Diffusion in a one-dimensional lattice with random asymmetric transition rates

Abstract

The authors study one-dimensional continuous-time random walks for which the pairs (W_n⁺, W_n+1^-) of nearest-neighbour transition rates are assumed to be independent, equally distributed random variables. The long-time asymptotic behaviour of the mean displacement, (x(t)), is determined exactly for a specific model system in which 'diodes' (u, 0) and 'two-way bonds' ( lambda v, v) occur with probabilities p and 1-p, respectively. For lambda nu F( beta ^-1ln t), where nu =ln(1-p)/ln lambda and beta =ln lambda , and where F is a periodic function with period 1. The mean displacement thus not only increases slower than linearly in time, but exhibits superimposed, non-decaying oscillations.