From Bosonic Grand-Canonical Ensembles to Nonlinear Gibbs Measures

TL;DR

This paper derives nonlinear Gibbs measures from many-body quantum mechanics in a mean-field limit, focusing on the defocusing cubic NLS on a 1D interval, and discusses their invariance under NLS flow.

Contribution

It provides a rigorous derivation of nonlinear Gibbs measures from quantum many-body systems for the first time in the context of the defocusing cubic NLS.

Findings

Derived Gibbs measure lives over H^{1/2} space

Measure is invariant under NLS flow

Focus on the 1D defocusing cubic NLS case

Abstract

In a recent paper, in collaboration with Mathieu Lewin and Phan Th{\`a}nh Nam, we showed that nonlinear Gibbs measures based on Gross-Pitaevskii like functionals could be derived from many-body quantum mechanics, in a mean-field limit. This text summarizes these findings. It focuses on the simplest, but most physically relevant, case we could treat so far, namely that of the defocusing cubic NLS functional on a 1D interval. The measure obtained in the limit, which (almost) lives over H^{1/2} , has been previously shown to be invariant under the NLS flow by Bourgain.