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Mathieu Lewin; Phan Thành Nam; Nicolas Rougerie
Derivation of nonlinear Gibbs measures from many-body quantum mechanics
(Dérivation de mesures de Gibbs non linéaires comme limites d’un modèle de mécanique quantique à $N$ corps)
Journal de l'École polytechnique — Mathématiques, 2 (2015), p. 65-115, doi: 10.5802/jep.18
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Class. Math.: 81V70, 35Q40
Keywords: Many-body quantum mechanics, Bose-Einstein condensation, mean-field limit, non-linear Schrödinger equation, non-linear Gibbs measure, quantum de Finetti theorem

Résumé - Abstract

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 $d\ge 2$.

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