硬圆弧密排
Dense packings of hard circular arcs
摘要：
本文研究了二维欧氏空间中全等硬无穷细圆弧的稠密填充问题。它侧重于那些可表示为主要的，其对向角$θ\in \left ( π, 2π\right ]$。与那些可表示为其对向角$θ\in \left [0, π\right]$的小圆弧不同，具有$θ\in \left ( π, 2π\right ]$的两个硬的无限细圆弧不可能任意地接近，一旦它们被布置在一种配置中，例如在彼此的顶部上，可无限地复制而不引入任何重叠。这使得这些硬凹颗粒，尽管是无限薄，最密集的包装有限的数量密度。这就提出了一个问题，即什么是这些密度填充，以及它们达到的数量密度是多少。在Monte Carlo数值模拟的支持下，本文的工作表明，可以解析地构造由硬主圆弧组成的紧致闭圆群，其中特定的、与$θ$相关的、数量的硬主圆弧（逆时针）相互交错.这些紧凑的封闭的圆形群，然后安排在一个三角形网格。这些解析构造的最密已知的包装相比，相应的结果的Monte Carlo数值模拟，以评估它们是否可以自发地出现。
英文摘要：
This work investigates dense packings of congruent hard infinitesimally--thin circular arcs in the two-dimensional Euclidean space. It focuses on those denotable as major whose subtended angle $θ\in \left ( π, 2π\right ]$. Differently than those denotable as minor whose subtended angle $θ\in \left [0, π\right]$, it is impossible for two hard infinitesimally-thin circular arcs with $θ\in \left ( π, 2π\right ]$ to arbitrarily closely approach once they are arranged in a configuration, e.g. on top of one another, replicable ad infinitum without introducing any overlap. This makes these hard concave particles, in spite of being infinitesimally thin, most densely pack with a finite number density. This raises the question as to what are these densest packings and what is the number density that they achieve. Supported by Monte Carlo numerical simulations, this work shows that one can analytically construct compact closed circular groups of hard major circular arcs in which a specific, $θ$-dependent, number of them (anti-)clockwise intertwine. These compact closed circular groups then arrange on a triangular lattice. These analytically constructed densest-known packings are compared to corresponding results of Monte Carlo numerical simulations to assess whether they can spontaneously turn up.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
