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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.
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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 

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References

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