Probability Density of Random Walks on Random Fractals: Stretched Gaussians and Multifractal Features

Abstract

The question how the classical laws of diffusion are changed on fractal structures and in particular on percolation clusters, has attracted much attention in the last years [1–3]. A central role plays the configurational average of the probability density 〈P(r, t)〉of a random walker, which gives the probability to find the random walker at time t at a distance r from its starting point at t = 0. Its form characterizes the localization of diffusion on fractal structures, and is relevant to several other physical problems of interest such as quantum localization [4] or self-avoiding random walks on fractals [1,5]. From 〈P(r, t)〉 the diffusion constant and the conductivity can be obtained [1], while its Fourier transform represents the scattering function which is also experimentally accessible.