Limites de champ moyen bosoniques {\`a} temp{\'e}rature positive

Abstract

I present recent results in quantum statistical mechanics, obtained in joint works with Mathieu Lewin and Phan Th{\`a}nh Nam. We consider a certain mean-field limit of the grand-canonical ensemble for a Bose gas at positive temperature. In this limit, the reduced density matrices of the quantum theory converge to their analogues in classical field theory, given by a non-linear Gibbs measure. In particular, we deal with cases where the latter must be defined via a renormalization procedure. The corresponding renormalization at the level of the original quantum grand-canonical model, with non-commuting fields, is one of the important difficulties.