propertythird remarkable propertyof stable distributionsof stable distributions

Property 3. Power law asymptotics x (-1-) (“heavy tails”)

Particular casesParticular cases

1. 1. Gauss, Gauss, = 2 = 2

All moments are finiteAll moments are finite

2. Cauchy, 2. Cauchy, = 1 = 1

, 0q

x q

Moments of order q < 1 are finiteMoments of order q < 1 are finite

CorollaryCorollary:: descriptiondescription of highly non-equilibrium processesof highly non-equilibrium processes possessing large bursts and/or outlierspossessing large bursts and/or outliers

_1_1

Due to the three remarkable properties of stable distributions it is now believed that the Léévy statistics provides a framework for the description of many natural phenomena in physical, chemical, biological, economical, … systems from a general common point of view

Electric Field Distribution in an Ionized Electric Field Distribution in an Ionized Gas (Holtzmark 1919)Gas (Holtzmark 1919)

Application:Application: spectral lines broadening in plasmas spectral lines broadening in plasmas

3i

ii

erE

r

3i

ii

erE

r

Nn const

V

Nn const

V

1 2 ... nE E E E

1 2 ... nE E E E

3 dimensional

symmetric

stable

distribution

32

32

ikEdkW E W k e

3

2ikEdk

W E W k e

3

2a kW k e

32a k

W k e

52

1~W E

E

5

2

1~W E

E

Related examplesRelated examples

gravity field of stars (=3/2)

electric fields of dipoles (= 1) and quadrupoles (= 3/4)

velocity field of point vortices in fully developed turbulence (=3/2) ……

(to name only a few)

also: asymmetric Levy stable distributions in the first passage theory, single molecule line shape cumulants in glasses …

2 10 dim

(1900), (1905)

Normal diffusion R R c Dt t

Bachelier Einstein

2 1

0 dim

(1900), (1905)

Normal diffusion R R c Dt t

Bachelier Einstein

20 , 1

Anomalous diffusion

R R t 20 , 1

Anomalous diffusion

R R t

1

Subdiffusion

1

Subdiffusion

Normal vs Anomalous DiffusionNormal vs Anomalous Diffusion

turbulent plasmas

transport on fractals

contamimants in

underground water

amorphous solids

convective patterns

polymeric systems

deterministic maps

1000nGauss

3000nCauchy

( , , )

turbulent media

gases fluids plasmas

Hamiltonian chaotic

systems

deterministic maps

foraging movement

financial markets

1

Superdiffusion

1

Superdiffusion

Self-Similar Structure of the Brownian and Self-Similar Structure of the Brownian and the Lévy Motionsthe Lévy Motions

2( )

Normal diffusion

X t tµ2( )

Normal diffusion

X t tµ 2( ) ,

1.66 1

Levy Superdiffusion

X t tm

m

µ

= >

2( ) ,

1.66 1

Levy Superdiffusion

X t tm

m

µ

= >

xx100100

xx100100 xx100100

xx100100

““If, veritably, one took the position from If, veritably, one took the position from second to second, each of these rectilinear second to second, each of these rectilinear segments would be replaced by a polygonal segments would be replaced by a polygonal contour of 30 edges, contour of 30 edges, each being as complicatedeach being as complicated as the reproduced designas the reproduced design, and so forth.” , and so forth.” J.B. PerrenJ.B. Perren

““The stochastic process which we call linear The stochastic process which we call linear Brownian motion is a schematic representa-Brownian motion is a schematic representa-tion which describes well the properties of real tion which describes well the properties of real Brownian motion, observable on a small Brownian motion, observable on a small enough, but not infinitely small scale, and enough, but not infinitely small scale, and which assumes that the which assumes that the same properties existsame properties exist on whatever scaleon whatever scale.” .” P.P. LP.P. Léévyvy

NORMAL vs ANOMALOUS DIFFUSIONNORMAL vs ANOMALOUS DIFFUSION

Brownian motion of a small grain of Brownian motion of a small grain of puttyputty

Foraging movement of spider Foraging movement of spider monkeysmonkeys

(Jean Baptiste Perrin: 1909) Gabriel Ramos-Fernandes et al. (2004)

2( )R t t2( )R t t 2 1,7( )R t t

2 1,7( )R t t

Classical example of superdiffusionClassical example of superdiffusion

RichardsonRichardson: diffusion of passive admixture in locally isotropic turbulence (1927): diffusion of passive admixture in locally isotropic turbulence (1927)

2 3( )l 2 3( )l

a)a) Frenkiel and Katz (1956), Frenkiel and Katz (1956), b)b) Seneca (1955),Seneca (1955),c)c) Kellogg (1956)Kellogg (1956)

2( )l versus versus (Gifford, (Gifford,

1957)1957)

MoninMonin: space fractional diffusion equation (1955, 1956): space fractional diffusion equation (1955, 1956)

1/ 3

2 2 21/ 3

2 2 2

( , ) ( ) ( , )

,

p l D p l

D cx y z

1/ 3

2 2 21/ 3

2 2 2

( , ) ( ) ( , )

,

p l D p l

D cx y z

1/ 3( ) linear integral operator with linear integral operator with complicated kernel (Monin)complicated kernel (Monin)

Diffusion of tracers in fluid flows.Diffusion of tracers in fluid flows.Large scale structures (eddies, jets or convection rolls) dominate the transport.Large scale structures (eddies, jets or convection rolls) dominate the transport.

Example. Experiments Example. Experiments in a rapidly rotating in a rapidly rotating annulus (Swinney et al.).annulus (Swinney et al.).

Ordered flow:Ordered flow:Levy diffusionLevy diffusion

(flights and traps)(flights and traps)

Weakly turbulent Weakly turbulent flow:flow:

Gaussian diffusionGaussian diffusion

Anomalous Diffusion inAnomalous Diffusion in Channeling Channeling

Fig.1. NORMAL DIFFUSION: Fig.1. NORMAL DIFFUSION: randomly distributed atom randomly distributed atom stringsstrings

Fig.2. SUPERDIFFUSION: positively and negatively Fig.2. SUPERDIFFUSION: positively and negatively charged particles in a periodic field of atom strings along charged particles in a periodic field of atom strings along the axis <100> in a silicon crystal.the axis <100> in a silicon crystal.

Anomalous diffusion in Lorentz gases with infinite «horizon»(Bouchaud and Le Doussal; Zacherl et al.)(Bouchaud and Le Doussal; Zacherl et al.)

Polymer adsorption and self - avoiding Levy flights(de Gennes; Bouchaud)(de Gennes; Bouchaud)

Anomalous diffusion in a linear array of convection rolls(Pomeau et al.; Cardoso and Tabeling)(Pomeau et al.; Cardoso and Tabeling)

Distribution of the length the pathsDistribution of the length the paths

The typical displacementThe typical displacement

3

1( ) ,

Rp l l

l

3

1( ) ,

Rp l l

l

X X t t 2 1 2 1 2 1 2/ / /(ln )X X t t 2 1 2 1 2 1 2/ / /(ln )

Distribution of the size of the loops decays as a power lawDistribution of the size of the loops decays as a power law the projection of the chain’s conformation on the wall: the projection of the chain’s conformation on the wall: self - self - avoiding Levy flightavoiding Levy flight (two adsorbed monomers cannot occupy the (two adsorbed monomers cannot occupy the same site)same site) at the adsorption thresholdat the adsorption threshold p l

l( ) ,

11

1 p ll

( ) , 1

11

each roll acts as a trap with a releaseeach roll acts as a trap with a releasetime distribution decaying as time distribution decaying as

( ) 1

1

( ) 1

1

X t 1 3/X t 1 3/

Impulsive Noises Modeled with the Lévy Impulsive Noises Modeled with the Lévy Stable DistributionsStable Distributions

Naturally serve for the description of the processesNaturally serve for the description of the processeswith large outliers, far from equilibriumwith large outliers, far from equilibrium

Examples include :

economic stock prices and current exchange rates (1963) radio frequency electromagnetic noise underwater acoustic noise noise in telephone networks biomedical signals stochastic climate dynamics turbulence in the edge plasmas of thermonuclear devices (Uragan 2M, ADITYA, 2003; Heliotron J, 2005 ...)

Examples include :

economic stock prices and current exchange rates (1963) radio frequency electromagnetic noise underwater acoustic noise noise in telephone networks biomedical signals stochastic climate dynamics turbulence in the edge plasmas of thermonuclear devices (Uragan 2M, ADITYA, 2003; Heliotron J, 2005 ...)

LLéévy noises and Lvy noises and Léévy motionvy motion

LÉVY NOISES

Lévy index outliers

LÉVY MOTION: successive additions of the noise values

Lévy index “flights” become longer

EXPERIMENTAL OBSERVATION OF LEXPERIMENTAL OBSERVATION OF LÉÉVY VY FLIGHTS IN WIND VELOCITYFLIGHTS IN WIND VELOCITY((Lammefjord, denmark, Lammefjord, denmark, 2006)2006)

Measuring masts Velocity increments Distribution of increments

250

min

250

min

25 m

in25

min

2.5

min

2.5

min

Lévy Noise vs Gaussian NoiseLévy Noise vs Gaussian Noise

Lévy turbulence in „Uragan 3M“Lévy turbulence in „Uragan 3M“

DEM/USD exchange rateDEM/USD exchange rate

V.Gonchar V.Gonchar et.al., 2003et.al., 2003

V.Gonchar, A. V.Gonchar, A. Chechkin, Chechkin,

20012001

««Lévy TurbulenceLévy Turbulence» » in boundaryin boundary plasmaplasma of stellarator of stellarator « «UraganUragan 3М» ( 3М» (IPP KIPTIPP KIPT))

Helical magnetic coilsHelical magnetic coils((from abovefrom above))

Poincare section of magnetic linePoincare section of magnetic line, ЛЗ – , ЛЗ – Langmuir proveLangmuir prove

eV100eV,500)0(,m10

ms,60kW,200

0.3,π2/)(T,7.0

m1.0,m1,9,3

ie3-18

e

RF

φ

0

TTn

P

aB

aRml

Ion saturation currentIon saturation current

at different probe positionsat different probe positions::

аа) ) RR = = 111 с111 сmm ; ; бб) ) RR = 112 с = 112 сmm ; ;

вв) ) RR = 112.5 с = 112.5 сmm ; ; гг) ) RR = 113 с = 113 сmm..

««Lévy TurbulenceLévy Turbulence» » in boundaryin boundary plasmaplasma of of stellarator stellarator « «UraganUragan 3М» ( 3М» (IPP KIPTIPP KIPT))

1. 1. Kurtosis and “chi-square” criteriumKurtosis and “chi-square” criterium: : evidence of non-Gaussianityevidence of non-Gaussianity..

2. 2. Modified method of percentilesModified method of percentiles: : LLéévy index of turbulent fluctuationsvy index of turbulent fluctuations

Ion saturation currentIon saturation current nn Floating potentialFloating potential

Fluctuations of density and potential measured in boundary plasma of stellarator Fluctuations of density and potential measured in boundary plasma of stellarator “Uragan 3M” obey L“Uragan 3M” obey Léévy statisticsvy statistics ( (SeptemberSeptember 2002) 2002)

similar conclusions about the Lsimilar conclusions about the Léévy statistics of plasma fluctuations have been vy statistics of plasma fluctuations have been drawn fordrawn for

ADITYAADITYA ( (MarchMarch 2003) 2003) LL-2-2MM ( ( October October 2003)2003) LHDLHD, , TJTJ--IIII ( (DecemberDecember 2003) 2003) Heliotron J (March 2004)Heliotron J (March 2004)

““bursty” character of fluctuations in other devices bursty” character of fluctuations in other devices ( (DIIIDIII--DD, , TCABRTCABR, …), …)

““Levy turbulence”Levy turbulence” is a widely spread phenomenon is a widely spread phenomenon

Models are strongly needed !Models are strongly needed !

_2_2

problem:problem:

KINETIC THEORY

FOR NON-GAUSSIAN LÉVY WORLD ???

KINETIC EQUATIONS WITH FRACTIONAL DERIVATIVES …

Fractional Kinetics Fractional Kinetics “Strange” Kinetics “Strange” Kinetics

M.F. Schlesinger, G.M. Zaslavsky, J. Klafter, M.F. Schlesinger, G.M. Zaslavsky, J. Klafter, NatureNature (1993)) : connected with deviations (1993)) : connected with deviations

“Normal” Kinetics Fractional Kinetics

Diffusion law

Relaxation Exponential Non-exponential

“Lévy flights in time”

Stationarity Maxwell-Boltzmann

equilibrium

Confined Lévy flights

as non-Boltzmann

stationarity

2( )x t t2( )x t t

2( ) , 1x t t 2( ) , 1x t t

2 ( ) ln , 0x t t 2 ( ) ln , 0x t t

Types of “Fractionalisation”Types of “Fractionalisation”

time – fractionaltime – fractional

space / velocity – fractionalspace / velocity – fractional

multi-dimensional / anisotropicmulti-dimensional / anisotropic time-space / velocity fractional time-space / velocity fractional

Caputo/Riemann-LiouvilleCaputo/Riemann-Liouville derivative derivative

RieszRiesz derivative derivative

Question Question :: Derivation ? Derivation ?Answer:Answer:Generalized Master Equation, Generalized Master Equation, Generalized Langevin Equation Generalized Langevin Equation

, 0 1t t

, 0 1

t t

2

2, 0 2

x x

2

2, 0 2

x x

2

2v v

2

2v v

BUT:BUT:OUT OF THE SCOPEOUT OF THE SCOPE

OF THE TALK !OF THE TALK !

2

2f dU f

f Dt x dx x

2

2f dU f

f Dt x dx x

FOKKER-FOKKER-PLANCK EQPLANCK EQ

x /f dU ff D

t x dx x

f dU ff D

t x dx x

)(tYdx

dU

dt

dx )(tY

dx

dU

dt

dx

:Riesz fractional derivative:

integrodifferential operator

= 2 : Fokker - Planck equation (FPE)

0 < < 2 : Fractional FPE (FFPE)

Kinetic equations for the stochastic systems Kinetic equations for the stochastic systems driven by white Lévy noisedriven by white Lévy noise

(assumptions: overdamped case, or strong friction limit, 1-dim, D = intensity of the noise = const: no Ito-Stratonovich dilemma!)

Langevin description, x(t)

U(x) : potential energy, Y(t) : white noise, : the Lévy index = 2 : white Gaussian noise0 < < 2 : white Lévy noise

Kinetic description, f(x,t)

( 1)/22

( 1)ˆexp( ) ( ) cos 1 arctan2

1

d dk ikx k k xd x x

( 1)/22

( 1)ˆexp( ) ( ) cos 1 arctan2

1

d dk ikx k k xd x x

)exp()(ˆ,1

1)(

2kk

xx

)exp()(ˆ,

1

1)(

2kk

xx

ˆ ˆ( ) ( )FT FTd d

k k ik kd x dx

ˆ ˆ( ) ( )FT FTd d

k k ik kd x dx

2 2

22 2

ˆ( )2

ikxd dk dk k e

dxd x

2 2

22 2

ˆ( )2

ikxd dk dk k e

dxd x

)(ˆ kkxd

d FT

)(ˆ kk

xd

d FT

)()()(ˆ xxedxkFT

ikx

)()()(ˆ xxedxkFT

ikx

Definition of the Riesz fractional deriva-tive Definition of the Riesz fractional deriva-tive via its Fourier representationvia its Fourier representation

Indeed, .But:

Example:Example:

tkDtkf exp),(ˆ tkDtkf exp),(ˆ

ˆ ˆ ˆ, ( ,0) 1f D k f f kt

ˆ ˆ ˆ, ( ,0) 1f D k f f kt

, , ( ,0) ( )f f

D D const f x xt x

, , ( ,0) ( )

f fD D const f x x

t x

Space-fractional diffusion equation,Space-fractional diffusion equation, UU=0: =0: free free Lévy flightsLévy flights

After Fourier transform:

Solution in Fourier - space:characteristic functionof free Lévy flights

Characteristic diffusion displacementCharacteristic diffusion displacement 1/x Dt

1/

x Dt

superdiffusion: superdiffusion: faster than faster than tt1/21/2

( 0 ( 0 < < < 2 ) < 2 )

Free Lévy flights vs Brownian motion in Free Lévy flights vs Brownian motion in semi-infinite domainsemi-infinite domain

Brownian Motion, = 2 Lévy Motion, 0<<2

First passage time PDF

Method of images

2 / 4 3/ 23

( )4

a Dtap t e t

Dt

2 / 4 3/ 2

3( )

4

a Dtap t e t

Dt

3/ 2( )p t t3/ 2( )p t t

( , ) ( , ) ( , )imf x t W x a t W x a t ( , ) ( , ) ( , )imf x t W x a t W x a t 1 1/

3/ 2

( )imp t t

t

1 1/

3/ 2

( )imp t t

t

Sparre Andersen universalitySparre Andersen universality

Method of images breaks down for Method of images breaks down for Lévy flights (A. Chechkin et. al., 2004)Lévy flights (A. Chechkin et. al., 2004)

Free Lévy flights vs Brownian motion in Free Lévy flights vs Brownian motion in semi-infinite domain: Leapoversemi-infinite domain: Leapover

(T. Koren. A. Chechkin, Y. Klafter, 2007)(T. Koren. A. Chechkin, Y. Klafter, 2007)

Brownian Motion, = 2 Lévy Motion, 0<<2

Leapover: how large is the leap over the boundary by the Lévy stable process ?

No leapover 1 / 2( )a af 1 / 2( )a af

Leapover PDF decays slower than Leapover PDF decays slower than Lévy flight PDFLévy flight PDF

2 4 2 2( ) ; ; , 0, 0, 0,1,2,...

2 4 2 2

max bx bxU x a b m

m

2 4 2 2( ) ; ; , 0, 0, 0,1,2,...

2 4 2 2

max bx bxU x a b m

m

1)(,

)(exp)( xdxf

D

xUCxf

1)(,

)(exp)( xdxf

D

xUCxf

02

2

dx

fdDf

dx

dU

dx

d 02

2

dx

fdDf

dx

dU

dx

d

Reminder: Stationary solution of the FPE in Stationary solution of the FPE in external field, external field, = 2 = 2

Reminder: Stationary solution of the FPE in Stationary solution of the FPE in external field, external field, = 2 = 2

Equation for the stationary PDF:

Boltzmann’ solution:

Two properties of stationary PDF:1. Unimodality (one hump at the origin).2. Exponential decay at large values of x.

(1)

(2)

2 21

sin /2 ( )( ) , , ( )Cf x C x dx x f x

x

2 21

sin /2 ( )( ) , , ( )Cf x C x dx x f x

x

kkf exp)(ˆ

kkf exp)(ˆ)(ˆ)sgn(

ˆ 1 kfkkdk

fd )(ˆ)sgn(ˆ 1 kfkk

dk

fd

)(ˆ)( kfxfFT )(ˆ)( kfxf

FT

FTdk

d x

FTdk

d x

2( ) ,

2

xU x

2( ) ,

2

xU x

0d dU d f

fdx dx d x

0d dU d f

fdx dx d x

Lévy flights in a hLévy flights in a harmonic potential, 1armonic potential, 1 < 2 < 2 FFPE for the stationary PDF (dimensionless units):

(Remind: pass to the Fourier space )

Equation for the characteristic function:

: symmetric stable law

Two properties of stationary PDF:

2. Slowly decaying tails:

Harmonic force is not “strong” enough to “confine” Lévy flightsHarmonic force is not “strong” enough to “confine” Lévy flights

1. Unimodality (one hump at the origin).

)(ˆ)sgn(ˆ 13

3kfkk

dk

fd )(ˆ)sgn(ˆ 13

3kfkk

dk

fd

)(ˆ)( kfxfFT )(ˆ)( kfxf

FT

k

xd

d FT:reminder

k

xd

d FT:reminder

3 0d d f

x fdx d x

3 0d d f

x fdx d x

3 ( )dx

x Y tdt

3 ( )dx

x Y tdt

Confined Lévy flights. Quartic potential Confined Lévy flights. Quartic potential well, well, UU xx44

(A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L. Tanatarov, 2002)(A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L. Tanatarov, 2002)

Langevin equation: Fractional FPE for the stationary PDF:

Equation for the characteristic function:

+ normalization + symmetry + boundary conditions …

(1)

(2)

3

3

3ˆ 2ˆ ˆsgn( ) ( ) ( ) exp cos2 2 63

k kd f k f k f kdk

3

3

3ˆ 2ˆ ˆsgn( ) ( ) ( ) exp cos2 2 63

k kd f k f k f kdk

)1(

1)(

42 xxxf

)1(

1)(

42 xxxf

Confined Confined Lévy flights. Quartic potential, Lévy flights. Quartic potential, UU xx44. Cauchy case, . Cauchy case, = 1 = 1

Equation for the characteristic function:

PDF:

Two properties of stationary PDF

min 0x 1. Bimodality: local minimum at , two maxima at

2. Steep power law asymptotics with finite variance:

max 1/ 2x

4( )f x x

2 2 ( ) 1x dx x f x

2 2 ( ) 1x dx x f x

2xcc cr 2xcc cr

2xcc cr 2xcc cr

4crc 4crc

1sin 2C 1sin 2C

xx

Cxf

c,)(

1 x

x

Cxf

c,)(

1

2, cxU c 2, cxU c

2, cxU c 2, cxU c

is non-unimodal. Proved with the use of the “hypersingular” representation of the Riesz derivative.

Proposition 2.Proposition 2. Stationary PDF for the Lévy flights in external field

Two propositionsTwo propositions(A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L. Tanatarov, 2004)(A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L. Tanatarov, 2004)

Proposition 1Proposition 1.. Stationary PDF for the Lévy flights in external field

has power-law asymptotics,

is a “universal constant”, i.e., it does not depend on c.

Critical exponent ccr:

: “confined”

Proved with the use of the representation of the Riesz derivative in terms of left- and right Liouville-Weyl derivatives.

,...2,1,0,22 mxU m ,...2,1,0,22 mxU m)(tY

dx

dU

dt

dx )(tY

dx

dU

dt

dx

Numerical simulationNumerical simulation

Langevin equation with a white Lévy noise,

,

Illustration:Illustration: Typical sample paths

Potential wells Brownian motion Lévy motion

mm walls are steeper walls are steeper Lévy flights are shorter Lévy flights are shorter confined Lévy flightsconfined Lévy flights

Quartic potential Quartic potential UU xx44, , = 1.2. = 1.2. Time Time evolution of the PDFevolution of the PDF

Potential well Potential well UU xx5.55.5, , = 1.2. Trimodal = 1.2. Trimodal transient state in ttransient state in time evolution of the PDFime evolution of the PDF

Potential well Potential well UU xx5.55.5, , = 1.2. Trimodal = 1.2. Trimodal transient state in a finer scaletransient state in a finer scale

Confined Lévy flights in bistable potential: Confined Lévy flights in bistable potential: Lévy noise induced transitions Lévy noise induced transitions (stochastic climate dynamics) (stochastic climate dynamics)

(A. Sliusarenko et al., 2007)(A. Sliusarenko et al., 2007)

( )dx dU

Y tdt dx ( )

dx dUY t

dt dx 2 4/ 2 / 4U x x x 2 4/ 2 / 4U x x x

1exp , 1

4GaussT D DD

1exp , 1

4GaussT D DD

Fig.1. Escape of the trajectory overthe barrier, schematic view. Fig.2. Typical trajectories for different .

1 expp t t TT

1 expp t t TT

CT

D

C

TD

Confined Lévy flights in bistable potential: Confined Lévy flights in bistable potential: Kramers’ problemKramers’ problem

0.5 1.0 1.5 2.0

0.98

1.00

1.02

1.04

1.06

1.08

1.10

1.12

1.14

0.0 0.5 1.0 1.5 2.0

0.0

0.5

1.0

1.5

2.0

lg C

()

1.0 1.5 2.0 2.5 3.0 3.5

1

2

3

4

5

lg T

esc

-lg D

Completely unusual behavior of T at small D’s:

Escape probability p(t) as functionof walking time t:

0 500 1000 1500 2000-12

-11

-10

-9

-8

-7

-6

-5

ln p

(t)

t

A.V. Chechkin et al., 2007A.V. Chechkin et al., 2007

Damped Lévy flights in velocity space.Damped Lévy flights in velocity space. Damping by dissipative non-linearity Damping by dissipative non-linearity(posed by B. West, V. Seshadri (1982), Chechkin et al. 2005)(posed by B. West, V. Seshadri (1982), Chechkin et al. 2005)

( ) ( ) ( ) ( )Langevin equation V dt V V t dt L dt ( ) ( ) ( ) ( )Langevin equation V dt V V t dt L dt

FFPEFFPE ,P V t P

V V P Dt V V

,P V t PV V P D

t V V

0 = 1.0, 1 = 0.0001, 2 = 0 (curve 1)

0 = 1.0, 1 = 0, 2 = 0.000001 (curve 2), = 1.0, slopes = -1, -3, -5

0,)()( 42

210 nVVVV 0,)()( 4

22

10 nVVVV

Damped Lévy flights: Damped Lévy flights:

22ndnd Moments Moments 44thth Moments Moments

“ “PLT Lévy flights” : PDFs resembles Lévy stable distribution in the central partPLT Lévy flights” : PDFs resembles Lévy stable distribution in the central part

at greater scales the asymptotics decay in a power-law way, but faster, than the Lévy at greater scales the asymptotics decay in a power-law way, but faster, than the Lévy

stable ones, therefore, stable ones, therefore, the Central Limit Theorem is applied the Central Limit Theorem is applied

at large times the PDF tends to Gaussian, however, at large times the PDF tends to Gaussian, however, sometimes very slowlysometimes very slowly

2x

Power-law truncated Lévy flightsPower-law truncated Lévy flights(I. Sokolov, A. Chechkin, Y. Klafter, 2004)(I. Sokolov, A. Chechkin, Y. Klafter, 2004)

Particular case of Particular case of distributed order space fractional diffusion equationdistributed order space fractional diffusion equation:: 2 2

2 2( , ) ( , )

1| |

f x t f x tC D

tx x

2 2

2 2( , ) ( , )

1| |

f x t f x tC D

tx x

0 < 0 < < 2 < 2 2

22

0

ˆ( ) 2

k

fx t Dt

k

22

20

ˆ( ) 2

k

fx t Dt

k

5

(5 )sin / 2( , ) ,

DC tf x t x

x

5

(5 )sin / 2( , ) ,

DC tf x t x

x

The Lévy distribution is truncated by a power law with a power The Lévy distribution is truncated by a power law with a power between 3 and 5. Due to the finiteness of the second moment the between 3 and 5. Due to the finiteness of the second moment the PDF PDF f(x,t)f(x,t) slowly converges to a Gaussian slowly converges to a Gaussian

Probability of return to the origin: Probability of return to the origin: | | 1/(0, ) , 0 22

k tdkf t e t

| | 1/(0, ) , 0 22

k tdkf t e t

The slope –1/The slope –1/ in the double logarithmic scale changes into -1/2 for the Gaussian in the double logarithmic scale changes into -1/2 for the Gaussian

TheoryTheory Experiment on the ADITYA tokamakExperiment on the ADITYA tokamak

Experiment on the ADITYA: probabilities of return to the origin for different radial positions of the Experiment on the ADITYA: probabilities of return to the origin for different radial positions of the probes [R.Jha et al. Phys. Plasmas.2003.Vol.10.No.3.PP.699-704]. Insets: rescaled PDFs.probes [R.Jha et al. Phys. Plasmas.2003.Vol.10.No.3.PP.699-704]. Insets: rescaled PDFs.

Power-law truncated Lévy flights: Theory Power-law truncated Lévy flights: Theory and experimentand experiment

Evolution of the PDF of the PLT Lévy Evolution of the PDF of the PLT Lévy processprocess (I. Sokolov et al., in press)(I. Sokolov et al., in press)

PDF at small time = 0.001PDF at small time = 0.001

PDF at large time = 1000PDF at large time = 1000

Scale-free models provide an efficient description of a broad variety of

processes in complex systems

This phenomenological fact is corroborated by the observation that the

power-law properties of Lévy processes strongly persist, mathematically,

by the existence of the Generalized Central Limit Theorem due to which

Lévy stable laws become fundamental

All naturally occurring power laws in fractal or dynamic patterns are

finite

Here we present examples of regularisation of Lévy processes in

presence of

― non-dissipative non-linearity : confined Lévy flights,

― dissipative non-linearity : damped Lévy flights,

― power law truncation of free Lévy flights

These models might be helpful when discussing applicability of the

Lévy random processes in different physical, chemical, bio..... , financial,

… systems

Scale-free models provide an efficient description of a broad variety of

processes in complex systems

This phenomenological fact is corroborated by the observation that the

power-law properties of Lévy processes strongly persist, mathematically,

by the existence of the Generalized Central Limit Theorem due to which

Lévy stable laws become fundamental

All naturally occurring power laws in fractal or dynamic patterns are

finite

Here we present examples of regularisation of Lévy processes in

presence of

― non-dissipative non-linearity : confined Lévy flights,

― dissipative non-linearity : damped Lévy flights,

― power law truncation of free Lévy flights

These models might be helpful when discussing applicability of the

Lévy random processes in different physical, chemical, bio..... , financial,

… systems

Intermediate zIntermediate zÁÁver ver

From flights in space – to flights in timeFrom flights in space – to flights in timeHarvey Scher and Elliott W. Montroll, Anomalous transit-time dispersion in Harvey Scher and Elliott W. Montroll, Anomalous transit-time dispersion in amorphous solids. amorphous solids. Physical ReviewPhysical Review BB, vol.12, No.6, 2455-2477 (1975)., vol.12, No.6, 2455-2477 (1975).

Measuring photocurrentMeasuring photocurrent

Idealized transient - Idealized transient - current current tracetrace

Experimental and theoretical plotsExperimental and theoretical plots, , log-log scale log-log scale

LLéévy flights in timevy flights in timeModelModel: : jumps between traps, waiting time jumps between traps, waiting time distributed with distributed with ww(().).

Two possibilitiesTwo possibilities::

1. 1. Duration of experimentDuration of experiment >> >> typical waiting time in a traptypical waiting time in a trap = < = <> > normal normal diffusiondiffusion..

2. 2. Slow decayingSlow decaying asymptoticsasymptotics:: SLOW anomalousSLOW anomalous diffusion diffusion..

(1 )( ) ,0 1,w const

Propagator for normal diffusionPropagator for normal diffusion Propagator for subdiffusionPropagator for subdiffusion

Time-fractional Fokker-Planck equationTime-fractional Fokker-Planck equationfor Lfor Léévy flights in timevy flights in time

2

2f dU f

f Dx dxt x

2

2f dU f

f Dx dxt x

1

0

( ) ( ) ( ) ( ) (0)stt s dt t e s s st

1: (0)s

t

11

( ) , 0 1,w

Fractional Caputo derivative:Fractional Caputo derivative:

SolutionSolution:: separation of variablesseparation of variables ( , ) ( ) ( )n n nf x t T t xTime evolutionTime evolution:: Mittag Mittag –– Leffler relaxation Leffler relaxation

1 1

exp , 11( )

(1 ) , 1

nn

n n

n n

tt

T t E t

t t

Stretched exponential lawStretched exponential law

Slow power law relaxation Slow power law relaxation

General ZGeneral ZÁÁverver

Theory of LTheory of Lévy stable probability laws :vy stable probability laws :

Mathematical basis for the description of a broad range of non-Brownian Mathematical basis for the description of a broad range of non-Brownian random phenomena random phenomena

Kinetic theory of LKinetic theory of Lévy random motion:vy random motion:Based on diffusion-kinetic equations with fractional derivatives in time, space Based on diffusion-kinetic equations with fractional derivatives in time, space and/or velocityand/or velocity

Approach based on LApproach based on Lévy random motion and fractional kinetic vy random motion and fractional kinetic equations:equations:

already proved its effectiveness in different physical, bio, … financial already proved its effectiveness in different physical, bio, … financial problemsproblems

A. Chechkin, V. Gonchar, J. Klafter and R. Metzler, Fundamentals of Lévy Flight Processes. Adv. Chem. Phys. 133B, (2006).

BUT: THEORY IS ONLY AT THE BEGINNINGBUT: THEORY IS ONLY AT THE BEGINNING

Б.В. Гнеденко, А.Н. КолмогоровБ.В. Гнеденко, А.Н. Колмогоров “Предельные распределения для сумм “Предельные распределения для сумм

независимых случайных величин” независимых случайных величин” ( (19491949))::

““Все эти законы распределения, называемые “устойчивыми” …Все эти законы распределения, называемые “устойчивыми” …заслуживают, как нам кажется, самого серьезного заслуживают, как нам кажется, самого серьезного внимания. Вероятно, круг реальных прикладных задач, в внимания. Вероятно, круг реальных прикладных задач, в которых они будут играть существенную роль, окажется со которых они будут играть существенную роль, окажется со временем достаточно широким.”временем достаточно широким.”

“ …“ … “ “нормальная” сходимость к ненормальным (т.е. нормальная” сходимость к ненормальным (т.е. отличным от гауссовского –А.Ч.) устойчивым законам … , отличным от гауссовского –А.Ч.) устойчивым законам … , несомненно, уже сейчас должна быть рассмотрена во несомненно, уже сейчас должна быть рассмотрена во всяком большом руководстве, рассчитывающем достаточно всяком большом руководстве, рассчитывающем достаточно хорошо вооружить, например, работника в области хорошо вооружить, например, работника в области статистической физики.”статистической физики.”

Ю.Л. Климонтович (2002), R. Balescu (2007)

Dyakuem za pozornost’ !Dyakuem za pozornost’ !