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 $\text{ln}⟨r^2⟩$ vs $\text{ln}t$ plot is a straight line with the value of the slope varying with $p,\Delta ,d$, where $\Delta \equiv W_2/W_1$ and $d$ is the dimension, i.e., $⟨r^2⟩\propto t^{1+\eta \left(p,\Delta ,d\right)}$ with $\eta >0$ for $\Delta >1$. In other terms, all the diffusion $\left(\mathcal{D}\equiv ⟨r^2⟩/2t\propto t^{\eta }\right)$ 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 ${\mathcal{D}}_{\text{eff}}$, which was shown to scale as ${\left(p_c-p\right)}^{1/2}$. However, the anomalous nature (time dependence) of $\mathcal{D}\left(t\right)$ 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.

• Received 5 November 2007

DOI:https://doi.org/10.1103/PhysRevE.77.031119