Anomalous diffusion and charge relaxation on comb model: exact solutions

The random walks on the comb structure are considered. It is shown that due to fingers a diffusion has an anomalous character, that is an r.m.s. displacement depends on time by a power way with exponent (1)/(2). The generalized diffusion equation for an anomalous case is deduced. It essentially differs from a usual diffusion equation in the continuity equation form: instead of the first time derivative, the time derivative of fractal order (1)/(2) appears. In the second part the charge relaxation on the comb structure is studied. A non-Maxwell character is established. The reason is that the electric field has three components, but a charge may relax only along some conducting lines. (C) 2000 Elsevier Science B.V. All rights reserved.