The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains
The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains
复制标题
具有最大不重叠球形颗粒的泊松胚芽模型的体积分数
DOI：
--
复制
发表时间：
1999
影响因子：
1.2
通讯作者：
D. Stoyan
中科院分区：
文献类型：
--
作者：
Daryl J. Daley;H. Stoyan;D. Stoyan
本文考虑了一个随机系统的芽-粒模型，该系统由非重叠的球体组成，d = 1，2，3。球的中心（即“胚芽”的“颗粒”）形成一个平稳的泊松过程;球的结果从一个均匀的增长过程开始在同一时刻在所有点的径向方向和停止任何球时，它接触任何其他领域。上限和下限推导出的球体所占的空间的体积分数;模拟产生的值为0.632，0.349和0.186 d = 1，2和3。模拟还提供了随机选择的球体的体积的分布函数的尾部的估计;这些尾部与两个指数分布的尾部进行比较，其中一个是下限，并且在原点处是渐近线，另一个具有与模拟分布相同的平均值。分布尾部的上界也是原点处的渐近线，但比这两种指数分布的尾部更重。Daley、Mallow和Shepp发现了一维情况的更详细信息;总结了相关信息，包括体积分数1 - e-1 = 0.63212和晶粒体积分布e-y exp（e-y - 1）的尾部，该尾部比指数边界更接近d = 2和3的模拟尾部。
This paper considers a germ-grain model for a random system of non-overlapping spheres in ℝ d for d = 1, 2 and 3. The centres of the spheres (i.e. the ‘germs’ for the ‘grains’) form a stationary Poisson process; the spheres result from a uniform growth process which starts at the same instant in all points in the radial direction and stops for any sphere when it touches any other sphere. Upper and lower bounds are derived for the volume fraction of space occupied by the spheres; simulation yields the values 0.632, 0.349 and 0.186 for d = 1, 2 and 3. The simulations also provide an estimate of the tail of the distribution function of the volume of a randomly chosen sphere; these tails are compared with those of two exponential distributions, of which one is a lower bound and is an asymptote at the origin, and the other has the same mean as the simulated distribution. An upper bound on the tail of the distribution is also an asymptote at the origin but has a heavier tail than either of these exponential distributions. More detailed information for the one-dimensional case has been found by Daley, Mallows and Shepp; relevant information is summarized, including the volume fraction 1 - e-1 = 0.63212 and the tail of the grain volume distribution e-y exp(e-y - 1), which is closer to the simulated tails for d = 2 and 3 than the exponential bounds.