We consider a system of N trapped bosons with repulsive interactions in a combined semiclassical mean-field limit at positive temperature. We show that the free energy is well approximated by the minimum of the Hartree free energy functional – a natural extension of the Hartree energy functional to positive temperatures. The Hartree free energy functional converges in the same limit to a semiclassical free energy functional, and we show that the system displays Bose–Einstein condensation if and only if it occurs in the semiclassical free energy functional. This allows us to show that for weak coupling the critical temperature decreases due to the repulsive interactions.

我们考虑在正温度下，在结合的半经典均值场极限中具有排斥相互作用的N个捕获的玻色子的系统。我们表明，自由能量可以通过Hartree自由能函数的最小值（Hartree能量函数对正温度的自然扩展）来很好地近似。Hartree自由能泛函在一个半经典自由能泛函中以相同的极限收敛，并且我们证明了当且仅当系统出现在半经典自由能泛函中时，系统才会显示Bose-Einstein凝聚。这使我们能够表明，对于弱耦合，临界温度由于排斥相互作用而降低。