Fokker-Planck equation with memory: the crossover from ballistic to diffusive processes in many-particle systems and incompressible media

TL;DR

This paper develops a non-Markovian Fokker-Planck equation to describe the crossover from ballistic to diffusive behavior in many-particle systems and incompressible media, providing solutions that improve upon traditional models especially at short times.

Contribution

It introduces a non-Markovian generalization of the Fokker-Planck equation with kinetic coefficients derived from observable quantities, capturing the ballistic-to-diffusive transition more accurately.

Findings

Solutions match long-time predictions of continuous random walk theory

Model accurately describes short-time ballistic behavior

Provides a unified framework for diffusive processes in complex media

Abstract

The unified description of diffusion processes that cross over from a ballistic behavior at short times to normal or anomalous diffusion (sub- or superdiffusion) at longer times is constructed on the basis of a non-Markovian generalization of the Fokker-Planck equation. The necessary non-Markovian kinetic coefficients are determined by the observable quantities (mean- and mean square displacements). Solutions of the non-Markovian equation describing diffusive processes in the physical space are obtained. For long times, these solutions agree with the predictions of the continuous random walk theory; they are, however, much superior at shorter times when the effect of the ballistic behavior is crucial.