L é vy Flights and Fractional Kinetic Equations
description
Transcript of L é vy Flights and Fractional Kinetic Equations
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’ !