Levy Flights in External Force Fields: Langevin and Fractional Fokker-Planck Equations, and their Solutions

TL;DR

This paper investigates Levy flights under external forces using Langevin and fractional Fokker-Planck equations, analyzing specific force cases, and exploring non-Gibbsian stationary states and connections to Tsallis statistics.

Contribution

It introduces a detailed analysis of Levy flights in external force fields through both Langevin and fractional Fokker-Planck frameworks, including numerical validation and non-equilibrium stationary states.

Findings

Stationary solutions deviate from Boltzmann distribution under Hookean force.

Numerical simulations confirm analytical results.

Potential link between Levy flights and Tsallis's q-statistics.

Abstract

We consider Levy flights subject to external force fields. This anomalous transport process is described by two approaches, a Langevin equation with Levy noise and the corresponding generalized Fokker-Planck equation containing a fractional derivative in space. The cases of free flights, constant force and linear Hookean force are analyzed in detail, and we corroborate our findings with results from numerical simulations. We discuss the non-Gibbsian character of the stationary solution for the case of the Hookean force, i.e. the deviation from Boltzmann equilibrium for long times. The possible connection to Tsallis's q-statistics is studied.