In previous chapters, we have seen examples of anomalous diffusion in the CTRW model, in Lévy flights, and in long-range correlated walks. Diffusion in fractal lattices is also anomalous. Here we consider nearest-neighbor random walks in the Sierpinski gasket, for which an exact solution is possible. We analyze the problem and solve it following several different approaches. The analysis not only illustrates anomalous diffusion in a simple way, but also stresses important aspects of diffusion theory, such as the relation to conductivity and elasticity.

Anomalous diffusion

Imagine a random walk in the Sierpinski gasket. At each step the walker moves randomly to one of the four nearest-neighbor sites on the gasket, with equal probabilities. We require the mean-square displacement after n steps, 〈r2(n)〉.

Naively, one should think that, since diffusion is regular (〈r2(n)〉 ∼ n) in all integer dimensions, so would be the case for fractals, since fractals may be regarded as mere extrapolations of regular space to noninteger dimensions. Surprisingly, this turns out to be wrong! Perhaps the best way to convince oneself of this fact is to perform numerical simulations of random walks on the Sierpinski gasket. The technique is described in Appendix A.

In Fig. 5.1 we show results obtained from exact enumeration. In fractals, however, the different sites are not equivalent. Regard the Sierpinski gasket as embedded in a regular two-dimensional triangular lattice. At each node of the gasket two bonds of the embedding triangular lattice are missing, but which bonds are missing varies from one gasket site to the next.