Department of Physics & Astronomy, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA.
Phys Rev E. 2017 Sep;96(3-1):032910. doi: 10.1103/PhysRevE.96.032910. Epub 2017 Sep 15.
The concept of a hyperuniformity disorder length h was recently introduced for analyzing volume fraction fluctuations for a set of measuring windows [Chieco et al., Phys. Rev. E 96, 032909 (2017).PLEEE81539-375510.1103/PhysRevE.96.032909]. This length permits a direct connection to the nature of disorder in the spatial configuration of the particles and provides a way to diagnose the degree of hyperuniformity in terms of the scaling of h and its value in comparison with established bounds. Here, this approach is generalized for extended particles, which are larger than the image resolution and can lie partially inside and partially outside the measuring windows. The starting point is an expression for the relative volume fraction variance in terms of four distinct volumes: that of the particle, the measuring window, the mean-squared overlap between particle and region, and the region over which particles have nonzero overlap with the measuring window. After establishing limiting behaviors for the relative variance, computational methods are developed for both continuum and pixelated particles. Exact results are presented for particles of special shape and for measuring windows of special shape, for which the equations are tractable. Comparison is made for other particle shapes, using simulated Poisson patterns. And the effects of polydispersity and image errors are discussed. For small measuring windows, both particle shape and spatial arrangement affect the form of the variance. For large regions, the variance scaling depends only on arrangement but particle shape sets the numerical proportionality. The combined understanding permit the measured variance to be translated to the spectrum of hyperuniformity lengths versus region size, as the quantifier of spatial arrangement. This program is demonstrated for a system of nonoverlapping particles at a series of increasing packing fractions as well as for an Einstein pattern of particles with several different extended shapes.
最近引入了超均匀性无序长度 h 的概念，用于分析一组测量窗口中的体积分数波动[Chieco 等人，Phys. Rev. E 96, 032909（2017）。PLEEE81539-375510.1103/PhysRevE.96.032909]。该长度允许与粒子空间配置的无序性质直接连接，并提供了一种根据 h 的缩放及其与已建立边界的比较值来诊断超均匀性程度的方法。在这里，该方法被推广到扩展粒子，这些粒子比图像分辨率大，可以部分位于测量窗口内和部分位于测量窗口外。出发点是用四个不同的体积来表示相对体积分数方差的表达式：粒子的体积、测量窗口的体积、粒子和区域之间的均方重叠体积，以及区域内的粒子与测量窗口有非零重叠的区域。在确定相对方差的极限行为之后，为连续粒子和像素化粒子开发了计算方法。对于具有特殊形状的粒子和具有特殊形状的测量窗口，给出了精确结果，因为这些方程是可处理的。使用模拟泊松模式对其他粒子形状进行了比较。并讨论了多分散性和图像误差的影响。对于小的测量窗口，粒子形状和空间排列都会影响方差的形式。对于大的区域，方差的缩放仅取决于排列，但粒子形状决定了数值比例。综合理解允许将测量的方差转换为超均匀性长度与区域大小的谱，作为空间排列的量化器。该程序演示了一系列增加的填充分数下非重叠粒子系统以及具有几种不同扩展形状的爱因斯坦粒子图案的情况。
