Nonlinear inhomogeneous Fokker-Planck equations: entropy and free-energy time evolution

TL;DR

This paper extends a free-energy formalism to inhomogeneous nonlinear Fokker-Planck equations, analyzing entropy production, stationary solutions, and the evolution of free energy in these complex systems.

Contribution

It introduces a generalized free-energy framework for inhomogeneous nonlinear Fokker-Planck equations, linking entropy, auxiliary potentials, and system coefficients.

Findings

Derived explicit free-energy functional for inhomogeneous equations

Analyzed entropy production during relaxation to equilibrium

Characterized stationary solutions of the equations

Abstract

We extend a recently introduced free-energy formalism for homogeneous Fokker-Planck equations to a wide, and physically appealing, class of inhomogeneous nonlinear Fokker-Planck equations. In our approach, the free-energy functional is expressed in terms of an entropic functional and an auxiliary potential, both derived from the coefficients of the equation. With reference to the introduced entropic functional, we discuss the entropy production in a relaxation process towards equilibrium. The properties of the stationary solutions of the considered Fokker-Planck equations are also discussed.