Presentation on theme: "Lévy Flights and Fractional Kinetic Equations"— Presentation transcript:
1Lévy Flights and Fractional Kinetic Equations IEP Kosice, 9 May 2007Lévy Flights and Fractional Kinetic EquationsA. ChechkinInstitute 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).
2LÉVY motion or “LÉVY flights” (B LÉVY motion or “LÉVY flights” (B. Mandelbrot): paradigm of non-brownian random motionMathematical Foundations:Brownian MotionLévy Motion1. Central Limit Theorem1. Generalized CentralLimit Theorem2. Properties of Gaussian probability laws and processes2. Properties of Lévystable laws and processes
4B.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.”
5First 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 indexGauss: = 2, cn = n1/2Corollary 1. Statistical self-similarity, random fractalMandelbrot: When does the whole look like its parts description of random fractal ptocessesCorollary 2. Normal diffusion law, Brownian motionor the rule of summing variancesCorollary 3. Superdiffusion, LévyQuestion: Xi are not stable r.v.
6second 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 quantitiesGauss: attracts f(x) with finitevarianceLévy: attract f(x) with infinitevariance
7third remarkable property of stable distributions Property 3. Power law asymptotics x (-1-) (“heavy tails”)Particular cases1. Gauss, = 2All moments are finite2. Cauchy, = 1Moments of order q < 1 are finiteCorollary: description of highly non-equilibrium processespossessing large bursts and/or outliers
8_1Due 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
9Electric Field Distribution in an Ionized Gas (Holtzmark 1919) Application: spectral lines broadening in plasmas
10Related 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 …
12Self-Similar Structure of the Brownian and the Lévy Motions x100x100“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. Perrenx100x100“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
13NORMAL vs ANOMALOUS DIFFUSION Brownian motion of a small grain of puttyForaging movement of spider monkeys(Jean Baptiste Perrin: 1909) Gabriel Ramos-Fernandes et al. (2004)
14Classical 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)
15Diffusion of tracers in fluid flows. Large scale structures (eddies, jets or convection rolls) dominate the transport.Example. Experimentsin a rapidly rotatingannulus (Swinney et al.).Ordered flow:Levy diffusion(flights and traps)Weakly turbulent flow:Gaussian diffusion
16Anomalous 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
17Anomalous diffusion in Lorentz gases with infinite «horizon» (Bouchaud and Le Doussal; Zacherl et al.) Distribution of the length the paths The typical displacementPolymer 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 thresholdAnomalous diffusion in a linear array of convection rolls(Pomeau et al.; Cardoso and Tabeling)each roll acts as a trap with a releasetime distribution decaying as
18Impulsive Noises Modeled with the Lévy Stable Distributions Naturally serve for the description of the processeswith large outliers, far from equilibriumExamples 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, )
19Lévy noises and Lévy motion successive additions of the noise valuesLévy index “flights” becomelongerLÉVY NOISESLévy index outliers
20EXPERIMENTAL OBSERVATION OF LÉVY FLIGHTS IN WIND VELOCITY (Lammefjord, denmark, 2006) Measuring masts Velocity increments Distribution of increments250 min25 min2.5 min
21Lévy Noise vs Gaussian Noise Lévy turbulence in „Uragan 3M“V.Gonchar et.al., 2003DEM/USD exchange rateV.Gonchar, A. Chechkin, 2001
22Ion 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 proveIon saturation currentat 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 fluctuationsIon saturation current nFloating 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 !
25problem: FOR NON-GAUSSIAN LÉVY WORLD ??? KINETIC THEORYFOR NON-GAUSSIAN LÉVY WORLD ??? KINETIC EQUATIONS WITH FRACTIONAL DERIVATIVES …
26Fractional Kinetics “Strange” Kinetics M.F. Schlesinger, G.M. Zaslavsky, J. Klafter, Nature (1993)) : connected with deviations“Normal” KineticsFractional KineticsDiffusion lawRelaxationExponentialNon-exponential“Lévy flights in time”StationarityMaxwell-BoltzmannequilibriumConfined Lévy flightsas non-Boltzmannstationarity
27Types of “Fractionalisation” FOKKER-PLANCK EQ time – fractionalCaputo/Riemann-Liouville derivative space / velocity – fractionalRiesz derivative multi-dimensional / anisotropic time-space / velocity fractionalQuestion : Derivation ?Answer:Generalized Master Equation,Generalized Langevin EquationOUT OF THE SCOPEOF THE TALK !BUT:
28Kinetic 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 noise0 < < 2 : white Lévy noiseKinetic description, f(x,t):Riesz fractional derivative: integrodifferential operator = 2 : Fokker - Planck equation (FPE)0 < < 2 : Fractional FPE (FFPE)
29Definition of the Riesz fractional deriva-tive via its Fourier representation Indeed,.But:Example:
30Space-fractional diffusion equation, U=0: free Lévy flights After Fourier transform:Solution in Fourier - space:characteristic functionof free Lévy flights superdiffusion: faster than t1/2Characteristic diffusion displacement( 0 < < 2 )
31Free Lévy flights vs Brownian motion in semi-infinite domain Lévy Motion, 0<<2First passage time PDFSparre Andersen universalityMethod of imagesMethod of images breaks down for Lévy flights (A. Chechkin et. al., 2004)
32Leapover 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, = 2Lévy Motion, 0<<2Leapover: how large is the leap over the boundary by the Lévy stable process ?No leapoverLeapover PDF decays slower than Lévy flight PDF
33Reminder: 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.
34Lévy 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 lawTwo 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
35Confined 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 …
36Confined Lévy flights. Quartic potential, U x4. Cauchy case, = 1 Equation for the characteristic function:PDF:Two properties of stationary PDFBimodality: local minimum at, two maxima at2. Steep power law asymptotics with finite variance:
37Two 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 fieldis non-unimodal. Proved with the use of the “hypersingular” representation of the Riesz derivative.Proposition 2. Stationary PDF for the Lévy flights in external fieldhas 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.
38Numerical 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 pathsPotential wells Brownian motion Lévy motion
39Quartic potential U x4, = 1.2. Time evolution of the PDF
40Potential well U x5.5, = 1.2. Trimodal transient state in time evolution of the PDF
41Potential well U x5.5, = 1.2. Trimodal transient state in a finer scale
42Confined Lévy flights in bistable potential: Lévy noise induced transitions (stochastic climate dynamics) (A. Sliusarenko et al., 2007)Fig.1. Escape of the trajectory overthe barrier, schematic view.Fig.2. Typical trajectories for different .
43Confined Lévy flights in bistable potential: Kramers’ problem Completely unusualbehaviorof T at small D’s:Escape probability p(t) as functionof walking time t:A.V. Chechkin et al., 2007
44Damped 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)FFPE0 = 1.0, 1 = , 2 = 0 (curve 1)0 = 1.0, 1 = 0, 2 = (curve 2), = 1.0, slopes = -1, -3, -5
46Power-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 partat 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 slowlyParticular 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
47Power-law truncated Lévy flights: Theory and experiment Experiment on the ADITYA tokamakExperiment 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.
48Evolution 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 = 0.001PDF at large time = 1000
49Intermediate zÁverScale-free models provide an efficient description of a broad variety of processes in complex systemsThis 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 fundamentalAll naturally occurring power laws in fractal or dynamic patterns are finiteHere we present examples of regularisation of Lévy processes in presence ofnon-dissipative non-linearity : confined Lévy flights,dissipative non-linearity : damped Lévy flights,power law truncation of free Lévy flightsThese models might be helpful when discussing applicability of the Lévy random processes in different physical, chemical, bio..... , financial, … systems
50From 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 photocurrentIdealized transient - current traceExperimental and theoretical plots, log-log scale
51Lévy 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 diffusionPropagator for subdiffusion
52Time-fractional Fokker-Planck equation for Lévy flights in time Fractional Caputo derivative:Solution: separation of variablesTime evolution: Mittag – Leffler relaxationStretched exponential lawSlow power law relaxation
53General ZÁver BUT: THEORY IS ONLY AT THE BEGINNING Theory of Lévy stable probability laws :Mathematical basis for the description of a broad range of non-Brownian random phenomenaKinetic theory of Lévy random motion:Based on diffusion-kinetic equations with fractional derivatives in time, space and/or velocityApproach based on Lévy random motion and fractional kinetic equations:already proved its effectiveness in different physical, bio, … financial problemsBUT: THEORY IS ONLY AT THE BEGINNINGA. Chechkin, V. Gonchar, J. Klafter and R. Metzler, Fundamentals of Lévy Flight Processes. Adv. Chem. Phys. 133B, (2006).
54Ю.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)