DIFFUSION IN LAYERED RANDOM FLOWS, POLYMERS, ELECTRONS IN RANDOM POTENTIALS, AND SPIN DEPOLARIZATION IN RANDOM-FIELDS

Several related models are studied in a common framework. We first reconsider the model of Matheron and de Marsilly for (anomalous) tracer dispersion in a stratified porous medium. In each horizontal layer the flow velocity is constant, parallel to the layer, and depends randomly on the vertical coordinate z. This model is mapped onto a d = 1 localization problem in a random potential and, equivalently, onto a d = 1 polymer. At large t the averaged distribution of horizontal displacements x takes the scaling form [P(x,t,z = 0)] = at-5/4Q(bxt-3/4), where Q(y) is independent of the details of the model. Q(y), a, and b are obtained exactly for a large class of models. From the Lifschitz tails of the localization problem we find in the region x much greater than t3/4, i.e., y --> infinity, that Q(y) approximately Absolute value of y exp(- C Absolute value of y4/3). We also obtain exactly in d = 1 the scaling functions for the local and total average magnetization of spins diffusing in a random magnetic field, by mapping onto a polymer problem, as well as the average local concentration for diffusion in the presence of random sources and sinks. These mappings are then used to study higher-dimensional extensions of these models.