Anomalous diffusion in one dimension

Balakrishnan, V. (1985) Anomalous diffusion in one dimension Physica A: Statistical Mechanics and its Applications, 132 (2-3). pp. 569-580. ISSN 0378-4371

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In view of the interest in the occurrence of anomalous diffusion (<r2(t)>~t2H, 0 < H < ½) in several physical circumstances, we study anomalous diffusion per se in terms of exactly solvable one-dimensional models. The basic idea is to exploit the fact that temporal correlations lead directly to anomalous diffusion, and provide solvable analogues of more realistic physical situations. We first derive a general equation for a deterministic trajectory xε(t) that comprehensively characterizes the diffusive motion, by finding the ε-quantiles of the time-dependent probability distribution. The class of all diffusion processes (or, equivalently, symmetric random walks) for which xε(t) ~ t½ , and, subsequently, xε(t) ~ tH , is identified. Explicit solutions are presented for families of such processes. Considering random walks whose step sequences in time are governed by renewal processes, and proceeding to the continuum limit, a true generalization of Brownian motion (the latter corresponds to the limiting value H= ½ ) is obtained explicitly: <x2(t)> ~ t2H; the diffusive spread of the initial condition is given by xε(t) ~ tH; and the first passage time from the origin to the point x has a stable Levy distribution with an exponent equal to H.

Item Type:Article
Source:Copyright of this article belongs to European Physical Society.
ID Code:1092
Deposited On:27 Sep 2010 04:31
Last Modified:11 May 2011 12:22

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