Anomalous Diffusion on Fractal Networks

Abstract

This chapter is concerned with the problem of diffusion on fractal networks, which plays a central role when we investigate dynamical properties of fractals. The relation between vibrational excitations on fractal networks and diffusion is also discussed. We start by describing in detail the diffusion of random walkers on a percolating network. In uniform systems, the mean-squared displacement (r ² (t)) of a random walker is proportional to the time t, i.e., (r ² (t)) α t, for any Euclidean dimension d. How does (r ² (t)) behave in the case of fractal percolating networks? For this, de Gennes [5.1] posed the following problem called an ant in the labyrinth:

An ant parachutes down onto an occupied site of the infinite cluster of a percolating network. At every time unit, the ant makes one attempt to jump to one of its adjacent sites. If that site is occupied, it moves there. If it is empty, the ant stays at its original site. What is the ensemble-averaged squared distance that the ant travels in time t?