Anomalous diffusion in one dimension

In view of the interest in the occurrence of anomalous diffusion (〈r2(t)〉 ∼ t2H, 0 < H < 1 2) 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) ∼ t 1 2, 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 = 1 2) 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 Lévy distribution with an exponent equal to H. © 1985.