TRANSPORT-PROPERTIES ON A RANDOM COMB

We study the random walk of a particle in a random comb structure, bot h in the presence of a biasing field and in the field-free case. We sh ow that the mean-field treatment of the quenched disorder can be exact ly mapped on to a continuous time random walk (CTRW) on the backbone o f the comb, with a definite waiting time density. We find an exact exp ression for this central quantity. The Green function for the CTRW is then obtained, Its first and second moments determine the drift and di ffusion at all times. We show that the drift velocity v vanishes asymp totically for power-law and stretched-exponential distributions of bra nch lengths on the comb, whatever be the biasing field strength. For a n 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, T he 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 th e mean branch length is finite, and present a simple formula for the e ffective diffusion constant; these results are extended to regular (no nrandom) combs as well, The physical reason for anomalous drift or dif fusion is traced to the properties of the distribution of a first pass age time (on a finite chain) that controls the effective waiting time density of the CTRW.