General properties of nonlinear mean field Fokker-Planck equations

TL;DR

This paper reviews the fundamental properties of nonlinear mean field Fokker-Planck equations, exploring their derivation, limits, and specific examples related to different entropy forms, to extend thermodynamic concepts.

Contribution

It provides a comprehensive review of nonlinear mean field Fokker-Planck equations, including derivations, limits, and explicit examples for various entropies, advancing the understanding of non-Boltzmannian distributions.

Findings

Derived the passage from generalized Kramers to Smoluchowski equations.

Provided explicit examples for Boltzmann, Tsallis, and Fermi-Dirac entropies.

Analyzed properties of nonlinear Fokker-Planck equations in dissipative systems.

Abstract

Recently, several authors have tried to extend the usual concepts of thermodynamics and kinetic theory in order to deal with distributions that can be non-Boltzmannian. For dissipative systems described by the canonical ensemble, this leads to the notion of nonlinear Fokker-Planck equation (T.D. Frank, Non Linear Fokker-Planck Equations, Springer, Berlin, 2005). In this paper, we review general properties of nonlinear mean field Fokker-Planck equations, consider the passage from the generalized Kramers to the generalized Smoluchowski equation in the strong friction limit, and provide explicit examples for Boltzmann, Tsallis and Fermi-Dirac entropies.