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

TL;DR

This paper introduces a mathematical model for anomalous diffusion based on a generalized Fisher's information measure derived from a non-standard entropy function, leading to a differential equation that describes superdiffusive and subdiffusive behaviors.

Contribution

It develops a new Fisher information measure from a generalized entropy and derives a differential equation modeling anomalous diffusion with explicit q-dependent solutions.

Findings

The model generalizes classical diffusion equations to include anomalous diffusion behaviors.

The mean squared displacement scales as t^{1/q} with a q-dependent constant.

Solutions encompass superdiffusive and subdiffusive regimes based on q values.

Abstract

Starting with the relative entropy based on a previously proposed entropy function $S_q[p]=\int dx p(x)(-\ln p(x))^q$, 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 $q\to 1$ 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 $<x^2>\sim K^{1/q}t^{1/q}$, where the parameter $q$ takes values $q=\frac{2n-1}{2n+1}$ (superdiffusion) and $q=\frac{2n+1}{2n-1}$ (subdiffusion), $\forall n\geq 1$.