Numerical study of diffusion on a random-mixed-bond lattice

Abstract

Diffusion on lattices with random mixed bonds in two and three dimensions is reconsidered using a random walk (RW) algorithm, which is equivalent to the master equation. In this numerical study the main focus is on the simple case of two different transition rates W(1),W(2) along bonds between sites. Although analysis of diffusion and transport on this type of disordered medium, especially for the case of one-bond pure percolation (i.e., W(1)=0 ), comprises a sizable subliterature, we exhibit additional basic results for the two-bond case: When the probability p of W(2) replacing W(1) in a lattice of W(1) bonds is below the percolation threshold p(c) , the mean square displacement r(2) is a nonlinear function of time t . A best fit to the lnr[(2) vs ln t plot is a straight line with the value of the slope varying with p,Delta,d , where Delta identical with W(2)/W(1) and d is the dimension, i.e., r(2) proportional, variant t(1+eta(p,Delta,d)) with eta>0 for Delta>1 . In other terms, all the diffusion (D identical with(r)(2)/2t proportional, variant t(eta)) is anomalous superdiffusion for p<p(c) and Delta>1 for d=2,3 . Previous work in the literature for d=2 with a different RW algorithm established an effective diffusion constant D(eff) , which was shown to scale as (p(c)-p)(1/2) . However, the anomalous nature (time dependence) of D(t) becomes manifest with an expanded regime of t , increased range of Delta , and the use of our algorithm. The nature of the superdiffusion is related to the percolation cluster geometry and Lévy walks.