Derivation of nonlinear Gibbs measures from many-body quantum mechanics

Mathieu Lewin; Phan Thành Nam; Nicolas Rougerie

doi:10.5802/jep.18

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]

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, Tome 2 (2015), pp. 65-115.

Résumé
Abstract

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.

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 $d \geq 2$ .

Reçu le : 2015-01-05
Accepté le : 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

Affiliations des auteurs :

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

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     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, Tome 2 (2015), pp. 65-115. doi : 10.5802/jep.18. https://jep.centre-mersenne.org/articles/10.5802/jep.18/

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