Clustering properties of d-dimensional overlapping spheres.
Clustering properties of d-dimensional overlapping spheres.
复制标题
d 维重叠球体的聚类特性。
DOI：
--
复制
发表时间：
1996
期刊：
影响因子：
--
通讯作者：
S. Torquato
中科院分区：
文献类型：
--
作者：
J. Quintanilla;S. Torquato
连续渗滤的典型模型是一个空间不相关、一维、二维和三维@1‐14# 大小相等的球体系统。在此模型中，大小相等的球体以平稳泊松过程的点为中心。由于允许球体重叠，因此形成了各种尺寸和体积的簇，如图 1 所示。该模型被赋予了多种名称，包括“完全可穿透球体”、“随机重叠球体”、“瑞士奶酪模型”和“泊松斑点模型”。此后，我们将该模型称为重叠球体。该模型 @3‐12,15,16# 已通过分析和数值确定了某些类型的“连通性”函数和相关量。在本文中，我们提出了 k 聚体〜由 k 个球体组成的簇的平均数密度 nk 的积分表示！以及 d 维重叠球体的 k 聚体的平均体积 v k 。 nk 已用于估计各种重叠粒子系统@1# 的渗流阈值和临界指数，严格限制平均簇密度@11#，并研究渗流模型@17# 中的表面张力。我们使用构造性范例来有效地评估 nk 和 v k 的这些积分表达式。为了说明这一范式，我们得出一维量 pk 和 v k 的精确分析结果。然而，在更高维度中，这些量是无法通过分析评估的积分，我们必须满足于数值评估。我们发现构造性范式产生的积分没有先前工作@16#中固有的冗余，因此可以更有效地对这些积分进行数值评估。执行这些数值积分的工作量随着 k 和维数的增加而增加，因此这些积分的有效计算变得势在必行。通过这种方法，我们能够计算 n4 并正确评估任意球体数密度 r 的 v 2、v 3 和 v 4。我们对这些量的评估与我们也执行的蒙特卡罗模拟非常一致。我们还将研究与系统内簇的平均粒子数和平均体积相关的各种簇统计数据。我们将考虑 Q，随机选择的簇中包含的平均颗粒数，S，随机选择的颗粒的簇中的平均颗粒数，VQ，随机选择的簇的平均体积，以及 VS，包含随机选择的颗粒的簇的平均体积。 “粒子平均簇数”S 和“粒子平均簇体积”VS 都在渗透阈值处发散~无限大小和体积的簇存在的最小密度@15,16#!。然而，“平均簇数”Q 和“平均簇体积”VQ 在空间维度 d>2 的渗透阈值下仍然有限。聚类统计量 S，有时称为平均聚类大小，是渗流理论中的一个众所周知的量，并且已被研究
A prototypical model of continuum percolation is a system of spatially uncorrelated, equal-sized spheres in one, two, and three dimensions @1‐14#. In this model, spheres of equal size are centered on the points of a stationary Poisson process. Since the spheres are allowed to overlap, clusters of various sizes and volumes are formed, as depicted in Fig. 1. This model has been given a variety of names, including ‘‘fully penetrable spheres,’’ ‘‘randomly overlapping spheres,’’ the ‘‘Swiss-cheese model,’’ and the ‘‘Poisson blob model.’’ We shall henceforth refer to this model as overlapping spheres. Certain types of ‘‘connectedness’’ functions and related quantities have been analytically and numerically determined for this model @3‐12,15,16#. In this paper we present integral representations of the average number density nk of a k-mer ~a cluster comprised of k spheres! and the average volume v k of a k-mer for overlapping spheres in d dimensions. The nk have been used to estimate the percolation threshold and critical exponent of various overlapping particle systems @1#, rigorously bound the mean cluster density @11#, and study surface tension in percolation models @17#. We use a constructive paradigm to efficiently evaluate these integral expressions for nk and v k . To illustrate this paradigm, we derive exact analytical results for the quantities pk and v k in one dimension. In higher dimensions, however, these quantities are integrals that cannot be evaluated analytically and we have to settle for numerical evaluation. We find that the constructive paradigm yields integrals free of the redundancies inherent in previous work @16# and so these integrals can be numerically evaluated more efficiently. The effort to perform these numerical integrations increases as k and the number of dimensions increase, and therefore efficient computation of these integrals becomes imperative. With this approach, we are able to compute n4 and correctly evaluate v 2, v 3, and v 4 for any sphere number density r. Our evaluations of these quantities are in excellent agreement with Monte Carlo simulations that we also perform. We also will study various cluster statistics related to the average number of particles and average volume of the clusters within the system. We will consider Q, the average number of particles contained in a cluster chosen at random, S, the average number of particles in the cluster of a particle chosen at random, VQ , the average volume of a randomly chosen cluster, and VS , the average volume of the cluster containing a randomly chosen particle. The ‘‘particleaveraged cluster number’’ S and the ‘‘particle-averaged cluster volume’’ VS both diverge at the percolation threshold ~the minimum density at which a cluster of infinite size and volume exists @15,16#!. However, the ‘‘average cluster number’’ Q and ‘‘average cluster volume’’ VQ remain finite at the percolation threshold for spatial dimension d>2. The cluster statistic S, sometimes called the mean cluster size, is a wellknown quantity in percolation theory and has been studied