The harmonic mean renormalises random diffusion across a spatial multigrid

Abstract

Most methods for modelling dynamics posit just two time scales: a fast and a slow scale. But many applications, such as the diffusion in a random media considered here, possess a wide variety of space-time scales. Consider the microscale diffusion on a one dimensional lattice with arbitrary diffusion coefficients between adjacent lattice points. I develop a slow manifold approach to model the diffusion, with some rigorous support, on a lattice that is coarser by a factor of four: the coarser scale effective diffusion coefficients are the harmonic mean of fine scale coefficients. Then iterating the analytic mapping of random diffusion from the finer grid to the coarser grid generates a hierarchy of models on a spatial multigrid across a wide range of space-time scales, all with rigorous support. The one step harmonic mean renormalises to harmonic means for the effective diffusion coefficients across the entire hierarchy.