Abstract
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 ad=1 localization problem in a random potential and, equivalently, onto ad=1 polymer. At larget theaveraged distribution of horizontal displacementsx takes the scaling form [P(x, t, z=0)]=at −5/4 Q(bxt −3/4), whereQ(y) is independent of the details of the model.Q(y),a, andb are obtained exactly for a large class of models. From the Lifschitz tails of the localization problem we find in the regionx≫t 3/4, i.e.,y→∞, thatQ(y)∼¦y¦ exp(−C¦y¦4/3). We also obtain exactly ind=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.
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Le Doussal, P. Diffusion in layered random flows, polymers, electrons in random potentials, and spin depolarization in random fields. J Stat Phys 69, 917–954 (1992). https://doi.org/10.1007/BF01058756
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DOI: https://doi.org/10.1007/BF01058756