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 φ. Second, 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.

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