Consequences of the H-Theorem from Nonlinear Fokker-Planck Equations

TL;DR

This paper derives a class of nonlinear Fokker-Planck equations from master equations, demonstrates the H-theorem for these systems, and explores their connection to various entropic forms, including Boltzmann-Gibbs entropy.

Contribution

It introduces a general framework linking nonlinear Fokker-Planck equations with entropy maximization and the H-theorem, expanding understanding of their interrelations.

Findings

H-theorem proven for nonlinear Fokker-Planck equations with external potential

Relation established between Fokker-Planck equations and entropic forms

Boltzmann-Gibbs entropy linked to nonlinear Fokker-Planck equations

Abstract

A general type of nonlinear Fokker-Planck equation is derived directly from a master equation, by introducing generalized transition rates. The H-theorem is demonstrated for systems that follow those classes of nonlinear Fokker-Planck equations, in the presence of an external potential. For that, a relation involving terms of Fokker-Planck equations and general entropic forms is proposed. It is shown that, at equilibrium, this relation is equivalent to the maximum-entropy principle. Families of Fokker-Planck equations may be related to a single type of entropy, and so, the correspondence between well-known entropic forms and their associated Fokker-Planck equations is explored. It is shown that the Boltzmann-Gibbs entropy, apart from its connection with the standard -- linear Fokker-Planck equation -- may be also related to a family of nonlinear Fokker-Planck equations.