Diffusion on fractals
The issue of diffusion on fractals is addressed with stress on the question of how to generalize the diffusion equation for Euclidean lattices to the case of fractal lattices. Such an equation is proposed on the basis of scaling arguments. The solutions of this equation are interpreted as analytic envelopes of the exact probability distribution P(r,t) which is far from smooth. In fact it is shown that P(r,t) has discontinuities which appear self-similarly on all length scales. The exactly soluble examples of Sierpi\ifmmode \acute{n}\else \'{n}\fi{}ski gaskets embedded in general dimension d are analyzed. The predstudied by assuming that the sticking probability of the particles depends on the local orientation of the interface. Directional solidification is simulated by the deposition of particles undergoing biased random walks. Linearly stable patterns are generated if the basic features of the directional solidification experiments are taken into account. The resulting patterns are very similar to those observed experimentally.