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

M. Mangeat (1) , Thomas Guérin (1) , D. S Dean (1)

1 LOMA - Laboratoire Ondes et Matière d'Aquitaine

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. 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.

Domaines: Mécanique statistique [cond-mat.stat-mech]

M. Mangeat, Thomas Guérin, D. S Dean. Effective diffusivity of Brownian particles in a two dimensional square lattice of hard disks. Journal of Chemical Physics, 2020, 152 (23), pp.234109. ⟨10.1063/5.0009095⟩. ⟨hal-03045638⟩