Memory-Controlled Diffusion

TL;DR

This paper generalizes the Fokker-Planck equation to include non-linear and non-local memory effects, analyzing their impact on diffusion behavior and stationary solutions.

Contribution

It introduces a generalized equation incorporating memory effects, provides criteria for stationary solutions, and finds exact long-time solutions for non-linear memory diffusion.

Findings

Memory kernels restrict the conservation of probability density.

Finite localization occurs due to delay effects, halting further transport.

Higher order cumulants differ from standard diffusion, depending on memory strength.

Abstract

Memory effects require for their incorporation into random-walk models an extension of the conventional equations. The linear Fokker-Planck equation for the probability density $p(\vec r, t)$ is generalized to include non-linear and non-local spatial-temporal memory effects. The realization of the memory kernels are restricted due the conservation of the basic quantity $p$. A general criteria is given for the existence of stationary solutions. In case the memory kernel depends on $p$ polynomially the transport is prevented. Owing to the delay effects a finite amount of particles remains localized and the further transport is terminated. For diffusion with non-linear memory effects we find an exact solution in the long-time limit. Although the mean square displacement shows diffusive behavior, higher order cumulants exhibits differences to diffusion and they depend on the memory strength.