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.

17 pages, 5 figures

References in corpus (6)

• A possible deformed algebra and calculus inspired in nonextensive thermostatistics
• Color Superconductivity and Tsallis Statistics
• Exponential Families and MaxEnt Calculations for Entropy Measures of Statistical Physics
• Characteristic Function of the Tsallis $q$-Gaussian and Its Applications in Measurement and Metrology
• Deformed quantum harmonic oscillator with diffusion and dissipation
• Statistical Nuclear Spectroscopy with $q$-normal and bivariate $q$-normal distributions and $q$-Hermite polynomials