Analytic study of a model of diffusion on a random comblike structure

Abstract

We present an analytic study of a model of diffusion on a random comblike structure in which no bias field does exist neither along the backbone nor along the branches. For any given disordered structure, our analytic treatment allows to compute in an exact manner 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). The behaviours strongly depend on the distribution of the lengths of the branches. With an exponential distribution, transport is normal, while anomalous diffusion may take place for a power law distribution (when long branches are present with a sufficiently high weight).