Abstract
The two-point cluster functionC 2(r 1,r 2) provides a measure of clustering in continuum models of disordered many-particle systems and thus is a useful signature of the microstructure. For a two-phase disordered medium,C 2(r 1,r 2) is defined to be the probability of finding two points at positionsr 1 andr 2 in thesame cluster of one of the phases. An exact analytical expression is found for the two-point cluster functionC 2(r 1,r 2) of a one-dimensional continuumpercolation model of Poisson-distributed rods (for an arbitrary number density) using renewal theory. We also give asymptotic formulas for the tail probabilities. Along the way we find exact results for other cluster statistics of this continuum percolation model, such as the cluster size distribution, mean number of clusters, and two-point blocking function.
Similar content being viewed by others
References
S. Haan and R. Zwanzig,J. Phys. A: Math. Gen. 10:1123 (1977).
Y. C. Chiew and E. D. Glandt,J. Phys. A: Math. Gen. 16:2599 (1983).
G. Stell,J. Phys. A: Math. Gen. 17:L855 (1984).
T. DiSimone, S. Demoulini, and R. M. Stratt,J. Chem. Phys. 85:392 (1986).
S. Torquato, J. D. Beasley, and Y. C. Chiew,J. Chem. Phys. 88:6540 (1988).
J. Given and G. Stell,Physica A 161:152 (1989).
S. B. Lee and S. Torquato,J. Chem. Phys. 91:1173 (1989).
J. Given, I. C. Kim, S. Torquato, and G. Stell,J. Chem. Phys. 93:5128 (1990).
G. W. Milton,Appl. Phys. Lett. 52:5294 (1981).
G. W. Milton and N. Phan-Thien,Proc. R. Soc. Lond. A 380:305 (1982).
S. Torquato and G. Stell,J. Chem. Phys. 77:2071 (1982).
W. Feller,An Introduction to Probability Theory and Its Applications, Vol. II (Wiley, New York, 1966).
Z. W. Salsburg, R. W. Zwanzig, and J. G. Kirkwood,J. Chem. Phys. 21:1098 (1953).
A. D. J. Haymet,J. Chem. Phys. 80:3801 (1984).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Cinlar, E., Torquato, S. Exact determination of the two-point cluster function for one-dimensional continuum percolation. J Stat Phys 78, 827–839 (1995). https://doi.org/10.1007/BF02183690
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02183690