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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.) (English. French summary) Zbl 1322.81082

Summary: 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 \(d \geq 2\).

MSC:

81V70 Many-body theory; quantum Hall effect
35Q40 PDEs in connection with quantum mechanics
35Q55 NLS equations (nonlinear Schrödinger equations)
82B30 Statistical thermodynamics
81Q20 Semiclassical techniques, including WKB and Maslov methods applied to problems in quantum theory
94A17 Measures of information, entropy
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