EFFECTIVE-MEDIUM APPROXIMATION FOR RANDOM-WALKS WITH NON-MARKOVIAN DYNAMICAL DISORDER

We analyze a random walk, in which a walker makes transitions between sites in a lattice with rates whose values are determined by fluctuati ng medium variables. The values of these medium variables fluctuate be tween zero (blocked jump) and a nonzero value (permitted jump). The dy namics of the walker are determined by the fluctuations of these mediu m variables, but the medium variables are unaffected by the motion of the walker. The case in which the medium variables obey conventional r ate equations (a Pauli master equation) has been analyzed by Harrison and Zwanzig [Phys. Rev. A 32, 1072 (1985)] and by Sahimi [J. Phys. C 1 9, 1311 (1986)], who developed an effective medium approximation (EMA) solution for this model. We consider here a generalization of this mo del, in which the medium variables are non-Markovian, with dynamics go verned by a generalized master equation containing a memory kernel. A generalized EMA is developed for this case, whose accuracy is tested n umerically for a ''lattice'' of two sites. Calculations based on this approach are then presented for a lattice of infinite extent.