Strongly and weakly self-similar diffusion

Many dispersive processes have moments of displacements with large-t behavior [\x\(p)] similar to t(gammap). The study of gamma (p) as a function of p provides a more complete characterization of the process than does the single number gamma (2). Also at long times, the core of the concentration relaxes to a self-similar profile, while the large-x tails, consisting of particles which have experienced exceptional displacements, are not self-similar. Depending on the particular process, the effect of the tails can be negligible and then gamma (p) is a linear function of p (strong self-similarity). But if the tails are important then gamma (p) is a non-trivial function of p (weak self-similarity). In the weakly self-similar case, the low moments are determined by the self-similar core, while the high moments are determined by the non-self-similar tails. The popular exponent gamma (2) may be determined by either the core or the tails. As representatives of a large class of dispersive processes for which gamma (p), is a piecewise-linear function of p, we study two systems: a stochastic model, the "generalized telegraph model", and a deterministic area-preserving map, the "kicked Harper map". We also introduce a formula which enables one to obtain the moment [\x\(p)] from the Laplace-Fourier representation of the concentration. In the case of the generalized telegraph model, this formula provides analytic expressions for gamma (p). (C) 2001 Elsevier Science B.V. All rights reserved.