Work

Derivation of nonlinear Gibbs measures from many-body quantum mechanics

Year: 2015

Type: article

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 .

Source: Journal de l’École polytechnique — Mathématiques

DOI: 10.5802/jep.18

Authors: Mathieu Lewin, Phan Thành Nam, Nicolas Rougerie

Institutions: Centre de Recherche en Mathématiques de la Décision, Institute of Science and Technology Austria, Laboratoire de Physique et Modélisation des Milieux Condensés

Language: English

Topic: Advanced Mathematical Physics Problems

Subfield: Mathematical Physics

Field: Mathematics

Domain: Physical Sciences

Open Access status: diamond

Funders: European Commission, Agence Nationale de la Recherche, Institut Henri Poincaré, FP7 Ideas: European Research Council

Awards: Mathematics and Numerics of Infinite Quantum Systems, NOnlinear problems in Nuclear and Atomic Physics, Mathematical methods for the many-body problem in statistical and quantum mechanics, International IST Postdoctoral Fellowship Programme, G7240651115