Stationary solution and $H$ theorem for a generalized Fokker-Planck equation

TL;DR

This paper explores a broad class of generalized Fokker-Planck equations, demonstrating their stationary solutions, establishing an $H$ theorem with a generalized entropy, and connecting these results to known probability distributions and divergence measures.

Contribution

It introduces a unified framework for analyzing stationary solutions and $H$ theorems in generalized Fokker-Planck equations, including a novel generalized Tsallis entropy form.

Findings

Existence of stationary solutions depending on an effective potential

Verification of an $H$ theorem using a generalized Tsallis entropy

Recovery of stationary solutions through entropy maximization

Abstract

We investigate a family of generalized Fokker-Planck equations that contains Richardson and porous media equations as members. Considering a confining drift term that is related to an effective potential, we show that each equation of this family has a stationary solution that depends on this potential. This stationary solution encompasses several well-known probability distributions. Moreover, we verify an $H$ theorem for the generalized Fokker-Planck equations using free-energy-like functionals. We show that the energy-like part of each functional is based on the effective potential and the entropy-like part is a generalized Tsallis entropic form, which has an unusual dependence on the position and can be related to a generalization of the Kullback-Leibler divergence. We also verify that the optimization of this entropic-like form subjected to convenient constraints recovers the stationary solution. The analysis presented here includes several studies about $H$ theorems for other generalized Fokker-Planck equations as particular cases.