Anomalous diffusion in the nonasymptotic regime.

Abstract

We analyze some properties of the one-dimensional Lévy flights, assuming that the one-step transition rates depend on the flight length x as p(alpha)(x) equivalent to x(-(alpha+2)). For flights on a finite, (2M+1)-site lattice, we can define an effective, size-dependent, diffusion coefficient D(alpha)(M) equivalent to [M(1-alpha) - 1]/(1 - alpha) if alpha < 1, with D1(M) equivalent to ln(M). Using the generalization of statistical mechanics given by Tsallis, we show that for flights on infinite systems, the generalized displacement moments <x(R)> are well defined provided that alpha > R - 3. These moments exhibit a power-law singularity if alpha --> 1(-) and R>2/3. The short- and intermediate-time properties of the generalized mean-square displacement are then studied numerically. This work suggests the conditions under which the asymptotic analytical formulas (obtained in the literature by the use of the generalized central limit theorem) could be applied to finite-time experiments. These formulas should work much better if alpha is close to zero than in the alpha -->1(-) neighborhood.