Strongly and weakly self-similar diffusion

Abstract

Many dispersive processes have moments of displacements with large- t behavior 〈∣ x∣ p 〉∼ t γ p . The study of γ p as a function of p provides a more complete characterization of the process than does the single number γ 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 γ p is a linear function of p (strong self-similarity). But if the tails are important then γ 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 γ 2 may be determined by either the core or the tails. As representatives of a large class of dispersive processes for which γ 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 γ p .