Mean field approach for diffusion of interacting particles

TL;DR

This paper derives a nonlinear Fokker-Planck equation for interacting particles on a lattice, linking microscopic interactions to macroscopic diffusion behavior, and explores how barrier heights influence mobility and diffusion.

Contribution

It introduces a mean field framework connecting microscopic interaction energies with macroscopic diffusion properties in a lattice model.

Findings

The parameter $eta$ affects mobility and diffusion dependence on concentration.

Equilibrium solutions are unaffected by the barrier-related parameter $eta$.

Classical particle interactions can reproduce fermion and boson statistics.

Abstract

A nonlinear Fokker-Planck equation is obtained in the continuous limit of a one-dimensional lattice with an energy landscape of wells and barriers. Interaction is possible among particles in the same energy well. A parameter $\gamma$, related to the barrier's heights, is introduced. Its value is determinant for the functional dependence of the mobility and diffusion coefficient on particle concentration, but has no influence on the equilibrium solution. A relation between the mean field potential and the microscopic interaction energy is derived. The results are illustrated with classical particles with interactions that reproduce fermion and boson statistics.