Derivation of nonlinear Gibbs measures from many-body quantum mechanics

Mathieu Lewin; Phan Thành Nam; Nicolas Rougerie

doi:10.5802/jep.18

Mathieu Lewin ¹; Phan Thành Nam ²; Nicolas Rougerie ³

¹ CNRS & Université Paris-Dauphine, CEREMADE (UMR 7534) Place de Lattre de Tassigny, F-75775 Paris Cedex 16, France
² IST Austria Am Campus 1, 3400 Klosterneuburg, Austria
³ Université Grenoble 1 & CNRS, LPMMC (UMR 5493) B.P. 166, F-38042 Grenoble, France

Journal de l’École polytechnique — Mathématiques, Volume 2 (2015), pp. 65-115.

Abstract
Résumé

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 $d \geq 2$ .

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.

Received: 2015-01-05
Accepted: 2015-05-03

MR Zbl

DOI: 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
Mot clés : Mécanique quantique à $N$ corps, condensation de Bose-Einstein, limite de champ moyen, équation de Schrödinger non linéaire, mesure de Gibbs non linéaire, théorème de de Finetti quantique

Author's affiliations:

Mathieu Lewin ¹; Phan Thành Nam ²; Nicolas Rougerie ³

¹ CNRS & Université Paris-Dauphine, CEREMADE (UMR 7534) Place de Lattre de Tassigny, F-75775 Paris Cedex 16, France
² IST Austria Am Campus 1, 3400 Klosterneuburg, Austria
³ Université Grenoble 1 & CNRS, LPMMC (UMR 5493) B.P. 166, F-38042 Grenoble, France

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Mathieu Lewin and Phan Th\`anh Nam and Nicolas Rougerie},
     title = {Derivation of nonlinear {Gibbs} measures from many-body quantum mechanics},
     journal = {Journal de l{\textquoteright}\'Ecole polytechnique {\textemdash} Math\'ematiques},
     pages = {65--115},
     publisher = {\'Ecole polytechnique},
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Mathieu Lewin; Phan Thành Nam; Nicolas Rougerie. 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. https://jep.centre-mersenne.org/articles/10.5802/jep.18/

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