热力学态几何
Geometry of Thermodynamic States
摘要：
本文将一种新的几何统计估计方法应用于经典统计力学的正则分布。在这个方案中，热力学状态，或者等价地，统计力学状态，可以根据实希尔伯特空间${\cal H}$中单位球${\cal S}$的子流形${\cal M}$的几何来简明地表征。然后，热力学变量的测量对应于将${\cal H}$中的状态矢量还原为本征态，其中跃迁概率是玻尔兹曼权重。我们导出了一组共轭热力学变量在平衡热力学状态下的测不准关系。这些都是量子力学中阿南丹-阿哈罗诺夫关系的一个惊人的热力学类似物的结果。因此，我们能够提供一个解决方案的争论周围的“温度波动”的规范合奏的地位。通过考虑其状态空间中的热力学轨迹的曲率，然后我们能够导出一系列高阶方差界，我们明确地计算到二阶。
英文摘要：
A novel geometric formalism for statistical estimation is applied here to the canonical distribution of classical statistical mechanics. In this scheme thermodynamic states, or equivalently, statistical mechanical states, can be characterised concisely in terms of the geometry of a submanifold ${\cal M}$ of the unit sphere ${\cal S}$ in a real Hilbert space ${\cal H}$. The measurement of a thermodynamic variable then corresponds to the reduction of a state vector in ${\cal H}$ to an eigenstate, where the transition probability is the Boltzmann weight. We derive a set of uncertainty relations for conjugate thermodynamic variables in the equilibrium thermodynamic states. These follow as a consequence of a striking thermodynamic analogue of the Anandan-Aharonov relations in quantum mechanics. As a result we are able to provide a resolution to the controversy surrounding the status of `temperature fluctuations' in the canonical ensemble. By consideration of the curvature of the thermodynamic trajectory in its state space we are then able to derive a series of higher order variance bounds, which we calculate explicitly to second order.
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