Transport in the spatially tempered, fractional Fokker–Planck equation

Abstract

A study of truncated Lévy 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.