Anomalous transport in lattice and continuum percolating systems

Abstract

Anomalous diffusion is studied on a certain class of percolation models in which the diffusion properties arise both from the underlying network and from the singular distribution of transfer rates. Scaling arguments parallel to those of Kogut and Straley lead to nonuniversal modifications of the dynamical exponents. The scaling results are demonstrated in a Sierpiński honeycomb and in the effective-medium theory and explicit values for the exponents are obtained. The diffusion-coefficient exponent and the spectral dimension are found to be 2/(1-a) and 2(1-a)/(2-a), respectively, (0<a<1) in the effective-medium theory, with values being significantly different from the corresponding ones in ordinary percolation. As a concrete example, the frequency-dependent conductivity has been calculated for a three-dimensional cubic network and several interesting crossover behaviors are obtained.