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

Abstract

Starting with the relative entropy based on a previously proposed entropy function S q [ p ] = ∫ d x p ( x ) × ( − ln 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 of the functions in the Barenblatt–Pattle solution. We find that the mean squared displacement, up to a q-dependent constant, has a time dependence according to 〈 x 2 〉 ∼ K 1 / q t 1 / q , where the parameter q takes values q = 2 n − 1 2 n + 1 (superdiffusion) and q = 2 n + 1 2 n − 1 (subdiffusion), ∀ n ⩾ 1 .