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

Starting with the relative entropy based on a previously proposed entropy function Sq[p]=∫dxp(x)×(−lnp(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→1q→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 〈x2〉∼K1/qt1/q〈x2〉∼K1/qt1/q, where the parameter q   takes values q=2n−12n+1 (superdiffusion) and q=2n+12n−1 (subdiffusion), ∀n⩾1∀n⩾1.