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

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