Strongly and weakly self-similar diffusion
(Ferrari, Raffaele and Manfroi, Aldo J. and Young, William R.), Physica D: Nonlinear Phenomena, vol. 154, no. 1-2, pp. pages, 2001.
Many dispersive processes have moments of displacements with large-t behavior ???|x|$backslash$n p??? ??? t$backslash$n ??p. The study of ??$backslash$n p as a function of p provides a more complete characterization of the process than does the single number ??$backslash$n 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 ??$backslash$n p is a linear function of p (strong self-similarity). But if the tails are important then ??$backslash$n 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 ??$backslash$n 2 may be determined by either the core or the tails. As representatives of a large class of dispersive processes for which ??$backslash$n 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|$backslash$n p??? from the Laplace-Fourier representation of the concentration. In the case of the generalized telegraph model, this formula provides analytic expressions for ??$backslash$n p. ?? 2001 Elsevier Science B.V.
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