Equilibrium states for the Bose gas

TL;DR

This paper rigorously computes the generating functional for the free Bose gas with attractive boundary conditions, analyzing condensate localization and ensemble equivalence in the thermodynamic limit.

Contribution

It provides a rigorous calculation of the generating functional and explores condensate localization and ensemble non-equivalence in the Bose gas.

Findings

Explicit form of the Kac function obtained

Non-equivalence of ensembles demonstrated in the condensate region

Condensate density is zero locally despite non-equivalence

Abstract

The generating functional of the cyclic representation of the CCR (Canonical Commutation Relations) representation for the thermodynamic limit of the grand canonical ensemble of the free Bose gas with attractive boundary conditions is rigorously computed. We use it to study the condensate localization as a function of the homothety point for the thermodynamic limit using a sequence of growing convex containers. The Kac function is explicitly obtained proving non-equivalence of ensembles in the condensate region in spite of the condensate density being zero locally.