Stochastic transport in disordered systems

Abstract

We develop a theory of stochastic transport in disordered media, starting from a linear master equation with random transition rates. A Green function formalism is employed to reduce the basic equation to a form suitable for the construction of a class of effective medium approximations (EMAs). The lowest order EMA, developed in detail here, corresponds to recent approximations proposed by Odagaki and Lax [Phys. Rev. B 24, 5284 (1981], Summerfield [Solid State Commun. 39, 401 (1981)], and Webman [Phys. Rev. Lett. 47, 1496 (1981)]. It yields an effective transition rate Wm which can be identified as the memory kernel of a generalized master equation, and used to define an associated continuous-time random walk on a uniform lattice. The long-time behavior of the mean-squared displacement arising from an initially localized state can be found from Wm, as can diffusion constants in any case where the long-time behavior of the system is diffusive. Detailed calculations are presented for seven lattice systems in one, two, and three dimensions, and for a variety of probability density functions f(w) for the transitions rates. For percolation-type densities, i.e., those with only a fraction p<1 of the bonds transmitting, the EMA predicts three distinct kinds of behavior: localization, ‘‘fractal’’ transport with slower than linear growth of the mean-squared displacement, and diffusion in the cases p<pc, p=pc, p≳pc, respectively, where pc is the bond percolation threshold of the lattice. Depending on the form of f(w) near w=0, critical exponents may take values independent of f(w) (‘‘universality’’) or heavily dependent on f(w) (‘‘nonuniversality’’).