Showing papers by "Kurt E. Shuler published in 1990"
TL;DR: In this paper, the authors study diffusion in lattices with periodic and random arrangements of anisotropic scatterers and show that the diffusion constant is bounded above and below at an arbitrary densityρ by the diffusion constants for an appropriately chosen periodic arrangement of scatterer at the same density.
Abstract: We study diffusion in lattices with periodic and random arrangements of anisotropic scatterers. We show, using both analytical techniques based upon our previous work on asymptotic properties of multistate random walks and computer calculation, that the diffusion constant for the random arrangement of scatterers is bounded above and below at an arbitrary densityρ by the diffusion constant for an appropriately chosen periodic arrangement of scatterers at the same density. We also investigate the accuracy of the low-density expansion for the diffusion constant up to second order in the density for a lattice with randomly distributed anisotropic scatterers. Comparison of the analytical results with numerical calculations shows that the accuracy of the density expansion depends crucially on the degree of anisotropy of the scatterers. Finally, we discuss a monotonicity law for the diffusion constant with respect to variation of the transition rates, in analogy with the Rayleigh monotonicity law for the effective resistance of electric networks. As an immediate corollary we obtain that the diffusion constant, averaged over all realizations of the random arrangement of anisotropic scatterers at densityρ, is a monotone function of the density.
1 citations
TL;DR: In this article, the authors show that the presence of a symmetric field can change this result, both in making the degree of asymmetry depend on time, and driving the random walk toward a more symmetric state.
Abstract: Any ensemble of random walks with symmetric transition probabilities will have symmetric properties. However, any single realization of such a random walk may be asymmetric. In an earlier paper, Weiss and Weissman developed a measure of asymmetry and applied it to random walks in the absence of a field, showing that the degree of asymmetry (in the diffusion limit) is independent of time and that the most probable degree of asymmetry corresponds to the maximum possible. We show in the present paper how the presence of a symmetric field can change this result, both in making the degree of asymmetry depend on time, and driving the random walk toward a more symmetric state.
1 citations