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Comptes Rendus Physique
Volume 7, n° 3-4
pages 318-330 (avril-mai 2006)
Doi : 10.1016/j.crhy.2006.01.004
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 Chavanis
Laboratoire de physique théorique, université Paul-Sabatier, 118, route de Narbonne, 31062 Toulouse, France 

Abstract

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. To cite this article: P.-H. Chavanis, C. R. Physique 7 (2006).

The full text of this article is available in PDF format.
Résumé

Je présente une classe générale dʼéquations 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 lʼastrophysique 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 dʼinteraction 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 dʼautres) sont des cas particuliers de cette classe générale dʼéquations de Fokker-Planck non-linéaires en champ moyen. Pour citer cet article : P.-H. Chavanis, C. R. Physique 7 (2006).

The full text of this article is available in PDF format.

Keywords : Generalized Fokker-Planck equations, Vlasov equation, Long-range interactions

Mots-clés : Classe générale dʼéquations de Fokker-Planck, Équation de Vlasov, Interaction à longue portée




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