标题： 重心Voronoi镶嵌作为静电平衡：凸域中的广义汤姆森问题 显示英文标题

标题： Centroidal Voronoi Tessellations as Electrostatic Equilibria: A Generalized Thomson Problem in Convex Domains

摘要： 我们提出了一种变分框架，在其中质心Voronoi镶嵌（CVTs）作为广义静电能泛函的局部极小值出现。通过将凸域内的点分布建模为与连续边界电荷平衡的排斥电荷，我们证明了由此产生的平衡构型收敛于CVT结构。我们通过证明CVTs最小化经典质心能量和静电势来实现这一点，从而建立了几何量化与势论之间的联系。最后，我们引入了一种基于玻尔兹曼统计和随机游走动力学的全局CVT优化热力学退火方案。通过引入时间步长变化的方案（更快或更慢的冷却），我们展示了可以恢复质心能量泛函（因此是静电势）的极小值集合。通过恢复对应于每个极小值的一组生成器位置，我们可以创建一个晶格延续，从而提供一个可定制的单一极小值搜索框架。 显示英文摘要

摘要： We present a variational framework in which Centroidal Voronoi Tessellations (CVTs) arise as local minimizers of a generalized electrostatic energy functional. By modeling interior point distributions in a convex domain as repelling charges balanced against a continuous boundary charge, we show that the resulting equilibrium configurations converge to CVT structures. We prove this by showing that CVTs minimize both the classical centroidal energy and the electrostatic potential, establishing a connection between geometric quantization and potential theory. Finally, we introduce a thermodynamic annealing scheme for global CVT optimization, rooted in Boltzmann statistics and random walk dynamics. By introducing a scheme for varying time steps (faster or slower cooling) we show that the set of minima of the centroid energy functional (and therefore the electrostatic potential) can be recovered. By recovering a set of generator locations corresponding to each minimum we can create a lattice continuation that allows for a customizable framework for individual minimum seeking.