多体自由体积的闭式解析形式与单分散硬圆盘热力学
Closed analytical form of many-body free volume and thermodynamics of monodisperse hard disks
- Institute of Physics of the National Academy of Sciences of Ukraine（乌克兰国家科学院物理研究所）
- Center for Theoretical Physics, Polish Academy of Sciences（波兰科学院理论物理中心）
- Yukhnovskii Institute for Condensed Matter Physics of the National Academy of Sciences of Ukraine（乌克兰国家科学院尤赫诺夫斯基凝聚态物理研究所）
- Institute of Applied Mathematics and Fundamental Sciences, Lviv National Polytechnic University（利沃夫国立理工大学应用数学与基础科学研究所）
- Faculty of Chemistry and Chemical Technology, University of Ljubljana（卢布尔雅那大学化学与化学技术学院）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文提出硬圆盘自由体积的闭式解析表示，通过最多五个排除圆盘的交叠面积计算热力学，并验证其状态方程，揭示中间密度混合流体区及六角有序化度量。
AI中文摘要：
硬圆盘模型是排除体积物理、液体结构和二维有序化的基本参考系统。然而，尽管其表面简单，其热力学的解析表述一直困难，因为圆盘中心可用的自由体积具有高度非平凡的多体几何。这里我们证明该几何具有闭式且精确的表示：对于任何给定的硬圆盘构型，自由体积可以通过最多五个排除圆盘的交叠面积解析表达。这提供了从粒子坐标到构型配分函数和熵的直接几何路径。我们证明N圆盘配分函数可分解为条件自由体积的乘积，并确定其两个极限渐近形式：由广泛空腔控制的低密度流体区和由密集私有胞控制的高密度流体区。利用从代表性平衡粒子构型计算的几何度量，所得理论在几乎整个密度范围直至接近堆积密度时复现了既定的硬圆盘状态方程。在中间密度下，分析揭示了一个混合流体区，其中局域笼状缺陷在液体六角共存之前的密度区间内提供了额外的构型熵。最后，五圆盘交叠面积被证明是六角有序化的敏感局部标量度量，补充了取向序参数，并强调多体相关性的核心作用。
英文摘要：
The hard disk model is a fundamental reference system for excluded volume physics, liquid structure, and 2D ordering. Yet, despite its apparent simplicity, an analytical formulation of its thermodynamics has remained difficult because the free volume available to disks centers has a highly nontrivial many-body geometry. Here we show that this geometry admits a closed and exact representation: for any given hard disk configuration, the free volume can be analytically expressed through intersection areas of up to five exclusion disks. This provides a direct geometrical route from particle coordinates to the configurational partition function and entropy. We prove that the N disk partition function factorizes into a product of conditional free volumes and identify its two limiting asymptotic forms: the low density fluid regime controlled by extensive cavities and the high density fluid regime controlled by intensive private cells. Using the geometric measures computed from representative equilibrium particle configurations, the resulting theory reproduces the established hard disk equation of state over almost the entire density range up to close packing. At intermediate densities, the analysis reveals a mixed fluid regime in which localized caged defects provide an additional configurational entropy within the density interval preceding liquid hexatic coexistence. Finally, the five disk intersection area is shown to be a sensitive local scalar measure of hexagonal ordering, complementing the orientational order parameter and emphasizing the central role of many-body correlations.
