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

Abstract

Dérivation de la théorie de Hartree pour des gaz de bosons dans le régime de champ moyen RésuméDans 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 → ∞, 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. AbstractThis 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 → ∞, 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.

Ground states of large bosonic systems : the Gross–Pitaevskii limit revisited

“…This theorem validates the mean-field approximation for a large class of trapped Bose gases, in particular (see [13] and references therein) when the strength of the interaction is proportional to the inverse of the particle number, case β = 0 in (1.6). However, when the interaction becomes stronger, the mean-field approximation is harder to justify.…”

“…In the present paper we will provide alternative proofs of the energy lower bound and the convergence of states using the quantum de Finetti theorem in the same spirit as in [13,14]. Our proofs are conceptually and technically simpler than those provided in [18].…”

Bose gases, Bose–Einstein condensation, and the Bogoliubov approximation

“…22. For a constant φ, corresponding to a homogeneous system, the resulting Hartree energy is then simply equal to 1 2 N v. It is also known that starting from a product state of the form (45), a solution to the timedependent Schrödinger equation i∂ t = H N stays roughly a product at later times, with the factors in the limit N → ∞ determined by the time-dependent Hartree equation…”

Dynamics of sound waves in an interacting Bose gas

“…Among other results, they have shown that the Vlasov equation effectively describes a classical manybody system while the Hartree equation describes a Bose gas in the mean-field limit. After Hepp's initial work [7] there has been a lot of effort to arrive at a mathematically rigorous understanding of quantum-mechanical mean-field limits; regarding the dynamics see, e.g., [18,16,13,5,6,9], and regarding ground state see, e.g., [17,4,10] and furthermore [11] for an elaborate overview.…”