Non-Linear Langevin and Fractional Fokker-Planck Equations for Anomalous Diffusion by Levy Stable Processes

TL;DR

This paper develops numerical methods to solve non-linear fractional Fokker-Planck equations incorporating Levy stable processes, to model and analyze anomalous diffusion phenomena with non-local transport effects.

Contribution

It introduces a numerical approach for solving fractional Fokker-Planck equations with Levy fluctuations, advancing the modeling of anomalous diffusion in velocity space.

Findings

Transport coefficient increases as fractality decreases.

Numerical solutions effectively model non-local transport effects.

Distribution functions exhibit properties consistent with Levy stable processes.

Abstract

The~numerical solutions to a non-linear Fractional Fokker--Planck (FFP) equation are studied estimating the generalized diffusion coefficients. The~aim is to model anomalous diffusion using an FFP description with fractional velocity derivatives and Langevin dynamics where L\'{e}vy fluctuations are introduced to model the effect of non-local transport due to fractional diffusion in velocity space. Distribution functions are found using numerical means for varying degrees of fractionality of the stable L\'{e}vy distribution as solutions to the FFP equation. The~statistical properties of the distribution functions are assessed by a generalized normalized expectation measure and entropy and modified transport coefficient. The~transport coefficient significantly increases with decreasing fractality which is corroborated by analysis of experimental data.