Long Cycles in a Perturbed Mean Field Model of a Boson Gas

TL;DR

This paper rigorously establishes the link between Bose condensation and the emergence of long cycles in a perturbed mean field Bose gas model, using large deviations theory to connect cycle structure with condensate density.

Contribution

It provides a precise mathematical formulation of the relationship between Bose condensation and long cycles, proving their equivalence in a perturbed mean field model.

Findings

Long cycles correspond to Bose condensate density.

Bose condensation implies the presence of infinite cycles.

The relation is validated using large deviations theory.

Abstract

In this paper we give a precise mathematical formulation of the relation between Bose condensation and long cycles and prove its validity for the perturbed mean field model of a Bose gas. We decompose the total density $\rho=\rho_{{\rm short}}+\rho_{{\rm long}}$ into the number density of particles belonging to cycles of finite length ($\rho_{{\rm short}}$) and to infinitely long cycles ($\rho_{{\rm long}}$) in the thermodynamic limit. For this model we prove that when there is Bose condensation, $\rho_{{\rm long}}$ is different from zero and identical to the condensate density. This is achieved through an application of the theory of large deviations. We discuss the possible equivalence of $\rho_{{\rm long}}\neq 0$ with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas.