Deformed Fokker-Planck equation: inhomogeneous medium with a position-dependent mass

TL;DR

This paper develops a generalized deformed Fokker-Planck equation for inhomogeneous media with position-dependent mass, using deformed derivatives, and explores its consistency with diffusion, Langevin dynamics, and entropy concepts.

Contribution

It introduces a deformed FPE framework for inhomogeneous media with position-dependent parameters, unifying diffusion, Langevin, and entropy formalisms in a generalized derivative space.

Findings

Deformed FPE is equivalent to a constant diffusion coefficient FPE in a deformed space.

The formalism aligns with the diffusion equation for inhomogeneous media.

Connections between superstatistics and position-dependent Langevin equations are established.

Abstract

We present the Fokker-Planck equation (FPE) for an inhomogeneous medium with a position-dependent mass particle by making use of the Langevin equation, in the context of a generalized deformed derivative for an arbitrary deformation space where the linear (nonlinear) character of the FPE is associated with the employed deformed linear (nonlinear) derivative. The FPE for an inhomogeneous medium with a position-dependent diffusion coefficient is equivalent to a deformed FPE within a deformed space, described by generalized derivatives, and constant diffusion coefficient. The deformed FPE is consistent with the diffusion equation for inhomogeneous media when the temperature and the mobility have the same position-dependent functional form as well as with the nonlinear Langevin approach. The deformed version of the H-theorem permits to express the Boltzmann-Gibbs entropic functional as a sum of two contributions, one from the particles and the other from the inhomogeneous medium. The formalism is illustrated with the infinite square well and the confining potential with linear drift coefficient. Connections between superstatistics and position-dependent Langevin equations are also discussed.