# Derivation of nonlinear Gibbs measures from many-body quantum mechanics

```@article{Lewin2014DerivationON,
title={Derivation of nonlinear Gibbs measures from many-body quantum mechanics},
author={Mathieu Lewin and Phan Th{\`a}nh Nam and Nicolas Rougerie},
journal={arXiv: Mathematical Physics},
year={2014}
}```
• Published 1 October 2014
• Physics
• arXiv: Mathematical Physics
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…
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## References

SHOWING 1-10 OF 88 REFERENCES
Exponential Relaxation to Equilibrium for a One-Dimensional Focusing Non-Linear Schrödinger Equation with Noise
• Mathematics, Physics
• 2014
We construct generalized grand-canonical- and canonical Gibbs measures for a Hamiltonian system described in terms of a complex scalar field that is defined on a circle and satisfies a nonlinear
The mean-field approximation and the non-linear Schrödinger functional for trapped Bose gases
• Physics, Mathematics
• 2014
We study the ground state of a trapped Bose gas, starting from the full many-body Schrodinger Hamiltonian, and derive the nonlinear Schrodinger energy functional in the limit of large particle
Mean field limit for bosons and propagation of Wigner measures
• Physics
• 2009
We consider the N-body Schrodinger dynamics of bosons in the mean field limit with a bounded pair-interaction potential. According to the previous work [Ammari, Z. and Nier, F., “Mean field limit for
REMARKS ON THE QUANTUM DE FINETTI THEOREM FOR BOSONIC SYSTEMS
• Mathematics
• 2013
The quantum de Finetti theorem asserts that the k-body density matrices of a N-body bosonic state approach a convex combination of Hartree states (pure tensor powers) when N is large and k fixed. In
Théorèmes de de Finetti, limites de champ moyen et condensation de Bose-Einstein
These lecture notes treat the mean-field approximation for equilibrium states of N body systems in classical and quantum statistical mechanics. A general strategy to justify effective models based on
Long time dynamics for the one dimensional non linear Schr\"odinger equation
• Mathematics
• 2010
In this article, we first present the construction of Gibbs measures associated to nonlinear Schr\"odinger equations with harmonic potential. Then we show that the corresponding Cauchy problem is
Examples of Bosonic de Finetti States over Finite Dimensional Hilbert Spaces
According to the Quantum de Finetti Theorem, locally normal infinite particle states with Bose–Einstein symmetry can be represented as mixtures of infinite tensor powers of vector states. This note
Statistical mechanics of the nonlinear Schrödinger equation
• Mathematics
• 1988
AbstractWe investigate the statistical mechanics of a complex fieldø whose dynamics is governed by the nonlinear Schrödinger equation. Such fields describe, in suitable idealizations, Langmuir waves