Analytic study of a model of diffusion on random comb-like structures

Abstract

An analytic study of a model of diffusion on random comb-like structures in which a bias field may or not exist along the backbone is presented. First, when no bias is present, the method allows to compute in an exact manner, for any given disordered structure, the asymptotic behaviour at large time of the probability of presence of the particle at its initial site and on the backbone, and of the particle position and dispersion. The expressions of these quantities are shown to coincide asymptotically with those derived in simple «mean-field» treatments. The results for any given sample do not depend on the particular configuration (self-averaging property). When a bias field is present along the backbone, one can also compute directly in an exact manner the asymptotic behaviour at large time of the disorder average of the probability of presence of the particle at its initial site. As for the particle position and dispersion, they can be computed in a periodized system of arbitrary periodN. The corresponding quantities for the random system can then be obtained by taking the limitN→∞. As a result, in both cases the behaviours strongly depend on the distribution of the lengths of the branches. With an exponential distribution transport is normal while anomalous drift and diffusion may take place for a power law distribution (when long branches are present with sufficiently high weights).