Anomalous diffusion in one dimension

Abstract

In view of the interest in the occurrence of anomalous diffusion (<r²(t)>~t^2H, 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 x_ε(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 x_ε(t) ~ t^½ , and, subsequently, x_ε(t) ~ t^H , 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)> ~ t^2H; the diffusive spread of the initial condition is given by x_ε(t) ~ t^H; and the first passage time from the origin to the point x has a stable Levy distribution with an exponent equal to H.