标题： 统计力学的几何化 显示英文标题

标题： Geometrisation of Statistical Mechanics

摘要： 经典和量子统计力学在这里用射影几何的语言来表述，以提供统计物理的统一几何框架。 在回顾了经典统计热力学的希尔伯特空间表述之后，我们引入射影几何作为分析统计物理概率方面的基础。 特别是，在$RP^{n}$上指定一个规范对偶性会诱导出统计力学状态空间上的黎曼度量。 在典型系综的情况下，我们证明平衡热态是由该度量下的哈密顿梯度流决定的。 这种流被简洁地表征为它在状态流形上诱导了一个射影自同构。 通过引入随机态的概念来研究热系统的测量问题。 然后将一般方法扩展以包括平衡热态的量子力学动力学。 在这种情况下，相关状态空间是复射影空间，这里被视为带有自然法博-施泰因度量的实流形。 量子热力学的一个显著特征是状态空间中热轨迹的固有多重性，这与无限温度态的非唯一性有关。 随后，我们提出了一种标准KMS关系的几何表征，该关系通常在$C^{*}$-代数的背景下被考虑。 详细研究了处于热浴中的量子自旋二分之一粒子的例子。 显示英文摘要

摘要： Classical and quantum statistical mechanics are cast here in the language of projective geometry to provide a unified geometrical framework for statistical physics. After reviewing the Hilbert space formulation of classical statistical thermodynamics, we introduce projective geometry as a basis for analysing probabilistic aspects of statistical physics. In particular, the specification of a canonical polarity on $RP^{n}$ induces a Riemannian metric on the state space of statistical mechanics. In the case of the canonical ensemble, we show that equilibrium thermal states are determined by the Hamiltonian gradient flow with respect to this metric. This flow is concisely characterised by the fact that it induces a projective automorphism on the state manifold. The measurement problem for thermal systems is studied by the introduction of the concept of a random state. The general methodology is then extended to include the quantum mechanical dynamics of equilibrium thermal states. In this case the relevant state space is complex projective space, here regarded as a real manifold endowed with the natural Fubini-Study metric. A distinguishing feature of quantum thermal dynamics is the inherent multiplicity of thermal trajectories in the state space, associated with the nonuniqueness of the infinite temperature state. We are then led to formulate a geometric characterisation of the standard KMS-relation often considered in the context of $C^{*}$-algebras. The example of a quantum spin one-half particle in heat bath is studied in detail.