Solvable models of temporally correlated random walk on a lattice

Abstract

We seek the conditional probability functionP(m,t) for the position of a particle executing a random walk on a lattice, governed by the distribution W(n, t) specifying the probability of n jumps or steps occurring in timet. Uncorrelated diffusion occurs when W is a Poisson distribution. The solutions corresponding to two different families of distributions W are found and discussed. The Poissonian is a limiting case in each of these families. This permits a quantitative investigation of the effects, on the diffusion process, of varying degrees of temporal correlation in the step sequences. In the first part, the step sequences are regarded as realizations of an ongoing renewal process with a probability density ψ(t) for the time interval between successive jumps.W is constructed in terms of ψ using the continuous-time random walk approach. The theory is then specialized to the case when ψ belongs to the class of special Erlangian density functions. In the second part,W is taken to belong to the family of negative binomial distributions, ranging from the geometric (most correlated) to the Poissonian (uncorrelated). Various aspects such as the continuum limit, the master equation for P, the asymptotic behaviour of P, etc., are discussed.