Comptes Rendus
no. 3-4
Nonlinear mean-field Fokker–Planck equations and their applications in physics, astrophysics and biology
[Équations de Fokker–Planck non-linéaires en champ moyen et leurs applications en physique, astrophysique et biologie]
Pierre-Henri Chavanis1
1 Laboratoire de physique théorique, université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse, France
Comptes Rendus. Physique, Volume 7 (2006) no. 3-4, pp. 318-330.

Je présente une classe générale d'équations de Fokker–Planck non-linéaires en champ moyen [P.-H. Chavanis, Phys. Rev. E 68 (2003) 036108] et montre leurs applications dans différents domaines de la physique, de l'astrophysique et de la biologie. Ces équations sont associées à des fonctionnelles entropiques généralisées et à des distributions non-Boltzmanniennes (Fermi–Dirac, Bose–Einstein, Tsallis, …). De plus, elles incluent un potentiel d'interaction binaire arbitraire. Je souligne des analogies entre différents domaines (turbulence bidimensionnelle, systèmes auto-gravitants, théorie des electrolytes de Debye–Hückel, milieux poreux, chimiotactie des populations bactériennes, condensation de Bose–Einstein, modèle BMF, équations de Cahn–Hilliard, …) qui étaient auparavant déconnectés. Tous ces exemples (et probablement beaucoup d'autres) sont des cas particuliers de cette classe générale d'équations de Fokker–Planck non-linéaires en champ moyen.

We discuss a general class of nonlinear mean-field Fokker–Planck equations [P.-H. Chavanis, Phys. Rev. E 68 (2003) 036108] 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–Hückel 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.

Publié le :
DOI : 10.1016/j.crhy.2006.01.004
Keywords: Generalized Fokker–Planck equations, Vlasov equation, Long-range interactions
Mot clés : Classe générale d'équations de Fokker–Planck, Équation de Vlasov, Interaction à longue portée
Pierre-Henri Chavanis 1

1 Laboratoire de physique théorique, université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse, France 
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Pierre-Henri Chavanis. Nonlinear mean-field Fokker–Planck equations and their applications in physics, astrophysics and biology. Comptes Rendus. Physique, Volume 7 (2006) no. 3-4, pp. 318-330. doi : 10.1016/j.crhy.2006.01.004. https://comptes-rendus.academie-sciences.fr/physique/articles/10.1016/j.crhy.2006.01.004/

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