Derivation of nonlinear Gibbs measures from many-body quantum mechanics

Mathieu Lewin (1) , Phan Thành Nam (2) , Nicolas Rougerie (3)

1 CEREMADE - CEntre de REcherches en MAthématiques de la DEcision
2 IST Austria - Institute of Science and Technology [Klosterneuburg, Austria]
3 LPM2C - Laboratoire de physique et modélisation des milieux condensés

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 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>1.

Domaines: Physique mathématique [math-ph]; Physique mathématique [math-ph]; Equations aux dérivées partielles [math.AP]

Mathieu Lewin, Phan Thành Nam, Nicolas Rougerie. Derivation of nonlinear Gibbs measures from many-body quantum mechanics. Journal de l'École polytechnique — Mathématiques, 2015, 2, pp.65-115. ⟨10.5802/jep.18⟩. ⟨hal-01070599v3⟩