Théorèmes de de Finetti, limites de champ moyen et condensation de Bose-Einstein
arXiv:1409.1182
Abstract
These lecture notes treat the mean-field approximation for equilibrium states of N body systems in classical and quantum statistical mechanics. A general strategy to justify effective models based on assumptions of statistical independence of the particles is in presented in detail. The main tools are a structure theorems of de Finetti that describe large N limits of states accessible to the systems in question, exploiting the indistinguishablity of particles. The focus is on quantum aspects, particularly the mean-field approximation for the ground state of a large system of bosons, in connection with Bose-Einstein condensation: structure of reduced density matrices of a large bosonic system, localization methods in Fock space, derivation of Hartree and non-linear Schrödinger effective energy functionals.
These are the lectures notes from my cours Peccot at the Collège de France, given in March-April 2014. They are in french for the moment, some english translation should be available soon(er or later). Two appendices contain each an unpublished result obtained in collaboration with Mathieu Lewin. Further small corrections and a few more references in this version.
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