On the canonical distributions of a thermal particle in a generalized velocity-dependent potential

We consider a thermal particle diffusing in velocity-space under a generalized velocity-dependent potential characterized by the inverse hyperbolic sine function of the particle velocity v and the control parameter v. This velocity-dependent potential can be considered as a deformation of Rayleigh's dissipation function. The stationary state of the corresponding Fokker-Planck equation is shown to be a canonical probability distribution. Furthermore an appropriate re-parameterization relates this stationary state with the ?-deformed Gaussian. A possible interpretation of the deformation parameter ? is proposed.