论文标题
在局部量子吉布斯状态
On local quantum Gibbs states
论文作者
论文摘要
我们在这项工作中解决了在本地约束下最小化量子熵的问题。我们假设宏观量（例如粒子密度，电流和动能）的每个点都固定在$ \ rm^d $的每个点上，并在$ l^2（\ rm^d）上寻找密度运算符$最小化熵功能。这样的最小化器称为当地吉布斯州。这种设置与规定全局约束的经典问题处于约束，其中固定了粒子，总电流和总能量的总数。例如，从量子动力学推导流体模型时出现了这个问题。我们证明，在相当普遍的条件下，熵承认了独特的约束最小化器。由于缺乏紧凑性，证据的主要困难是表明最小化序列的限制满足了局部能量限制。我们通过引入更简单的辅助最小化问题并使用涉及熵的单调论点来解决这个问题。
We address in this work the problem of minimizing quantum entropies under local constraints. We suppose macroscopic quantities such as the particle density, current, and kinetic energy are fixed at each point of $\Rm^d$, and look for a density operator over $L^2(\Rm^d)$ minimizing an entropy functional. Such minimizers are referred to as a local Gibbs states. This setting is in constrast with the classical problem of prescribing global constraints, where the total number of particles, total current, and total energy in the system are fixed. The question arises for instance in the derivation of fluid models from quantum dynamics. We prove, under fairly general conditions, that the entropy admits a unique constrained minimizer. Due to a lack of compactness, the main difficulty in the proof is to show that limits of minimizing sequences satisfy the local energy constraint. We tackle this issue by introducing a simpler auxiliary minimization problem and by using a monotonicity argument involving the entropy.
