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Lévy Flights and Fractional Kinetic Equations A. Chechkin Institute for Theoretical Physics of National Science Center «Kharkov Institute of Physics and Technology», Kharkov, UKRAINE IEP Kosice, 9 May 2007 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).

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

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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)

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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.”

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

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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 variance Lévy: attract f(x) with infinite variance variance

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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 possessing large bursts and/or outliers

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_1 é 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

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

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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 …

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Normal vs Anomalous Diffusion

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Self-Similar Structure of the Brownian and the Lévy Motions 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 J.B. Perren “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

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

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

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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

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

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Anomalous diffusion in Lorentz gases with infinite «horizon» (Bouchaud and Le Doussal; Zacherl et al.) Polymer adsorption and self - avoiding Levy flights (de Gennes; Bouchaud) Anomalous diffusion in a linear array of convection rolls (Pomeau et al.; Cardoso and Tabeling) Distribution of the length the paths The typical displacement 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 each roll acts as a trap with a release time distribution decaying as

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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, 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...)

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Lévy noises and Lévy motion LÉVY NOISES Lévy index outliers LÉVY MOTION: successive additions of the noise values Lévy index “flights” become longer

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EXPERIMENTAL OBSERVATION OF LÉVY FLIGHTS IN WIND VELOCITY (Lammefjord, denmark, 2006) Measuring masts Velocity increments Distribution of increments 250 min 25 min 2.5 min

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

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«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 ; а) R = 111 сm ; б) R = 112 сm ; в) R = 112.5 сm ; г) R = 113 сm. в) R = 112.5 сm ; г) R = 113 сm.

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«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

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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 ! Models are strongly needed ! _2

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problem: KINETIC THEORY FOR NON-GAUSSIAN LÉVY WORLD ??? KINETIC EQUATIONS WITH FRACTIONAL DERIVATIVES …

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Fractional Kinetics “Strange” Kinetics M.F. Schlesinger, G.M. Zaslavsky, J. Klafter, Nature (1993)) : connected with deviations “Normal” KineticsFractional Kinetics Diffusion law RelaxationExponentialNon-exponential “Lévy flights in time” StationarityMaxwell-Boltzmann equilibrium Confined Lévy flights as non-Boltzmann stationarity

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

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:Riesz fractional derivative: integrodifferential operator = 2 : Fokker - Planck equation (FPE) 0 < < 2 : Fractional FPE (FFPE) 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)

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

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Space-fractional diffusion equation, U=0: free Lévy flights After Fourier transform: Solution in Fourier - space: characteristic function of free Lévy flights Characteristic diffusion displacement superdiffusion: faster than t 1/2 ( 0 < < 2 )

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

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Free Lévy flights vs Brownian motion in semi-infinite domain: Leapover (T. Koren. A. Chechkin, Y. Klafter, 2007) Brownian Motion, = 2Lévy 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

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

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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: 2. Slowly decaying tails: Harmonic force is not “strong” enough to “confine” Lévy flights 1. Unimodality (one hump at the origin).

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Confined Lévy flights. Quartic potential well, U x 4 (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)

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Confined Lévy flights. Quartic potential, U x 4. Cauchy case, = 1 Equation for the characteristic function: PDF: Two properties of stationary PDF 1. 1. Bimodality: local minimum at, two maxima at 2. Steep power law asymptotics with finite variance:

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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 propositions (A. Chechkin, V. Gonchar, Y. Klafter, R. Metzler, L. Tanatarov, 2004) Proposition 1. Proposition 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 c cr : : “confined” Proved with the use of the representation of the Riesz derivative in terms of left- and right Liouville-Weyl derivatives.

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

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Quartic potential U x 4, = 1.2. Time evolution of the PDF

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Potential well U x 5.5, = 1.2. Trimodal transient state in time evolution of the PDF

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Potential well U x 5.5, = 1.2. Trimodal transient state in a finer scale

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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 .

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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

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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 = 0.0001, 2 = 0 (curve 1) 0 = 1.0, 1 = 0, 2 = 0.000001 (curve 2), = 1.0, slopes = -1, -3, -5

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Damped Lévy flights: 2 nd Moments 4 th Moments

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

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Theory 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.699-704]. Insets: rescaled PDFs. Power-law truncated Lévy flights: Theory and experiment

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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

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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 zÁver

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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, 2455-2477 (1975). Measuring photocurrent Idealized transient - current trace Experimental and theoretical plots, log-log scale

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Lévy flights in time 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. SLOW anomalous diffusion. Propagator for normal diffusion Propagator for subdiffusion Propagator for subdiffusion

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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

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General ZÁver Theory of Lvy stable probability laws : Theory of Lévy stable probability laws : Mathematical basis for the description of a broad range of non-Brownian random phenomena Kinetic theory of Lvy random motion: Kinetic theory of Lévy random motion: Based on diffusion-kinetic equations with fractional derivatives in time, space and/or velocity Approach based on Lvy random motion and fractional kinetic equations: Approach based on Lévy random motion and fractional kinetic equations: already proved its effectiveness in different physical, bio, … financial problems already proved its effectiveness in different physical, bio, … financial problems 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 BEGINNING

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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)

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Dyakuem za pozornost’ !

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