Optimal packings of uniform spheres are solved problems in two and three dimensions. The main difference between them is that the two-dimensional ground state can be easily achieved by simple dynamical processes while in three dimensions, this is impossible due to the difference in the local and global optimal packings. In this paper we show experimentally and numerically that in 2 + ε dimensions, realized by a container which is in one dimension slightly wider than the spheres, the particles organize themselves in a triangular lattice, while touching either the front or rear side of the container. If these positions are denoted by up and down the packing problem can be mapped to a 1/2 spin system. At first it looks frustrated with spin-glass like configurations, but the system has a well defined ground state built up from isosceles triangles. When the system is agitated, it evolves very slowly towards the potential energy minimum through metastable states. We show that the dynamics is local and is driven by the optimization of the volumes of 7-particle configurations and by the vertical interaction between touching spheres.

中文翻译：

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几何约束下粒状系统中的压实填料

均匀球体的最佳堆积解决了二维和三维问题。它们之间的主要区别在于，可以通过简单的动力学过程轻松地获得二维基态，而在三个维度上，由于局部和全局最佳包装的不同，这是不可能的。在本文中，我们从实验和数值上证明了2 + ε通过一个尺寸比球体稍宽的容器实现尺寸，这些粒子在接触容器的前侧或后侧时以三角形格子的形式组织起来。如果这些位置由上下表示，则堆积问题可以映射到1/2自旋系统。乍一看，它看起来像旋转玻璃一样令人沮丧，但是该系统具有由等腰三角形构成的定义明确的基态。搅动系统时，它会通过亚稳态非常缓慢地朝着势能最小值发展。我们表明动力学是局部的，是由7粒子配置的体积的优化和接触球体之间的垂直相互作用所驱动的。