Thermodynamic limit and proof of condensation for trapped bosons

TL;DR

This paper investigates the behavior of trapped bosons in the thermodynamic limit, proving full condensation for noninteracting gases and partial condensation for interacting gases across all temperatures.

Contribution

It provides a rigorous proof of condensation phenomena for both noninteracting and interacting trapped bosons in the thermodynamic limit.

Findings

Noninteracting bosons exhibit 100% condensation into the ground state at all temperatures.

Interacting bosons show less-than-100% condensation, depending on interaction strength, at all temperatures.

No phase transition occurs in the noninteracting case across any dimension.

Abstract

We study condensation of trapped bosons in the limit when the number of particles tends to infinity. For the noninteracting gas we prove that there is no phase transition in any dimension, but in any dimension at any temperature the system is 100% condensated into the one-particle ground state. In the case of an interacting gas we show that for a family of suitably scaled pair interactions, the Gross-Pitaevskii scaling included, a less-than-100% condensation into a one-particle eigenstate, which may depend on the interaction strength, persists at all temperatures.