The nonextensive entropy approach versus the stochastic in describing subdiffusion

TL;DR

This paper introduces a stochastic interpretation of subdiffusion using Sharma-Mittal entropy, linking it to nonlinear equations and Langevin dynamics with memory effects, and explores parameter relationships.

Contribution

It presents a novel stochastic framework connecting Sharma-Mittal entropy to subdiffusion and Langevin equations with memory effects, including parameter relationships.

Findings

Solution equivalence between Sharma-Mittal and Gauss entropy cases

External noise modeled by Gamma distribution influences subdiffusion coefficient

Parameters q and r relate to alpha and mean of the noise distribution

Abstract

We have proposed a new stochastic interpretation of the sudiffusion described by the Sharma-Mittal entropy formalism which generates a nonlinear subdiffusion equation with natural order derivatives. We have shown that the solution to the diffusion equation generated by Gauss entropy (which is the particular case of Sharma-Mittal entropy) is the same as the solution of the Fokker-Planck (FP) equation generated by the Langevin generalised equation where the `long memory effect' is taken into account. The external noise which pertubates the subdiffusion coefficient (occuring in the solution of FP equation) according to the formula $D_\alpha\rightarrow D_\alpha/u$ where $u$ is a random variable described by the Gamma distribution, provides us with solutions of equations obtained from Sharma-Mittal entropy. We have also shown that the parameters $q$ and $r$ occuring in Sharma-Mittal entropy are controlled by the parameters $\alpha$ and $<u>$, respectively.