Diffusion in lattices with anisotropic scatterers

We study diffusion in lattices with periodic and random arrangements of anisotropic scatterers. We show, using both analytical techniques based upon our previous work on asymptotic properties of multistate random walks and computer calculation, that the diffusion constant for the random arrangement of scatterers is bounded above and below at an arbitrary density by the diffusion constant for an appropriately chosen periodic arrangement of scatterers at the same density. We also investigate the accuracy of the low-density expansion for the diffusion constant up to second order in the density for a lattice with randomly distributed anisotropic scatterers. Comparison of the analytical results with numerical calculations shows that the accuracy of the density expansion depends crucially on the degree of anisotropy of the scatterers. Finally, we discuss a monotonicity law for the diffusion constant with respect to variation of the transition rates, in analogy with the Rayleigh monotonicity law for the effective resistance of electric networks. As an immediate corollary we obtain that the diffusion constant, averaged over all realizations of the random arrangement of anisotropic scatterers at density, is a monotone function of the density.