统计力学中的重加权原理
The Reweighting Principle in Statistical Mechanics
AI总结：
研究概率测度重加权在统计力学中对条件设定、指数倾斜及系综变换的统一视角，通过最小相对熵更新等方法揭示相关热力学结构及系综关系，并将框架拓展到路径空间。
AI中文摘要：
概率测度的重加权为统计力学中的条件设定、指数倾斜以及更一般的系综变换提供了统一视角。我们表明指数倾斜和条件设定分别作为与软约束和硬约束相关的最小相对熵更新出现。它们的相对熵自然地继承互补的热力学结构……最后，我们概述了相同的信息理论框架如何自然地扩展到路径空间，为平衡热力学和条件随机动力学提供统一的概率描述。
英文摘要：
Reweighting of probability measures provides a unifying perspective on ensemble transformations in statistical mechanics. We distinguish two complementary classes of reweighting: soft constraints, which redistribute probability while preserving the support of the reference measure, and hard constraints, which impose support restrictions through conditioning. We show that exponential tilting and conditioning on an exact observable value arise as the minimum relative entropy updates associated with soft expectation constraints and hard exact-value constraints, respectively. Their relative entropies naturally inherit complementary thermodynamic structures: exponential tilting gives rise to the Legendre structure of the canonical ensemble and reduces, for a uniform reference measure, to Gibbs entropy, whereas conditioning reduces to Boltzmann entropy through the surprisal of the constrained macrostate. By introducing an enlarged probability space in which observables are treated as explicit random variables, we further show that canonical and microcanonical ensembles arise as marginal and conditional distributions of a common joint reference measure. In the thermodynamic limit, large-deviation concentration makes soft and hard constraints macroscopically equivalent, providing a probabilistic interpretation of canonical--microcanonical ensemble equivalence. Finally, we outline how the same information-theoretic framework naturally extends to path space, suggesting a unified probabilistic description of equilibrium statistical mechanics and conditioned stochastic dynamics.
