In this paper, we consider the problem of minimizing quantum free energies under the constraint that the density of particles is fixed at each point of ${\mathbb{R}}^d$, for any $d\ge 1$. We are more particularly interested in the characterization of the minimizer, which is a self-adjoint nonnegative trace class operator, and will show that it is solution to a nonlinear self-consistent problem. This question of deriving quantum statistical equilibria is at the heart of the quantum hydrodynamical models introduced by Degond and Ringhofer in [4]. An original feature of the problem is the local nature of constraint, i.e. it depends on position, while more classical models consider the total number of particles in the system to be fixed. This raises difficulties in the derivation of the Euler-Lagrange equations and in the characterization of the minimizer, which are tackled in part by a careful parameterization of the feasible set.

在本文中，我们考虑在粒子密度固定在每个点的约束下最小化量子自由能的问题。 ${\mathbb{电阻}}^d$，对于任何 $d\ge 1$. 我们对最小化器的表征更感兴趣，它是一个自伴随非负迹类算子，并将证明它是非线性自洽问题的解决方案。这个推导量子统计平衡的问题是 Degond 和 Ringhofer 在 [4] 中引入的量子流体动力学模型的核心。该问题的一个原始特征是约束的局部性质，即它取决于位置，而更经典的模型认为系统中的粒子总数是固定的。这给欧拉-拉格朗日方程的推导和极小值的表征带来了困难，部分通过可行集的仔细参数化来解决。