We consider a dilute, homogeneous Bose gas at positive temperature. The system is investigated in the Gross–Pitaevskii limit, where the scattering length a is so small that the interaction energy is of the same order of magnitude as the spectral gap of the Laplacian, and for temperatures that are comparable to the critical temperature of the ideal gas. We show that the difference between the specific free energy of the interacting system and the one of the ideal gas is to leading order given by $$4 \pi a \left( 2 \varrho ^2 - \varrho _0^2 \right) $$ 4 π a 2 ϱ 2 - ϱ 0 2 . Here $$\varrho $$ ϱ denotes the density of the system and $$\varrho _0$$ ϱ 0 is the expected condensate density of the ideal gas. Additionally, we show that the one-particle density matrix of any approximate minimizer of the Gibbs free energy functional is to leading order given by the one of the ideal gas. This in particular proves Bose–Einstein condensation with critical temperature given by the one of the ideal gas to leading order. One key ingredient of our proof is a novel use of the Gibbs variational principle that goes hand in hand with the c-number substitution.

中文翻译：

---

正温度下均质玻色气体的总-皮塔耶夫斯基极限

我们考虑在正温度下稀释的均质 Bose 气体。该系统在 Gross-Pitaevskii 极限下进行研究，其中散射长度 a 非常小，以至于相互作用能与拉普拉斯算子的光谱间隙处于同一数量级，并且温度与临界温度相当理想气体。我们表明相互作用系统的特定自由能与理想气体之一之间的差异是由 $$4 \pi a \left( 2 \varrho ^2 - \varrho _0^2 \right) $ 给出的领先顺序$ 4 π a 2 ϱ 2 - ϱ 0 2 . 这里 $$\varrho $$ ϱ 表示系统的密度，$$\varrho _0$$ ϱ 0 是理想气体的预期冷凝密度。此外，我们证明了吉布斯自由能泛函的任何近似极小值的单粒子密度矩阵是由理想气体之一给出的领先顺序。这特别证明了玻色-爱因斯坦凝聚具有由理想气体之一给出的临界温度领先。我们证明的一个关键要素是对 Gibbs 变分原理的新颖使用，该原理与 c 数替换密切相关。