The problem considered here is motivated by a work by Nachtergaele and Yau where the Euler equations of fluid dynamics are derived from many-body quantum mechanics, see (Commun Math Phys 243(3):485–540, 2003). A crucial concept in their work is that of local quantum Gibbs states, which are quantum statistical equilibria with prescribed particle, current, and energy densities at each point of space (here \({\mathbb {R}}^d\), \(d \ge 1\)). They assume that such local Gibbs states exist, and show that if the quantum system is initially in a local Gibbs state, then the system stays, in an appropriate asymptotic limit, in a Gibbs state with particle, current, and energy densities now solutions to the Euler equations. Our main contribution in this work is to prove that such local quantum Gibbs states can be constructed from prescribed densities under mild hypotheses, in both the fermionic and bosonic cases. The problem consists in minimizing the von Neumann entropy in the quantum grand canonical picture under constraints of local particle, current, and energy densities. The main mathematical difficulty is the lack of compactness of the minimizing sequences to pass to the limit in the constraints. The issue is solved by defining auxiliary constrained optimization problems, and by using some monotonicity properties of equilibrium entropies.

这里考虑的问题是由 Nachtergaele 和 Yau 的工作激发的，其中流体动力学的欧拉方程源自多体量子力学，参见 (Commun Math Phys 243(3):485–540, 2003)。他们工作中的一个关键概念是局部量子吉布斯态，它是量子统计平衡，在每个空间点都有规定的粒子、电流和能量密度（这里是\({\mathbb {R}}^d\) , \ (d \ge 1\)）。他们假设存在这样的局域吉布斯态，并表明如果量子系统最初处于局域吉布斯态，那么系统将保持在适当的渐近极限中的吉布斯态，现在粒子、电流和能量密度的解为欧拉方程。我们在这项工作中的主要贡献是证明在费米子和玻色子情况下，这种局部量子吉布斯态可以在温和的假设下从规定的密度构建。问题在于在局部粒子、电流和能量密度的约束下，最小化量子大正则图像中的冯诺依曼熵。主要的数学困难是最小化序列缺乏紧凑性以传递到约束中的极限。该问题通过定义辅助约束优化问题来解决，