Random walks on a fluctuating lattice

Abstract

In recent years, studies of diffusion in random media have been extended to include the effects of media in which the defects fluctuate randomly in time. Typically, the diffusive motion of particles in a static medium persists when the medium is allowed to fluctuate, with the diffusivity (diffusion constant) D depending on the character of the fluctuations. In the present work, we study random walks on lattices in which the bonds connecting vertices open and close randomly in time, and the walker is not allowed to cross a closed bond. Variations of the model studied here have been used to model the diffusion of CO through myoglobin, the transport of ions in polymer solutions, and conduction in hydrogenated amorphous silicon. The major objective in analyzing these systems is to find efficient methods for computing the diffusivity. In this dissertation, we focus mainly on methods of computing the diffusivity in our model. In addition, we study the critical behavior of the model and present a demonstration, valid for a restricted range of model parameters, that the distribution of the displacement converges in time to a Gaussian with width D. To compute the diffusivity, we use a numerical renormalization group (RG) method, power series expansions in model parameters, and Monte Carlo simulations. We choose a model with two parameters characterizing the bond fluctuations--the time scale of fluctuations tau and the mean open-bond density p. We calculate a series expansion of the diffusivity to about 10th order in the parameter nu = exp(.1/τ) on the hypercubic lattice Zᵈ for d = 1, 2, 3, as well as on the Bethe lattice. We compute the same power series expansion to 3rd order in ν for arbitrary d. We compute estimates of the diffusivity on the Bethe lattice using the RG methods and show by comparison to Monte Carlo data that the RG provides excellent quantitative predictions of D when τ is not too large.