为什么规范不变性适用于统计力学
Why gauge invariance applies to statistical mechanics
AI总结：
本文介绍经典多体系统平衡统计力学中的规范不变性，阐明其几何与代数基础，并通过蒙特卡洛模拟验证，提出将其作为构建精确结果与智能采样算法的原理。
摘要：
我们给出了最近发现的经典多体系统平衡统计力学规范不变性的入门性阐述[J. Müller等人，Phys. Rev. Lett. 133, 217101 (2024)]。规范变换是相空间上的一种非交换平移操作，它保持微分相空间体积元不变，从而保持吉布斯积分测度守恒。当进行热平均时，任何可观测量（包括热力学量和结构量）都是不变量。平移变换在经典力学意义上是正则的。它们还构成一个无限维群，其无穷小变换的生成元构建了一个非交换李代数。我们阐述了与坐标位移的底层几何以及诺特定理的联系。平移的空间局域化产生了满足对易关系的微分算子，我们在纯构型设置和完整相空间设置中分别描述了这些关系。标准算子微积分给出了针对一般可观测量的相应平衡超力相关求和规则。利用蒙特卡洛模拟，我们明确展示了有限平移下的规范不变性。我们主张将规范不变性作为一种统计力学构建原理，用于获得精确结果和制定智能采样算法。
英文摘要：
We give an introductory account of the recently identified gauge invariance of the equilibrium statistical mechanics of classical many-body systems [J. Müller et al., Phys. Rev. Lett. Phys. Rev. Lett. 133, 217101 (2024)]. The gauge transformation is a non-commutative shifting operation on phase space that keeps the differential phase space volume element and hence the Gibbs integration measure conserved. When thermally averaged any observable is an invariant, including thermodynamic and structural quantities. Shifting transformations are canonical in the sense of classical mechanics. They also form an infinite-dimensional group with generators of infinitesimal transformations that build a non-commutative Lie algebra. We lay out the connections with the underlying geometry of coordinate displacement and with Noether's theorem. Spatial localization of the shifting yields differential operators that satisfy commutator relationships, which we describe both in purely configurational and in full phase space setups. Standard operator calculus yields corresponding equilibrium hyperforce correlation sum rules for general observables and order parameters. Using Monte Carlos simulations we demonstrate explicitly the gauge invariance for finite shifting. We argue in favour of using the gauge invariance as a statistical mechanical construction principle for obtaining exact results and for formulating smart sampling algorithms.
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