Diffusion on two-dimensional percolation clusters: Influence of cluster anisotropy

This paper investigates the influence of geometric anisotropy of two-dimensional percolation clusters on diffusion. Monte Carlo simulation indicates that the ensemble of all random walks starting from any given origin is typically anisotropic as determined by the local geometry of the cluster. Single-particle (or tracer) diffusion is described by the position-correlation matrix (PCM). A mean-field equation is developed for the PCM of the random walker in terms of the moment of inertia tensor of the cluster as a function of the chemical distance from the origin. In contrast, the ensemble of all random walks starting from all possible origins, weighted by the stationary probabilities, leads to essentially isotropic diffusion due to self-averaging over many regions within the cluster. The diffusion of an ensemble of particles in the latter case may be characterized by the velocity-correlation matrix (VCM). It is proved that the VCM is symmetric when a detailed balance condition holds and stationary initial conditions are used. An evolution equation for the VCM is constructed using a generalized Langevin equation with a power-law friction kernel. The kernel is a phenomenological characterization of the diffusion process in a fractal medium.