Nonlinear mean-field Fokker-Planck equations and their applications in physics, astrophysics and biology

TL;DR

This paper explores a broad class of nonlinear mean-field Fokker-Planck equations, highlighting their applications across physics, astrophysics, and biology, and revealing connections between diverse phenomena through generalized entropic functionals.

Contribution

It introduces a unified framework for nonlinear mean-field Fokker-Planck equations applicable to various fields, linking previously disconnected topics via generalized entropic functionals.

Findings

Unified description of diverse systems using nonlinear Fokker-Planck equations

Connections between turbulence, astrophysics, and biological processes

Application of generalized entropies to model complex phenomena

Abstract

We discuss a general class of nonlinear mean-field Fokker-Planck equations [P.H. Chavanis, Phys. Rev. E, 68, 036108 (2003)] and show their applications in different domains of physics, astrophysics and biology. These equations are associated with generalized entropic functionals and non-Boltzmannian distributions (Fermi-Dirac, Bose-Einstein, Tsallis,...). They furthermore involve an arbitrary binary potential of interaction. We emphasize analogies between different topics (two-dimensional turbulence, self-gravitating systems, Debye-H\"uckel theory of electrolytes, porous media, chemotaxis of bacterial populations, Bose-Einstein condensation, BMF model, Cahn-Hilliard equations,...) which were previously disconnected. All these examples (and probably many others) are particular cases of this general class of nonlinear mean-field Fokker-Planck equations.