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

10 pages

References in corpus (3)

• Percolation transition in the Bose gas II
• Percolation, Bose-Einstein Condensation, and String Proliferation
• What do emulsification failure and Bose-Einstein condensation have in common?

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