Diffusion on a lattice: transition rates, interactions and memory effects

TL;DR

This paper studies particle diffusion on a 2D lattice considering interactions and memory effects, deriving formulas for transition rates and diffusivity, and applying them to soft core and extended hard core interactions.

Contribution

It introduces a new analytical framework linking transition rates, chemical potential, and memory effects in lattice diffusion with interactions.

Findings

Derived an expression for the correlation factor $f$ accounting for memory effects.

Provided formulas for transition rates based on excess chemical potential.

Applied the theory to soft core and extended hard core interactions.

Abstract

We analyze diffusion of particles on a two dimensional square lattice. Each lattice site contains an arbitrary number of particles. Interactions affect particles only in the same site, and are macroscopically represented by the excess chemical potential. In a recent work, a general expression for transition rates between neighboring cells as functions of the excess chemical potential was derived. With transition rates, the mean field tracer diffusivity, $D^\text{MF}$, is immediately obtained. The tracer diffusivity, $D = D^\text{MF} f$, contains the correlation factor $f$, representing memory effects. An analysis of the joint probability of having given numbers of particles at different sites when a force is applied to a tagged particle allows an approximate expression for $f$ to be derived. The expression is applied to soft core interaction (different values for the maximum number of particles in a site are considered) and extended hard core.