Zachary Chase E, Torquato Salvatore
Department of Chemistry, Princeton University, Princeton, New Jersey 08544, USA.
Phys Rev E Stat Nonlin Soft Matter Phys. 2011 May;83(5 Pt 1):051133. doi: 10.1103/PhysRevE.83.051133. Epub 2011 May 31.
We provide numerical constructions of one-dimensional hyperuniform many-particle distributions that exhibit unusual clustering and asymptotic local number density fluctuations growing more slowly than the volume of an observation window but faster than the surface area. Hyperuniformity, defined by vanishing infinite-wavelength local density fluctuations, provides a quantitative metric of global order within a many-particle configuration and signals the onset of an "inverted" critical point in which the direct correlation function becomes long ranged. By targeting a specified form of the structure factor at small wavenumbers (S(k)~k(α) for 0<α<1) using collective density variables, we are able to tailor the form of asymptotic local density fluctuations while simultaneously measuring the effect of imposing weak and strong constraints on the available degrees of freedom within the system. This procedure is equivalent to finding the (possibly disordered) classical ground state of an interacting many-particle system with up to four-body interactions. Even in one dimension, the long-range effective interactions induce clustering and nontrivial phase transitions in the resulting ground-state configurations. We provide an analytical connection between the fraction of constrained degrees of freedom within the system and the disorder-order phase transition for a class of target structure factors by examining the realizability of the constrained contribution to the pair correlation function. Our results explicitly demonstrate that disordered hyperuniform many-particle ground states, and therefore also point distributions, with substantial clustering can be constructed. We directly relate the local coordination structure of our point patterns to the distribution of the void space external to the particles, and we provide a scaling argument for the configurational entropy (analogous to spin-frustated system) of the disordered ground states. By emphasizing the intimate connection between geometrical constraints on the particle distribution and structural regularity, our work has direct implications for higher-dimensional systems, including an understanding of the appearance of hyperuniformity and quasi-long-range pair correlations in maximally random strictly jammed packings of hard spheres.
我们提供了一维超均匀多粒子分布的数值构造，这些分布呈现出不寻常的聚类现象，并且渐近局部数密度涨落的增长速度比观测窗口的体积慢，但比表面积快。超均匀性由无限波长局部密度涨落的消失来定义，它为多粒子构型中的全局有序提供了一个定量度量，并标志着“反转”临界点的出现，在该临界点处直接相关函数变得长程。通过使用集体密度变量在小波数下针对结构因子的特定形式（对于(0 < \alpha < 1)，(S(k) \sim k^{\alpha})），我们能够定制渐近局部密度涨落的形式，同时测量对系统内可用自由度施加弱约束和强约束的效果。这个过程等同于找到一个具有多达四体相互作用的相互作用多粒子系统的（可能无序的）经典基态。即使在一维中，长程有效相互作用也会在所得的基态构型中诱导聚类和非平凡的相变。通过研究对配对相关函数的约束贡献的可实现性，我们为一类目标结构因子提供了系统内受约束自由度的分数与无序 - 有序相变之间的解析联系。我们的结果明确表明，可以构建具有大量聚类的无序超均匀多粒子基态，因此也可以构建点分布。我们直接将点模式的局部配位结构与粒子外部的空隙空间分布联系起来，并为无序基态的构型熵（类似于自旋受挫系统）提供了一个标度论证。通过强调粒子分布的几何约束与结构规则性之间的紧密联系，我们的工作对高维系统有直接影响，包括对硬球最大随机严格堵塞堆积中超均匀性和准长程配对相关性出现的理解。
