Derivation of nonlinear Gibbs measures from many-body quantum mechanics
Lewin, Mathieu; Nam, Phan Thành; Rougerie, Nicolas (2015), Derivation of nonlinear Gibbs measures from many-body quantum mechanics, Journal de l'école Polytechnique. Mathématiques, 2, p. 65-115. 10.5802/jep.18
TypeArticle accepté pour publication ou publié
External document linkhttps://arxiv.org/abs/1410.0335v3
Journal nameJournal de l'école Polytechnique. Mathématiques
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Abstract (EN)We prove that nonlinear Gibbs measures can be obtained from the corresponding many-body, grand-canonical, quantum Gibbs states, in a mean-field limit where the temperature T diverges and the interaction strength behaves as 1/T. We proceed by characterizing the interacting Gibbs state as minimizing a functional counting the free-energy relatively to the non-interacting case. We then perform an infinite-dimensional analogue of phase-space semiclassical analysis, using fine properties of the quantum relative entropy, the link between quantum de Finetti measures and upper/lower symbols in a coherent state basis, as well as Berezin-Lieb type inequalities. Our results cover the measure built on the defocusing nonlinear Schrödinger functional on a finite interval, as well as smoother interactions in dimensions d≥2.
Subjects / Keywordsnonlinear Gibbs measures; quantum mechanics
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