Efficient numerical simulation of ac conduction in heterogeneous materials at low temperatures

Abstract

The problem of computing the effective frequency-dependent conductivity of heterogeneous materials at low temperatures is studied. In this problem the activation energies and, therefore, the local conductivities (or the transition rates in a master equation formulation) are broadly distributed, varying over many orders of magnitude. Such broad variations make the computations with large lattices that represent the materials very difficult. We use an efficient method, based on computing the wavelet scale and detail coefficients of the local conductivities, in order to compute the effective ac conductivity of such materials. The method identifies the high-conductance paths in a large lattice and reduces it to one that requires far less computation. Using the method, we compute the effective ac conductivity of a two-dimensional lattice in which the activation energies are distributed according to a probability distribution function (PDF). Five distinct PDFs are used, and the effective ac conductivity is computed over many orders of magnitude variations in the frequency. Depending on the size of the initial system, the speedup in the computations for two-dimensional systems varies anywhere from a factor of 35--40 to over 200.