Diffusion on lattices having random mixed bonds, in two or three dimensions, was reconsidered by using a random-walk algorithm which was equivalent to the master equation. The main focus was on the simple case of two different transition rates, W1, W2, along the bonds between sites. The analysis of diffusion and transport on this type of disordered medium, especially for the case of one-bond pure percolation (W1 = 0) was well known, but additional basic results for the two-bond case were obtained here. When the probability, p, of W2 replacing W1 in a lattice of W1-bonds was below the percolation threshold, pc, the mean square displacement, <r2>, was a non-linear function of time, t. A best fit to the ln[<r2>] versus ln[t] plot was a straight line; with a slope that depended upon p, Δ and d, where Δ ≡ W2/W1 and d was the dimension. That is, <r2> was proportional to t1+η(p,Δ,d), with η > 0 for Δ > 1. In other terms, all of the diffusion (D ≡ <r2>/2t proportional to tη) was anomalous super-diffusion for p < pc and Δ > 1 for d = 2, 3. Previous published work for d = 2, with a different random-walk algorithm, had established an effective diffusion constant, Deff, which was shown to scale as (pc - p)1/2. However, the anomalous nature (time-dependence) of D(t) became apparent with increased t, increased range of Δ, and the use of the present algorithm. The nature of the super-diffusion was related to the percolation-cluster geometry and Lévy walks.

Numerical Study of Diffusion on a Random-Mixed-Bond Lattice. D.Holder, H.Scher, B.Berkowitz: Physical Review E, 2008, 77[3], 031119