Bose-Einstein Condensation in a Dilute, Trapped Gas at Positive Temperature

TL;DR

This paper rigorously analyzes a dilute Bose gas in a trap at positive temperature, showing that the free energy difference aligns with the Gross-Pitaevskii functional and confirming Bose-Einstein condensation in this regime.

Contribution

It establishes a connection between the canonical free energy of an interacting Bose gas and the Gross-Pitaevskii functional at positive temperature, proving Bose-Einstein condensation.

Findings

Free energy difference matches GP energy functional minimization.

One-particle density matrix approximates noninteracting gas with GP condensate.

Bose-Einstein condensation occurs in the described thermodynamic limit.

Abstract

We consider an interacting, dilute Bose gas trapped in a harmonic potential at a positive temperature. The system is analyzed in a combination of a thermodynamic and a Gross-Pitaevskii (GP) limit where the trap frequency $\omega$, the temperature $T$ and the particle number $N$ are related by $N \sim (T / \omega)^{3} \to\infty$ while the scattering length is so small that the interaction energy per particle around the center of the trap is of the same order of magnitude as the spectral gap in the trap. We prove that the difference between the canonical free energy of the interacting gas and the one of the noninteracting system can be obtained by minimizing the GP energy functional. We also prove Bose-Einstein condensation in the following sense: The one-particle density matrix of any approximate minimizer of the canonical free energy functional is to leading order given by that of the noninteracting gas but with the free condensate wavefunction replaced by the GP minimizer.