Generalized Diffusion

TL;DR

This paper derives a nonlinear generalization of the Fokker-Planck equation for anomalous diffusion, incorporating effects of drift and external fields, and confirms results with Monte Carlo simulations.

Contribution

It introduces a nonlinear Fokker-Planck equation that generalizes classical models to include nonlinear jump probabilities and external influences, linking to the Porous Media equation.

Findings

Scaling solutions exist only for power-law nonlinearities.

The generalized equation reduces to the Porous Media equation under certain conditions.

Monte Carlo simulations confirm theoretical predictions.

Abstract

The Fokker-Planck equation for the probability $f(r,t)$ to find a random walker at position $r$ at time $t$ is derived for the case that the the probability to make jumps depends nonlinearly on $f(r,t)$. The result is a generalized form of the classical Fokker-Planck equation where the effects of drift, due to a violation of detailed balance, and of external fields are also considered. It is shown that in the absence of drift and external fields a scaling solution, describing anomalous diffusion, is only possible if the nonlinearity in the jump probability is of the power law type ($\sim f^{\eta }(r,t)$), in which case the generalized Fokker-Planck equation reduces to the well-known Porous Media equation. Monte-Carlo simulations are shown to confirm the theoretical results.