On the canonical distributions of a thermal particle in the weakly confining potential of special type

TL;DR

This paper analyzes the stationary distribution of a thermal particle in a weakly confining potential, revealing it to be a canonical distribution related to the -deformed Gaussian, with implications for statistical mechanics.

Contribution

It demonstrates that the stationary state in a weakly confining potential is a canonical distribution connected to the -deformed Gaussian, providing new insights into particle dynamics.

Findings

Stationary state is a canonical probability distribution.

Re-parametrization relates the state to the -deformed Gaussian.

Potential characterized by inverse hyperbolic sine function.

Abstract

We consider a thermal particle which is diffusing in velocity-space and in a weakly confining potential characterized by the inverse hyperbolic sine function of the particle velocity $v$ and the control parameter $v_c$. The stationary state of the Fokker-Planck equation is shown to be a canonical probability distribution. Furthermore an appropriate re-parametrization relates this stationary state with the $\kappa$-deformed Gaussian.