Zhang G, Torquato S
Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA.
Department of Chemistry, Department of Physics, Princeton Institute for the Science and Technology of Materials, and Program in Applied and Computational Mathematics, Princeton University, Princeton, New Jersey 08544, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2013 Nov;88(5):053312. doi: 10.1103/PhysRevE.88.053312. Epub 2013 Nov 25.
The study of the packing of hard hyperspheres in d-dimensional Euclidean space R^{d} has been a topic of great interest in statistical mechanics and condensed matter theory. While the densest known packings are ordered in sufficiently low dimensions, it has been suggested that in sufficiently large dimensions, the densest packings might be disordered. The random sequential addition (RSA) time-dependent packing process, in which congruent hard hyperspheres are randomly and sequentially placed into a system without interparticle overlap, is a useful packing model to study disorder in high dimensions. Of particular interest is the infinite-time saturation limit in which the available space for another sphere tends to zero. However, the associated saturation density has been determined in all previous investigations by extrapolating the density results for nearly saturated configurations to the saturation limit, which necessarily introduces numerical uncertainties. We have refined an algorithm devised by us [S. Torquato, O. U. Uche, and F. H. Stillinger, Phys. Rev. E 74, 061308 (2006)] to generate RSA packings of identical hyperspheres. The improved algorithm produce such packings that are guaranteed to contain no available space in a large simulation box using finite computational time with heretofore unattained precision and across the widest range of dimensions (2≤d≤8). We have also calculated the packing and covering densities, pair correlation function g(2)(r), and structure factor S(k) of the saturated RSA configurations. As the space dimension increases, we find that pair correlations markedly diminish, consistent with a recently proposed "decorrelation" principle, and the degree of "hyperuniformity" (suppression of infinite-wavelength density fluctuations) increases. We have also calculated the void exclusion probability in order to compute the so-called quantizer error of the RSA packings, which is related to the second moment of inertia of the average Voronoi cell. Our algorithm is easily generalizable to generate saturated RSA packings of nonspherical particles.
在d维欧几里得空间(R^{d})中对硬超球体堆积的研究一直是统计力学和凝聚态物质理论中备受关注的一个话题。虽然已知最致密的堆积在足够低的维度下是有序的，但有人提出，在足够大的维度下，最致密的堆积可能是无序的。随机顺序添加（RSA）随时间变化的堆积过程，即全等的硬超球体被随机且顺序地放置到一个没有粒子间重叠的系统中，是研究高维无序的一个有用的堆积模型。特别令人感兴趣的是无限时间饱和极限，在这个极限下，另一个球体的可用空间趋于零。然而，在所有先前的研究中，相关的饱和密度是通过将接近饱和构型的密度结果外推到饱和极限来确定的，这必然会引入数值上的不确定性。我们改进了我们自己设计的一种算法 [S. 托尔夸托、O. U. 乌切和F. H. 斯蒂林格，《物理评论E》74, 061308 (2006)] 来生成相同超球体的RSA堆积。改进后的算法能生成这样的堆积，即在一个大的模拟盒中，使用有限的计算时间，以迄今未达到的精度并在最宽的维度范围（2≤d≤8）内保证不存在可用空间。我们还计算了饱和RSA构型的堆积密度和覆盖密度、对关联函数(g^{(2)}(r))以及结构因子(S(k))。随着空间维度的增加，我们发现对关联显著减小，这与最近提出的“去相关”原理一致，并且“超均匀性”（抑制无限波长密度涨落）的程度增加。我们还计算了空隙排除概率，以便计算RSA堆积的所谓量化误差，它与平均沃罗诺伊胞的惯性矩的二阶矩有关。我们的算法很容易推广以生成非球形粒子的饱和RSA堆积。
