Two Types of q-Gaussian Distributions Used to Study the Diffusion in a Finite Region

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.