A simple mathematical model for anomalous diffusion via Fisher's information theory

Starting with the relative entropy based on a previously proposed entropy function S-q vertical bar P vertical bar = integral dx p(x) x (-In p(x))(q), we find the corresponding Fisher's information measure. After function redefinition we then maximize the Fisher information measure with respect to the new function and obtain a differential operator that reduces to a space coordinate second derivative in the q 1 limit. We then propose a simple differential equation for anomalous diffusion and show that its solutions are a generalization or the functions in the Barenblatt-Pattie solution. We find that the mean squared displacement, Lip to a q-dependent constant, has a time dependence according to < x(2)> similar to K(1/q)t(1/q), where the parameter q takes values q = 2n-1/sn+1 (superdiffusion) and q = 2n+1/2n-1 (subdiffusion), for all n >= 1. (C) 2009 Elsevier B.V. All rights reserved.