Department of Physics and Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104-6396, USA.
Complex Assemblies of Soft Matter, CNRS-Solvay-UPenn UMI 3254, Bristol, Pennsylvania 19007-3624, USA.
Phys Rev E. 2017 Sep;96(3-1):032909. doi: 10.1103/PhysRevE.96.032909. Epub 2017 Sep 15.
We introduce the concept of a "hyperuniformity disorder length" h that controls the variance of volume fraction fluctuations for randomly placed windows of fixed size. In particular, fluctuations are determined by the average number of particles within a distance h from the boundary of the window. We first compute special expectations and bounds in d dimensions, and then illustrate the range of behavior of h versus window size L by analyzing several different types of simulated two-dimensional pixel patterns-where particle positions are stored as a binary digital image in which pixels have value zero if empty and one if they contain a particle. The first are random binomial patterns, where pixels are randomly flipped from zero to one with probability equal to area fraction. These have long-ranged density fluctuations, and simulations confirm the exact result h=L/2. Next we consider vacancy patterns, where a fraction f of particles on a lattice are randomly removed. These also display long-range density fluctuations, but with h=(L/2)(f/d) for small f, and h=L/2 for f→1. And finally, for a hyperuniform system with no long-range density fluctuations, we consider "Einstein patterns," where each particle is independently displaced from a lattice site by a Gaussian-distributed amount. For these, at large L,h approaches a constant equal to about half the root-mean-square displacement in each dimension. Then we turn to gray-scale pixel patterns that represent simulated arrangements of polydisperse particles, where the volume of a particle is encoded in the value of its central pixel. And we discuss the continuum limit of point patterns, where pixel size vanishes. In general, we thus propose to quantify particle configurations not just by the scaling of the density fluctuation spectrum but rather by the real-space spectrum of h(L) versus L. We call this approach "hyperuniformity disorder length spectroscopy".
我们引入了“超均匀无序长度” h 的概念，用于控制随机放置的固定大小窗口中体积分数波动的方差。具体来说，波动取决于距离窗口边界 h 内的平均粒子数。我们首先在 d 维中计算了特殊的期望和界，然后通过分析几种不同类型的二维模拟像素模式来说明了 h 与窗口大小 L 的行为范围 - 其中粒子位置存储为二进制数字图像，其中像素的值为零表示为空，值为一表示包含粒子。第一种是随机二项式模式，其中像素以等于面积分数的概率从零随机翻转到一。这些具有长程密度波动，模拟结果证实了 h=L/2 的精确结果。接下来我们考虑空位模式，其中晶格上的一部分粒子以分数 f 随机移除。这些也显示了长程密度波动，但对于小 f，h=(L/2)(f/d)，对于 f→1，h=L/2。最后，对于没有长程密度波动的超均匀系统，我们考虑“爱因斯坦模式”，其中每个粒子都由高斯分布的量独立地从晶格位置位移。对于这些，在大 L 时，h 接近每个维度的均方根位移的一半左右的常数。然后我们转向灰度像素模式，它代表模拟的多分散粒子排列，其中粒子的体积由其中心像素的值表示。我们讨论了点模式的连续体极限，其中像素尺寸消失。一般来说，我们因此建议不仅通过密度波动谱的标度，而且通过 h(L)与 L 的实空间谱来量化粒子构型。我们将这种方法称为“超均匀无序长度光谱学”。
