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

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 $ρ=ρ_{\rm short}+ρ_{\rm long}$ into the number density of particles belonging to cycles of finite length ($ρ_{\rm short}$) and to infinitely long cycles ($ρ_{\rm long}$) in the thermodynamic limit. For this model we prove that when there is Bose condensation, $ρ_{\rm long}$ 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 $ρ_{\rm long}\neq 0$ with off-diagonal long range order and winding paths that occur in the path integral representation of the Bose gas.