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

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 ω, the temperature T and the particle number N are related by N ∼ (T /ω) 3 → ∞ 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.2.4. The condensate energy and the interaction between the condensate and the thermal cloud .

Gross–Pitaevskii Limit of a Homogeneous Bose Gas at Positive Temperature

“…In this system the condensate and the thermal cloud necessarily live on the same length scale and interactions between them are relevant. We prove similar statements as in the case of the trapped gas in [5], in particular, we show the existence of a BEC phase transition with critical temperature given by the one of the ideal gas to leading order.…”

“…In a more recent work [5], the trapped Bose gas at positive temperature is studied in a combination of thermodynamic limit in the trap and GP limit. It could be shown that the difference between the interacting free energy of the system and the free energy of the ideal gas is to leading order given by the minimium of the GP energy functional.…”

“…The first ingredient is an estimate showing that γ N is, when projected to high momentum modes, given by the 1-pdm of the ideal gas to leading order. This part of the proof is motivated by a similar proof in [5] and is based on certain lower bounds for the bosonic relative entropy (the difference between two free energies) quantifying its coercivity. One main novelty in this part of our proof is a new lower bound for the bosonic relative entropy that allows us to simplify this part substantially w.r.t.…”

“…Let f (x, y) = σ(x) − σ(y) − σ ′ (y)(x − y). In the proof of Lemma 4.1 in [5] it has been shown that there is a C > 0 such that…”

Ground state energy of a dilute two-dimensional Bose gas from the Bogoliubov free energy functional

“…There exist only very limited rigorous results about the free energy of a bosonic system starting from a many-body Schrödinger Hamiltonian. In fact, in the dilute regime, the homogeneous gas in three dimensions has been treated by Seiringer [35] and Yin [38] (see also [8,9] for recent developments for the trapped and Gross-Pitaevskii cases). The lower [35] and upper bound [38] prove the free energy asymptotics F (T, ρ) = F 0 (T, ρ) + 4πa 2ρ 2 − [ρ − ρ fc ] 2 + + o(aρ 2 ) as ρa 3 → 0.…”