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L é vy Flights and Fractional Kinetic Equations

IEP Kosice, 9 May 2007. L é vy 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)

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L é vy Flights and Fractional Kinetic Equations

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  1. IEP Kosice, 9 May 2007 Lévy Flightsand 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. Mandelbrot): paradigm of non-brownian random motion Mathematical Foundations:

  3. Paul Pierre Lévy (1886-1971) Lévystable distributionsinMathematical 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)

  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 propertyof stable distributions • Stability: distribution of sum ofindependent 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 summingvariances Corollary 3.Superdiffusion, Lévy Question: Xi are not stable r.v.

  6. secondremarkablepropertyof stabledistributions 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) withfinite variance Lévy: attract f(x) withinfinite variance

  7. third remarkable propertyof 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:descriptionof 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 complicatedas 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 existon whatever scale.” P.P. Lévy

  13. 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)

  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. Large scale structures (eddies, jets or convection rolls) dominate the transport. Diffusion of tracers in fluid flows. 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, 2005 ...)

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

  20. 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

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

  22. «Lévy Turbulence» in boundaryplasmaof 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 = 112.5 сm ; г) R = 113 сm.

  23. «Lévy Turbulence» in boundaryplasmaof 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 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 !

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

  26. Fractional Kinetics  “Strange” Kinetics M.F. Schlesinger, G.M. Zaslavsky, J. Klafter, Nature (1993)) : connected with deviations

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

  32. Free Lévy flights vs Brownian motion in semi-infinite domain: Leapover(T. Koren. A. Chechkin, Y. Klafter, 2007) 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. 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. 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 Langevin equation with a white Lévy noise, mwalls are steeper Lévy flights are shorter confined Lévy flights , Illustration:Typical sample paths Potential wellsBrownian motionLévy 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. Damping by dissipative non-linearity (posed by B. West, V. Seshadri (1982), Chechkin et al. 2005) FFPE

  45. Damped Lévy flights: 2nd Moments 4th Moments

  46. 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 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.

  48. 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. Harvey Scher and Elliott W. Montroll, Anomalous transit-time dispersion in amorphous solids. Physical ReviewB, vol.12, No.6, 2455-2477 (1975). From flights in space – to flights in time Measuring photocurrent Idealized transient - current trace Experimental and theoretical plots, log-log scale

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