Effective diffusivity of Brownian particles in a two dimensional square lattice of hard disks

TL;DR

This paper analyzes the effective diffusion constant of Brownian particles in a 2D square lattice of hard disks, deriving series expansions and approximations to describe behavior across all volume fractions.

Contribution

It introduces a Green's function approach and a variant of the Fick-Jacobs approximation to comprehensively model diffusion in the lattice.

Findings

Series expansion of diffusion constant in terms of disk volume fraction

Effective diffusion constant behavior accurately described for all volume fractions

New analytical methods connect diffusion and conductivity in the lattice

Abstract

We revisit the classic problem of the effective diffusion constant of a Brownian particle in a square lattice of reflecting impenetrable hard disks. This diffusion constant is also related to the effective conductivity of non-conducting and infinitely conductive disks in the same geometry. We show how a recently derived Green's function for the periodic lattice can be exploited to derive a series expansion of the diffusion constant in terms of the disk's volume fraction $\varphi$. Secondly we propose a variant of the Fick-Jacobs approximation to study the large volume fraction limit. This combination of analytical results is shown to describe the behavior of the diffusion constant for all volume fractions.