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

Research Report SRR97-007

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

D.J. Daley, D. Stoyan and H. Stoyan

Abstract: 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. The volume fraction of the space occupied by the spheres is bounded from above and below; 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 is available for the one-dimensional case, including close bounds on the tail of the volume distribution.

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