Two types of $q$-Gaussian distributions used to study the diffusion in a finite region

TL;DR

This paper investigates two types of $q$-Gaussian distributions, including a newly defined one, to model diffusion processes in finite regions, deriving related equations with position- and time-dependent coefficients and analyzing their properties.

Contribution

It introduces a new $q$-Gaussian distribution, derives a $q$-deformed diffusion equation with variable coefficients, and explores discrete versions of the diffusion-decay process.

Findings

Standard results recovered as $q$ approaches zero

Derived a linear Fokker-Planck equation with variable diffusion coefficient

Formulated a discrete diffusion-decay equation with non-uniform spatial intervals

Abstract

In this work, we explore both the ordinary $q$-Gaussian distribution and a new one defined here, determining both their mean and variance, and we use them to construct solutions of the $q$-deformed diffusion differential equation. This approach allows us to realize that the standard deviation of the distribution must be a function of time. In one case, we derive a linear Fokker-Planck equation within a finite region, revealing a new form of both the position- and time-dependent diffusion coefficient and the corresponding continuity equation. It is noteworthy that, in both cases, the conventional result is obtained when $q$ tends to zero. Furthermore, we derive the deformed diffusion-decay equation in a finite region, also determining the position- and time-dependent decay coefficient. A discrete version of this diffusion-decay equation is addressed, in which the discrete times have a uniform interval, while for the discrete positions the interval is not uniform.