Qiu Tian, Fei Zhaoyu, Pan Rui, Quan H T
School of Physics, Peking University, Beijing 100871, China.
Collaborative Innovation Center of Quantum Matter, Beijing 100871, China.
Phys Rev E. 2020 Mar;101(3-1):032113. doi: 10.1103/PhysRevE.101.032113.
Entropy is one of the most basic concepts in thermodynamics and statistical mechanics. The most widely used definition of statistical mechanical entropy for a quantum system is introduced by von Neumann. While in classical systems, the statistical mechanical entropy is defined by Gibbs. The relation between these two definitions of entropy is still not fully explored. In this work, we study this problem by employing the phase-space formulation of quantum mechanics. For those quantum states having well-defined classical counterparts, we study the quantum-classical correspondence and quantum corrections of the entropy. We expand the von Neumann entropy in powers of ℏ by using the phase-space formulation, and the zeroth-order term reproduces the Gibbs entropy. We also obtain the explicit expression of the quantum corrections of the entropy. Moreover, we find that for the thermodynamic equilibrium state, all terms odd in ℏ are exactly zero. As an application, we derive quantum corrections for the net work extraction during a quantum Carnot cycle. Our results bring important insights into the understanding of quantum entropy and may have potential applications in the study of quantum heat engines.
熵是热力学和统计力学中最基本的概念之一。冯·诺依曼引入了量子系统中统计力学熵最广泛使用的定义。而在经典系统中，统计力学熵由吉布斯定义。这两种熵的定义之间的关系仍未得到充分探索。在这项工作中，我们通过采用量子力学的相空间表述来研究这个问题。对于那些具有明确经典对应物的量子态，我们研究了熵的量子 - 经典对应关系和量子修正。我们利用相空间表述将冯·诺依曼熵按ħ的幂次展开，零阶项重现了吉布斯熵。我们还得到了熵的量子修正的显式表达式。此外，我们发现对于热力学平衡态，ħ的所有奇次项恰好为零。作为一个应用，我们推导了量子卡诺循环中净功提取的量子修正。我们的结果为理解量子熵带来了重要的见解，并且可能在量子热机的研究中有潜在应用。
