Derivation of Hartree’s theory for mean-field Bose gases

Lewin, Mathieu

doi:10.5802/jedp.103

Derivation of Hartree’s theory for mean-field Bose gases
[Dérivation de la théorie de Hartree pour des gaz de bosons dans le régime de champ moyen]

Lewin, Mathieu ¹

¹ CNRS & Université de Cergy-Pontoise (UMR 8088) 95000 Cergy-Pontoise, France.

Journées équations aux dérivées partielles (2013), article no. 7, 21 p.

Résumé
Abstract

Dans cet article, nous présentons des résultats obtenus avec Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty et Jan Philip Solovej. Nous considérons un système de $N$ bosons qui interagissent avec un potentiel d’intensité $1 / N$ (on parle de régime de champ moyen). Dans la limite où $N \to \infty$ , nous montrons que le premier ordre du développement des valeurs propres du Hamiltonien à $N$ corps est donné par la théorie non linéaire de Hartree, alors que l’ordre suivant est donné par l’opérateur de Bogoliubov. Nous discutons également en détails du phénomène de condensation de Bose-Einstein dans de tels systèmes.

This article is a review of recent results with Phan Thành Nam, Nicolas Rougerie, Sylvia Serfaty and Jan Philip Solovej. We consider a system of $N$ bosons with an interaction of intensity $1 / N$ (mean-field regime). In the limit $N \to \infty$ , we prove that the first order in the expansion of the eigenvalues of the many-particle Hamiltonian is given by the nonlinear Hartree theory, whereas the next order is predicted by the Bogoliubov Hamiltonian. We also discuss the occurrence of Bose-Einstein condensation in these systems.

DOI : 10.5802/jedp.103

Classification : 35Q40, 81Q99
Mots clés : Hartree theory, mean-field limit, Bose-Einstein condensation, quantum de Finetti theorem

Affiliations des auteurs :

Lewin, Mathieu ¹

¹ CNRS & Université de Cergy-Pontoise (UMR 8088) 95000 Cergy-Pontoise, France.

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Lewin, Mathieu. Derivation of Hartree’s theory for mean-field Bose gases. Journées équations aux dérivées partielles (2013), article  no. 7, 21 p. doi : 10.5802/jedp.103. http://www.numdam.org/articles/10.5802/jedp.103/

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