Derivation of nonlinear gibbs measures from many-body quantum mechanics

M. Lewin, N. Phan Thanh, N. Rougerie, Journal de l’Ecole Polytechnique - Mathematiques 2 (2015) 65–115.

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Journal Article | Published | English
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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 2.
Publishing Year
Date Published
2015-01-01
Journal Title
Journal de l'Ecole Polytechnique - Mathematiques
Volume
2
Page
65 - 115
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Lewin M, Phan Thanh N, Rougerie N. Derivation of nonlinear gibbs measures from many-body quantum mechanics. Journal de l’Ecole Polytechnique - Mathematiques. 2015;2:65-115. doi:10.5802/jep.18
Lewin, M., Phan Thanh, N., & Rougerie, N. (2015). Derivation of nonlinear gibbs measures from many-body quantum mechanics. Journal de l’Ecole Polytechnique - Mathematiques, 2, 65–115. https://doi.org/10.5802/jep.18
Lewin, Mathieu, Nam Phan Thanh, and Nicolas Rougerie. “Derivation of Nonlinear Gibbs Measures from Many-Body Quantum Mechanics.” Journal de l’Ecole Polytechnique - Mathematiques 2 (2015): 65–115. https://doi.org/10.5802/jep.18.
M. Lewin, N. Phan Thanh, and N. Rougerie, “Derivation of nonlinear gibbs measures from many-body quantum mechanics,” Journal de l’Ecole Polytechnique - Mathematiques, vol. 2, pp. 65–115, 2015.
Lewin M, Phan Thanh N, Rougerie N. 2015. Derivation of nonlinear gibbs measures from many-body quantum mechanics. Journal de l’Ecole Polytechnique - Mathematiques. 2, 65–115.
Lewin, Mathieu, et al. “Derivation of Nonlinear Gibbs Measures from Many-Body Quantum Mechanics.” Journal de l’Ecole Polytechnique - Mathematiques, vol. 2, Ecole Polytechnique, 2015, pp. 65–115, doi:10.5802/jep.18.
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