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 fluctuating medium variables. The values of these medium variables fluctuate between zero (blocked jump) and a nonzero value (permitted jump). The dynamics of the walker are determined by the fluctuations of these medium variables, but the medium variables are unaffected by the motion of the walker. The case in which the medium variables obey conventional rate equations (a Pauli master equation) has been analyzed by Harrison and Zwanzig [Phys. Rev. A 32, 1072 (1985)] and by Sahimi [J. Phys. C 19, 1311 (1986)], who developed an effective medium approximation (EMA) solution for this model. We consider here a generalization of this model, in which the medium variables are non-Markovian, with dynamics governed by a generalized master equation containing a memory kernel. A generalized EMA is developed for this case, whose accuracy is tested numerically for a ''lattice'' of two sites. Calculations based on this approach are then presented for a lattice of infinite extent.