Transport properties on a random comb

We study the random walk of a particle in a random comb structure, both in the presence of a biasing field and an the field-free case. We show that the mean-field treatment of the quenched disorder can be exactly mapped on to a continuous time random walk (CTRW) on the backbone of the comb, with a definite waiting time density. We find an exact expression for this central quantity. The Green function for the CTRW is then obtained. Its first and second moments determine the drift and diffusion at all times. We show that the drift velocity v vanishes asymptotically for power-law and stretched-exponential distributions of branch lengths on the comb, whatever be the biasing field strength. For an exponential branch-length distribution, v is a nonmonotonic function of the bias, increasing initially to a maximum and then decreasing to zero at a critical value. In the field-free case, anomalous diffusion occurs for a range of power-law distributions of the branch length. The corresponding exponent for the mean square displacement is obtained, as is the asymptotic form of the positional probability distribution for the random walk. We show that normal diffusion occurs whenever the mean branch length is finite, and present a simple formula for the effective diffusion constant; these results are extended to regular (nonrandom) combs as well. The physical reason for anomalous drift or diffusion is traced to the properties of the distribution of a first passage time (on a finite chain) that controls the effective waiting time density of the CTRW.