Hostname: page-component-77c89778f8-vpsfw Total loading time: 0 Render date: 2024-07-18T23:50:54.614Z Has data issue: false hasContentIssue false
  • English
  • Français

The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains

Part of: Inference from stochastic processes Stochastic processes Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

D. J. Daley ,
H. Stoyan  and
D. Stoyan
D. J. Daley*
Affiliation:
Australian National University
H. Stoyan*
Affiliation:
TU Bergakademie Freiberg
D. Stoyan*
Affiliation:
TU Bergakademie Freiberg
*
∗ Postal address: School of Mathematical Sciences, Australian National University, Canberra ACT 0200, Australia. Email address: daryl@maths.anu.edu.au
∗∗ Postal address: TU Bergakademie Freiberg, Institut für Stochastik, 09596 Freiberg, Germany.
∗∗ Postal address: TU Bergakademie Freiberg, Institut für Stochastik, 09596 Freiberg, Germany.
Get access Rights & Permissions [Opens in a new window]

Abstract

This paper considers a germ-grain model for a random system of non-overlapping spheres in ℝd 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-yexp(e-y - 1), which is closer to the simulated tails for d = 2 and 3 than the exponential bounds.

Keywords

Germ-grain modelgrowth modelnon-overlapping spheresdynamic lily-pond modelPoisson processStienen modelvolume fraction

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 62M30: Spatial processes 60G55: Point processes
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 31 , Issue 3 , September 1999 , pp. 610 - 624
Copyright
Copyright © Applied Probability Trust 1999 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Daley, D. J., Stoyan, D. and Stoyan, H. (1997). The volume fraction of a Poisson germ model with maximally non-overlapping spherical grains. Statistics research report No. SRR 006–97, Centre for Mathematics and its Applications, Australian National University, Canberra.Google Scholar
Daley, D. J., Mallows, C. L. and Shepp, L. A. (1999). A one-dimensional Poisson growth model with non-overlapping intervals. To appear in Stoch. Proc. Appl.Google Scholar
Diggle, P. J. (1983). Statistical Analysis of Spatial Point Patterns. Chapman and Hall, London.Google Scholar
Häggström, O. and Meester, R. (1996). Nearest neighbor and hard sphere models in continuum percolation. Rand. Struct. Algorithms 9, 295–315.3.0.CO;2-S>CrossRefGoogle Scholar
Herczynski, R. (1975). Distribution function for random distribution of spheres. Nature 255, 540–541.CrossRefGoogle Scholar
Pickard, D. K. (1982). Isolated nearest neighbors. J. Appl. Prob. 19, 444–449.CrossRefGoogle Scholar
Quine, M. P. and Watson, D. F. (1984). Radial simulation of n-dimensional Poisson processes. J. Appl. Prob. 21, 548–557.CrossRefGoogle Scholar
Schlather, M. and Stoyan, D. (1997). The covariance of the Stienen model. In Advances in Theory and Applications of Random Sets, ed. Jeulin, D.. World Scientific, Singapore, pp. 157–174.Google Scholar
Stienen, J. (1982). Die Vergroeberung von Karbiden in reinen Eisen-Kohlenstoff-Staehlen. Dissertation, RWTH Aachen, Germany.Google Scholar
Stoyan, D. (1990). Stereological formulae for a random system of non-overlapping spheres. Statistics 21, 131–136.CrossRefGoogle Scholar
Stoyan, D. (1998). Random sets: models and statistics. Internat. Statist. Rev. 66, 1–27.CrossRefGoogle Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1995). Stochastic Geometry and its Applications. Wiley, Chichester, UK.Google Scholar
Yao, Y-C. and Simons, G. (1996). A large-dimensional independent and identically distributed property for nearest neighbour counts in Poisson processes. Ann. Appl. Prob. 6, 561–571.CrossRefGoogle Scholar