From bosonic canonical ensembles to non-linear Gibbs measures

TL;DR

This paper analyzes the mean-field limit of a 1D bosonic canonical ensemble in a superharmonic trap, deriving a classical field theory with non-linear Schrödinger-Gibbs measures conditioned on mass.

Contribution

It establishes the canonical ensemble limit as a non-linear Gibbs measure, extending previous grand-canonical results to fixed mass scenarios with focusing interactions.

Findings

Derived the limit model as a non-linear Schrödinger-Gibbs measure.

Included focusing/attractive interactions in the analysis.

Extended previous grand-canonical results to the canonical setting.

Abstract

We study the mean-field limit of the 1D bosonic canonical ensemble in a superharmonic trap. This is the regime with temperature proportional to particle number, both diverging to infinity, and correspondingly scaled interactions. We prove that the limit model is a classical field theory based on a non-linear Schr{\"o}dinger-Gibbs measure conditioned on the L2 mass, thereby obtaining a canonical analogue of previous results for the grand-canonical ensemble. We take advantage of this set-up with fixed mass to include focusing/attractive interactions/non-linearities in our study.