Derivation of nonlinear Gibbs measures from many-body quantum mechanics
Journal de l’École polytechnique - Mathématiques, Volume 2  (2015), p. 65-115

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 d2.

Nous prouvons que certaines mesures de Gibbs non linéaires peuvent être obtenues à partir des états de Gibbs grand-canoniques du problème à N corps, dans une limite de champ moyen où la température T diverge et la constante de couplage se comporte comme 1/T. Nous commençons par caractériser les états de Gibbs en présence d’interactions comme minimiseurs d’une fonctionnelle comptant l’énergie libre relativement au cas sans interaction. Nous procédons ensuite à un analogue en dimension infinie d’une analyse semi-classique, en utilisant des propriétés fines de l’entropie relative quantique, le lien entre mesures de de Finetti et symboles supérieurs/inférieurs dans une base d’états cohérents, ainsi que des inégalités de type Berezin-Lieb. Nos résultats couvrent la mesure construite à partir de la fonctionnelle de Schrödinger non linéaire défocalisante sur un intervalle fini, ainsi que le cas d’interactions plus régulières en dimension supérieure.

DOI : https://doi.org/10.5802/jep.18
Classification:  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
@article{JEP_2015__2__65_0,
     author = {Lewin, Mathieu and Nam, Phan Th\`anh and Rougerie, Nicolas},
     title = {Derivation of nonlinear Gibbs measures from many-body quantum mechanics},
     journal = {Journal de l'\'Ecole polytechnique - Math\'ematiques},
     publisher = {Ecole polytechnique},
     volume = {2},
     year = {2015},
     pages = {65-115},
     doi = {10.5802/jep.18},
     language = {en},
     url = {http://www.numdam.org/item/JEP_2015__2__65_0}
}
Lewin, Mathieu; Nam, Phan Thành; Rougerie, Nicolas. Derivation of nonlinear Gibbs measures from many-body quantum mechanics. Journal de l’École polytechnique - Mathématiques, Volume 2 (2015) , pp. 65-115. doi : 10.5802/jep.18. http://www.numdam.org/item/JEP_2015__2__65_0/

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