Équations de champ moyen pour la dynamique quantique d'un grand nombre de particules

Gérard, Patrick

Séminaire Bourbaki : volume 2003/2004, exposés 924-937, Astérisque, no. 299 (2005), Exposé no. 930, pp. 147-164.

L’objet de cet exposé est de montrer comment l’évolution de Schrödinger pour le problème à $N$ corps quantique est approchée, lorsque $N$ tend vers l’infini, dans un régime convenable, par une évolution non-linéaire en dimension trois d’espace. On traitera le cas des bosons, qui conduit à l’équation de Schrödinger-Poisson, et celui des fermions, qui débouche sur le système de Hartree-Fock.

The purpose of this talk is to describe how the Schrödinger evolution for the quantum $N$-body problem is approximated, as $N$ tends to infinity, in a suitable regime, by a nonlinear evolution in three space dimensions. We shall discuss the case of bosons, which leads to the Schrödinger-Poisson equation, and the case of fermions, with its connections to the Hartree-Fock system.

Classification : 35Q40, 35Q55, 81Q05, 81V70
Mot clés : problème à $N$ corps quantique, équations de champ moyen, équation de Hartree, équation de Schrödinger-Poisson, système de Hartree-Fock, déterminants de Slater, hiérarchie BBGKY
Keywords: quantum $N$-body problem, mean field equations, Hartree equation, Schrödinger-Poisson equation, Hartree-Fock system, Slater determinants, BBGKY hierarchy

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