Nonvalidity of the telegrapher's diffusion equation in two and three dimensionsfor crystalline solids

Abstract

We use a classical analog of two-dimensional (2D) and 3D quantum S-matrix scattering theory to study classical mesoscopic diffusion in isotropic, crystalline, solids. The individual collisions include transmission, reflection, and lateral scattering probabilities. The resulting stochastic process is a second-order Markov process in phase space, which is known in the literature as 2D (3D) persistent random walk. In striking contrast with the 1D case, in the continuum limit, the 2D and 3D total densities do not satisfy the telegrapher's diffusion equation. We explain this fact deriving the anomalous Maxwell-Cattaneo equation in the case of discrete diffusion processes. We find that inertial memory, giving the forward scattering a preferential direction, breaks the x-y symmetry.