Diffusion on Fractals — Efficient algorithms to compute the random walk dimension

Abstract

Abstract Self-similar fractals are used as a simple model for porous media in order to describe diffusive processes. The diffusion or Brownian motion of particles on a fractal is approximated by random walks on pre-fractals. Since there are a lot of holes in the fractal, where a random walker is not allowed to move in, the mean square displacement scales with time t asymptotically as t2/dw, where the random walk dimension dw is usually greater than 2. This dimension is an important quantity to characterize diffusion properties. In this paper three efficient methods to calculate the random walk dimension of finitely ramified Sierpinski carpets are presented: First a simulation of random walks on pre-carpets, where an efficient storing scheme decreases the needed amount of memory and speeds up the computation. Secondly we iterate the master equation describing the time evolution of the probability distribution. Thirdly a resistance scaling algorithm is presented which yields a resistance scaling exponent. This exponent is related to the random walk dimension via the Einstein relation, using analogies between random walks on graphs and resistor networks.