Random walk through fractal environments

Abstract

We analyze random walk through fractal environments, embedded in 3-dimensional, permeable space. Particles travel freely and are scattered off into random directions when they hit the fractal. The statistical distribution of the flight increments (i.e. of the displacements between two consecutive hittings) is analytically derived from a common, practical definition of fractal dimension, and it turns out to approximate quite well a power-law in the case where the dimension D of the fractal is less than 2, there is though always a finite rate of unaffected escape. Random walks through fractal sets with D less or equal 2 can thus be considered as defective Levy walks. The distribution of jump increments for D > 2 is decaying exponentially. The diffusive behavior of the random walk is analyzed in the frame of continuous time random walk, which we generalize to include the case of defective distributions of walk-increments. It is shown that the particles undergo anomalous, enhanced diffusion for D_F < 2, the diffusion is dominated by the finite escape rate. Diffusion for D_F > 2 is normal for large times, enhanced though for small and intermediate times. In particular, it follows that fractals generated by a particular class of self-organized criticality (SOC) models give rise to enhanced diffusion. The analytical results are illustrated by Monte-Carlo simulations.