Derivation of nonlinear Gibbs measures from many-body quantum mechanics

21 May 2015  ·  Lewin Mathieu CEREMADE, Nam Phan Thành LPMMC, Rougerie Nicolas LPMMC ·

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 Schr{\"o}dinger functional on a finite interval, as well as smoother interactions in dimensions d\textgreater{}1. read more

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Mathematical Physics Analysis of PDEs Mathematical Physics