Application of the Widom insertion formula to transition rates in a lattice

TL;DR

This paper derives an expression for particle transition rates on a lattice using Widom's insertion formula, linking microscopic interactions to macroscopic diffusion properties in a mean-field regime.

Contribution

It introduces a novel approach combining detailed balance and Widom's insertion formula to calculate transition rates based on excess chemical potential.

Findings

Transition rates depend inversely on the thermodynamic factor.

The method applies to soft-core interactions and various excess chemical potential forms.

The approach reproduces known diffusion behaviors in lattice systems.

Abstract

We consider diffusion of particles on a lattice in the so-called dynamical mean-field regime (memory effects are neglected). Interactions are local, that is, only among particles at the same lattice site. It is shown that a statistical mechanics analysis that combines detailed balance and Widom's insertion formula allows for the derivation of an expression for transition rates in terms of the excess chemical potential. The rates reproduce the known dependence of self-diffusivity as the inverse of the thermodynamic factor. Soft-core interactions and general forms of the excess chemical potential (linear, quadratic, and cubic with the density) are considered.