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Lévy Flights and Fractional Kinetic Equations

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1 Lévy Flights and Fractional Kinetic Equations
IEP Kosice, 9 May 2007 Lévy Flights and Fractional Kinetic Equations A. Chechkin Institute for Theoretical Physics of National Science Center «Kharkov Institute of Physics and Technology», Kharkov, UKRAINE  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) In collaboration with: A. Chechkin, V. Gonchar, J. Klafter and R. Metzler, Fundamentals of Lévy Flight Processes. Adv. Chem. Phys. 133B, 439 (2006).

2 LÉVY motion or “LÉVY flights” (B
LÉVY motion or “LÉVY flights” (B. Mandelbrot): paradigm of non-brownian random motion Mathematical Foundations: 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

3 Paul Pierre Lévy ( ) Lévy stable distributions in Mathematical Monographs: Lévy (1925, 1937) Хинчин (1938) Gnedenko & Kolmogorov ( ) Feller ( ) Lukacs ( ) Zolotarev ( , 1997) Janiki & Weron (1994) Samorodnitski & Taqqu (1994)

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

5 First remarkable property of stable distributions
Stability: distribution of sum of independent identically distributed stable random variables = distribution of each variable (up to scaling factor) cn = n1/a , 0 <   2,  is Lèvy index Gauss:  = 2, cn = n1/2 Corollary 1. Statistical self-similarity, random fractal Mandelbrot: When does the whole look like its parts  description of random fractal ptocesses Corollary 2. Normal diffusion law, Brownian motion or the rule of summing variances Corollary 3. Superdiffusion, Lévy Question: Xi are not stable r.v.

6 second remarkable property of stable distributions
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 f(x) with finite variance Lévy: attract f(x) with infinite variance

7 third remarkable property of stable distributions
Property 3. Power law asymptotics  x (-1-) (“heavy tails”) Particular cases 1. Gauss,  = 2 All moments are finite 2. Cauchy,  = 1 Moments of order q < 1 are finite Corollary: description of highly non-equilibrium processes possessing large bursts and/or outliers

8 _1 Due to the three remarkable properties of stable distributions it is now believed that the Lévy statistics provides a framework for the description of many natural phenomena in physical, chemical, biological, economical, … systems from a general common point of view

9 Electric Field Distribution in an Ionized Gas (Holtzmark 1919)
Application: spectral lines broadening in plasmas

10 Related 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 …

11 Normal vs Anomalous Diffusion

12 Self-Similar Structure of the Brownian and the Lévy Motions
x100 x100 “If, veritably, one took the position from second to second, each of these rectilinear segments would be replaced by a polygonal contour of 30 edges, each being as complicated as the reproduced design, and so forth.” J.B. Perren x100 x100 “The stochastic process which we call linear Brownian motion is a schematic representa-tion which describes well the properties of real Brownian motion, observable on a small enough, but not infinitely small scale, and which assumes that the same properties exist on whatever scale.” P.P. Lévy

Brownian motion of a small grain of putty Foraging movement of spider monkeys (Jean Baptiste Perrin: 1909) Gabriel Ramos-Fernandes et al. (2004)

14 Classical example of superdiffusion
Richardson: diffusion of passive admixture in locally isotropic turbulence (1927) Frenkiel and Katz (1956), Seneca (1955), Kellogg (1956) versus  (Gifford, 1957) Monin: space fractional diffusion equation (1955, 1956) linear integral operator with complicated kernel (Monin)

15 Diffusion of tracers in fluid flows.
Large scale structures (eddies, jets or convection rolls) dominate the transport. Example. Experiments in a rapidly rotating annulus (Swinney et al.). Ordered flow: Levy diffusion (flights and traps) Weakly turbulent flow: Gaussian diffusion

16 Anomalous Diffusion in Channeling
Fig.2. SUPERDIFFUSION: positively and negatively charged particles in a periodic field of atom strings along the axis <100> in a silicon crystal. Fig.1. NORMAL DIFFUSION: randomly distributed atom strings

17 Anomalous diffusion in Lorentz gases with infinite «horizon»
(Bouchaud and Le Doussal; Zacherl et al.)  Distribution of the length the paths  The typical displacement Polymer adsorption and self - avoiding Levy flights (de Gennes; Bouchaud)  Distribution of the size of the loops decays as a power law  the projection of the chain’s conformation on the wall: self - avoiding Levy flight (two adsorbed monomers cannot occupy the same site)  at the adsorption threshold Anomalous diffusion in a linear array of convection rolls (Pomeau et al.; Cardoso and Tabeling) each roll acts as a trap with a release time distribution decaying as

18 Impulsive Noises Modeled with the Lévy Stable Distributions
Naturally serve for the description of the processes with 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, )

19 Lévy noises and Lévy motion
successive additions of the noise values Lévy index   “flights” become longer LÉVY NOISES Lévy index   outliers 

Measuring masts Velocity increments Distribution of increments 250 min 25 min 2.5 min

21 Lévy Noise vs Gaussian Noise
Lévy turbulence in „Uragan 3M“ V.Gonchar, 2003 DEM/USD exchange rate V.Gonchar, A. Chechkin, 2001

22 Ion saturation current at different probe positions:
«Lévy Turbulence» in boundary plasma of stellarator «Uragan 3М» (IPP KIPT) Helical magnetic coils (from above) Poincare section of magnetic line , ЛЗ – Langmuir prove Ion saturation current at different probe positions: а) R = 111 сm ; б) R = 112 сm ; в) R = сm ; г) R = 113 сm.

23 «Lévy Turbulence» in boundary plasma of stellarator «Uragan 3М» (IPP KIPT)
1. Kurtosis and “chi-square” criterium: evidence of non-Gaussianity. 2. Modified method of percentiles: Lévy index of turbulent fluctuations Ion saturation current  n Floating potential  

24 _2 Models are strongly needed !
 Fluctuations of density and potential measured in boundary plasma of stellarator “Uragan 3M” obey Lévy statistics (September 2002)  similar conclusions about the Lévy statistics of plasma fluctuations have been drawn for  ADITYA (March 2003)  L-2M ( October 2003)  LHD, TJ-II (December 2003)  Heliotron J (March 2004) “bursty” character of fluctuations in other devices (DIII-D, TCABR, …) “Levy turbulence” is a widely spread phenomenon  Models are strongly needed !


26 Fractional Kinetics  “Strange” Kinetics
M.F. Schlesinger, G.M. Zaslavsky, J. Klafter, Nature (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

27 Types of “Fractionalisation”
FOKKER-PLANCK EQ  time – fractional Caputo/Riemann-Liouville derivative  space / velocity – fractional Riesz derivative  multi-dimensional / anisotropic  time-space / velocity fractional Question : Derivation ? Answer: Generalized Master Equation, Generalized Langevin Equation OUT OF THE SCOPE OF THE TALK ! BUT:

28 Kinetic equations for the stochastic systems driven 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 noise 0 <  < 2 : white Lévy noise Kinetic description, f(x,t) :Riesz fractional derivative: integrodifferential operator  = 2 : Fokker - Planck equation (FPE) 0 <  < 2 : Fractional FPE (FFPE)

29 Definition of the Riesz fractional deriva-tive via its Fourier representation
Indeed, .But: Example:

30 Space-fractional diffusion equation, U=0: free Lévy flights
After Fourier transform: Solution in Fourier - space: characteristic function of free Lévy flights  superdiffusion: faster than t1/2 Characteristic diffusion displacement ( 0 <  < 2 )

31 Free Lévy flights vs Brownian motion in semi-infinite domain
Lévy Motion, 0<<2 First passage time PDF Sparre Andersen universality Method of images Method of images breaks down for Lévy flights (A. Chechkin et. al., 2004)

32 Leapover PDF decays slower than Lévy flight PDF
Free Lévy flights vs Brownian motion in semi-infinite domain: Leapover (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 Leapover PDF decays slower than Lévy flight PDF

33 Reminder: Stationary solution of the FPE in external field,  = 2
Equation for the stationary PDF: (1) Boltzmann’ solution: (2) Two properties of stationary PDF: 1. Unimodality (one hump at the origin). 2. Exponential decay at large values of x.

34 Lévy flights in a harmonic potential, 1  < 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: 1. Unimodality (one hump at the origin). 2. Slowly decaying tails: Harmonic force is not “strong” enough to “confine” Lévy flights

35 Confined Lévy flights. Quartic potential well, U  x4 (A. Chechkin, V
Confined Lévy flights. Quartic potential well, U  x4 (A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L. Tanatarov, 2002) Langevin equation:  Fractional FPE for the stationary PDF: (1) Equation for the characteristic function: (2) + normalization + symmetry + boundary conditions …

36 Confined Lévy flights. Quartic potential, U  x4. Cauchy case,  = 1
Equation for the characteristic function: PDF: Two properties of stationary PDF Bimodality: local minimum at , two maxima at 2. Steep power law asymptotics with finite variance:

37 Two propositions (A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L
Two propositions (A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L. Tanatarov, 2004) Proposition 1. Stationary PDF for the Lévy flights in external field is non-unimodal. Proved with the use of the “hypersingular” representation of the Riesz derivative. Proposition 2. 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.

38 Numerical simulation Illustration: Typical sample paths
Langevin equation with a white Lévy noise, m   walls are steeper  Lévy flights are shorter  confined Lévy flights , Illustration: Typical sample paths Potential wells Brownian motion Lévy motion

39 Quartic potential U  x4,  = 1.2. Time evolution of the PDF

40 Potential well U  x5.5,  = 1.2. Trimodal transient state in time evolution of the PDF

41 Potential well U  x5.5,  = 1.2. Trimodal transient state in a finer scale

42 Confined Lévy flights in bistable potential: Lévy noise induced transitions (stochastic climate dynamics) (A. Sliusarenko et al., 2007) Fig.1. Escape of the trajectory over the barrier, schematic view. Fig.2. Typical trajectories for different .

43 Confined Lévy flights in bistable potential: Kramers’ problem
Completely unusual behavior of T at small D’s: Escape probability p(t) as function of walking time t: A.V. Chechkin et al., 2007

44 Damped Lévy flights in velocity space
Damped Lévy flights in velocity space. Damping by dissipative non-linearity (posed by B. West, V. Seshadri (1982), Chechkin et al. 2005) FFPE 0 = 1.0, 1 = , 2 = 0 (curve 1) 0 = 1.0, 1 = 0, 2 = (curve 2),  = 1.0, slopes = -1, -3, -5

45 Damped Lévy flights: 2nd Moments 4th Moments

46 Power-law truncated Lévy flights (I. Sokolov, A. Chechkin, Y
Power-law truncated Lévy flights (I. Sokolov, A. Chechkin, Y. Klafter, 2004)  “PLT 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 stable ones, therefore,  the Central Limit Theorem is applied  at large times the PDF tends to Gaussian, however, sometimes very slowly Particular case of distributed order space fractional diffusion equation: 0 <  < 2  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 PDF f(x,t) slowly converges to a Gaussian  Probability of return to the origin: The slope –1/ in the double logarithmic scale changes into -1/2 for the Gaussian

47 Power-law truncated Lévy flights: Theory and experiment
Experiment on the ADITYA tokamak 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 ]. Insets: rescaled PDFs.

48 Evolution of the PDF of the PLT Lévy process (I. Sokolov et al
Evolution of the PDF of the PLT Lévy process (I. Sokolov et al., in press) PDF at small time = 0.001 PDF at large time = 1000

49 Intermediate zÁver 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

50 From flights in space – to flights in time
Harvey Scher and Elliott W. Montroll, Anomalous transit-time dispersion in amorphous solids. Physical Review B, vol.12, No.6, (1975). Measuring photocurrent Idealized transient - current trace Experimental and theoretical plots, log-log scale

51 Lévy flights in time Two possibilities:
Model: jumps between traps, waiting time  distributed with w(). Two possibilities: 1. Duration of experiment >> typical waiting time in a trap = <>  normal diffusion. 2. Slow decaying asymptotics:  SLOW anomalous diffusion. Propagator for normal diffusion Propagator for subdiffusion

52 Time-fractional Fokker-Planck equation for Lévy flights in time
Fractional Caputo derivative: Solution: separation of variables Time evolution: Mittag – Leffler relaxation Stretched exponential law Slow power law relaxation

Theory of Lévy stable probability laws : Mathematical basis for the description of a broad range of non-Brownian random phenomena Kinetic theory of Lévy random motion: Based on diffusion-kinetic equations with fractional derivatives in time, space and/or velocity Approach based on Lévy random motion and fractional kinetic equations: already proved its effectiveness in different physical, bio, … financial problems BUT: THEORY IS ONLY AT THE BEGINNING A. Chechkin, V. Gonchar, J. Klafter and R. Metzler, Fundamentals of Lévy Flight Processes. Adv. Chem. Phys. 133B, (2006).

54 Ю.Л. Климонтович (2002), R. Balescu (2007)
Б.В. Гнеденко, А.Н. Колмогоров “Предельные распределения для сумм независимых случайных величин” (1949): “Все эти законы распределения, называемые “устойчивыми” …заслуживают, как нам кажется, самого серьезного внимания. Вероятно, круг реальных прикладных задач, в которых они будут играть существенную роль, окажется со временем достаточно широким.” “ … “нормальная” сходимость к ненормальным (т.е. отличным от гауссовского –А.Ч.) устойчивым законам … , несомненно, уже сейчас должна быть рассмотрена во всяком большом руководстве, рассчитывающем достаточно хорошо вооружить, например, работника в области статистической физики.” Ю.Л. Климонтович (2002), R. Balescu (2007)

55 Dyakuem za pozornost’ !

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