Transport in the spatially tempered, fractional Fokker–Planck equation

A study of truncated Levy flights in super-diffusive transport in the presence of an external potential is presented. The study is based on the spatially tempered, fractional Fokker–Planck (TFFP) equation in which the fractional diffusion operator is replaced by a tempered fractional diffusion (TFD) operator. We focus on harmonic (quadratic) potentials and periodic potentials with broken spatial symmetry. The main objective is to study the dependence of the steady-state probability density function (PDF), and the current (in the case of periodic potentials) on the level of tempering, λ, and on the order of the fractional derivative in space, α. An expansion of the TFD operator for large λ is presented, and the corresponding equation for the coarse grained PDF is obtained. The steady-state PDF solution of the TFFP equation for a harmonic potential is computed numerically. In the limit λ → ∞, the PDF approaches the expected Boltzmann distribution. However, nontrivial departures from this distribution are observed for finite (λ > 0) truncations, and α ≠ 2. In the study of periodic potentials, we use two complementary numerical methods: a finite-difference scheme based on the Grunwald–Letnikov discretization of the truncated fractional derivatives and a Fourier-based spectral method. In the limit λ → ∞, the PDFs converges to the Boltzmann distribution and the current vanishes. However, for α ≠ 2, the PDF deviates from the Boltzmann distribution and a finite non-equilibrium ratchet current appears for any λ > 0. The current is observed to converge exponentially in time to the steady-state value. The steady-state current exhibits algebraical decay with λ, as J ∼ λ−ζ, for α ⩾ 1.75. However, for α ⩽ 1.5, the steady-state current decays exponentially with λ, as J ∼ e−ξλ. In the presence of an asymmetry in the TFD operator, the tempering can lead to a current reversal. A detailed numerical study is presented on the dependence of the current on λ and the physical parameters of the system.

[1]  Vincenzo Carbone,et al.  A possible truncated-Lévy-flight statistics recovered from interplanetary solar-wind velocity and magnetic-field fluctuations , 2002 .

[2]  Diego del-Castillo-Negrete,et al.  Fractional diffusion models of nonlocal transport , 2006 .

[3]  Á. Cartea,et al.  Fluid limit of the continuous-time random walk with general Lévy jump distribution functions. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[4]  Peter Imkeller,et al.  Lévy flights: transitions and meta-stability , 2006 .

[5]  Aleksei V. Chechkin,et al.  Stationary states of non-linear oscillators driven by Lévy noise , 2002 .

[6]  D. del-Castillo-Negrete,et al.  Truncation effects in superdiffusive front propagation with Lévy flights. , 2009, Physical review. E, Statistical, nonlinear, and soft matter physics.

[7]  J. Klafter,et al.  The random walk's guide to anomalous diffusion: a fractional dynamics approach , 2000 .

[8]  O. Marichev,et al.  Fractional Integrals and Derivatives: Theory and Applications , 1993 .

[9]  D. R. Kulkarni,et al.  Evidence of Lévy stable process in tokamak edge turbulence , 2001 .

[10]  J. Rosínski.  Tempering stable processes , 2007 .

[11]  I. Sokolov,et al.  Transport in a Lévy ratchet: group velocity and distribution spread. , 2008, Physical review. E, Statistical, nonlinear, and soft matter physics.

[12]  R. Mantegna,et al.  Scaling behaviour in the dynamics of an economic index , 1995, Nature.

[13]  Koponen,et al.  Analytic approach to the problem of convergence of truncated Lévy flights towards the Gaussian stochastic process. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[14]  Diego del-Castillo-Negrete,et al.  Asymmetric transport and non-Gaussian statistics of passive scalars in vortices in shear , 1998 .

[15]  Peter Hänggi,et al.  Escape driven by alpha-stable white noises. , 2007, Physical review. E, Statistical, nonlinear, and soft matter physics.

[16]  Stanley,et al.  Stochastic process with ultraslow convergence to a Gaussian: The truncated Lévy flight. , 1994, Physical review letters.

[17]  V E Lynch,et al.  Nondiffusive transport in plasma turbulence: a fractional diffusion approach. , 2005, Physical review letters.

[18]  A. V. Tour,et al.  Lévy anomalous diffusion and fractional Fokker–Planck equation , 2000, nlin/0001035.

[19]  Aleksei V. Chechkin,et al.  Barrier crossing of a Lévy flight , 2005 .

[20]  I. Podlubny.  Fractional differential equations , 1998 .

[21]  V. Gonchar,et al.  Fluctuation-driven directed transport in the presence of Lévy flights , 2007, 0710.0883.

[22]  Gennady Samorodnitsky,et al.  Tails of solutions of certain nonlinear stochastic differential equations driven by heavy tailed Lévy motions , 2003 .

[23]  P. Reimann.  Brownian motors: noisy transport far from equilibrium , 2000, cond-mat/0010237.

[24]  B. Dubrulle,et al.  Truncated Lévy laws and 2D turbulence , 1998 .

[25]  Sune Jespersen,et al.  LEVY FLIGHTS IN EXTERNAL FORCE FIELDS : LANGEVIN AND FRACTIONAL FOKKER-PLANCK EQUATIONS AND THEIR SOLUTIONS , 1999 .

[26]  P D Ditlevsen.  Anomalous jumping in a double-well potential. , 1999, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.