Self-repellent Brownian Bridges in an Interacting Bose Gas

TL;DR

This paper models an interacting Bose gas using Brownian bridges, deriving conditions for Bose-Einstein condensation and proving the existence of a condensate phase under certain conditions.

Contribution

It provides an explicit formula for the free energy and a criterion for phase transition in an interacting Bose gas using a novel path integral approach.

Findings

Explicit free energy formula derived

Criterion for Bose-Einstein condensation established

Condensate phase proven to exist for high dimensions and weak interactions

Abstract

We consider a model of $d$-dimensional interacting quantum Bose gas, expressed in terms of an ensemble of interacting Brownian bridges in a large box and undergoing the influence of all the interactions between the legs of each of the Brownian bridges. We study the thermodynamic limit of the system and give an explicit formula for the limiting free energy and a necessary and sufficient criterion for the occurrence of a condensation phase transition. For $d\geq 5$ and sufficiently small interaction, we prove that the condensate phase is not empty. The ideas of proof rely on the similarity of the interaction to that of the self-repellent random walk, and build on a lace expansion method conducive to treating {\it paths} undergoing mutual repellence within each bridge.