Levy Flights in External Force Fields: From Models to Equations

TL;DR

This paper explores various generalizations of the Fokker-Planck equation to model Levy processes in potential fields, highlighting different origins of Levy statistics and their impact on relaxation patterns, with applications to double-well potentials.

Contribution

It introduces new generalizations of the Fokker-Planck equation for Levy processes, analyzing their different derivations and effects on equilibrium and relaxation behaviors.

Findings

Levy statistics can arise from fractal temporal processes or scale-free spatial structures.

Different generalizations lead to distinct relaxation patterns despite reaching Boltzmann equilibrium.

Application to double-well potential demonstrates varied diffusion behaviors.

Abstract

We consider different generalizations of the Fokker-Planck-equation devised to describe Levy processes in potential force fields. We show that such generalizations can proceed along different lines. On one hand, Levy statistics can emerge from the fractal temporal nature of the underlying process, i.e. a high variability in the rate of microscopic events. On the other hand, they may be a direct consequence of the scale-free spatial structure on which the process evolves. Although both forms considered lead to Boltzmann equilibrium, the relaxation patterns are quite different. As an example, generalized diffusion in a double-well potential is considered.