Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-14T09:33:10.807Z Has data issue: false hasContentIssue false
  • English
  • Français

Clustering in a Continuum Percolation Model

Part of: Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

J. Quintanilla  and
S. Torquato
J. Quintanilla*
Affiliation:
Princeton University
S. Torquato*
Affiliation:
Princeton University
*
∗ Postal address for both authors: Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, USA.
∗ Postal address for both authors: Department of Civil Engineering and Operations Research, Princeton University, Princeton, NJ 08544, USA.
Get access Rights & Permissions [Opens in a new window]

Abstract

We study properties of the clusters of a system of fully penetrable balls, a model formed by centering equal-sized balls on the points of a Poisson process. We develop a formal expression for the density of connected clusters of k balls (called k-mers) in the system, first rigorously derived by Penrose [15]. Our integral expressions are free of inherent redundancies, making them more tractable for numerical evaluation. We also derive and evaluate an integral expression for the average volume of k-mers.

Keywords

BOOLEAN MODELCLUSTER PROPERTIES

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 29 , Issue 2 , June 1997 , pp. 327 - 336
Copyright
Copyright © Applied Probability Trust 1997 

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.)

Footnotes

This work was supported in part by the US Department of Energy, Office of Basic Energy Sciences under Grant No. DE-FG02-92ER14275, and by the MRSEC Program of the National Science Foundation under Award Number DMR-9400362.

J.Q. was partially supported under a National Science Foundation Graduate Research Fellowship. His present address is: Department of Mathematics, University of North Texas, Denton, Texas 76203, USA.

References

[1] Chiew, Y. C. and Glandt, E. D. (1983) Percolation behaviour of permeable and of adhesive spheres. J. Phys. A: Math. Gen. 16, 2599–2608.Google Scholar
[2] Çinlar, E. and Torquato, S. (1995) Exact determination of the two-point cluster function for one-dimensional continuum percolation. J. Statist. Phys. 78, 827–839.Google Scholar
[3] Disimone, T., Demoulini, S. and Stratt, R. M. (1986) A theory of percolation in liquids. J. Chem. Phys. 85, 391–400.Google Scholar
[4] Gawlinski, E. T. and Stanley, H. E. (1981) Continuum percolation in two dimensions: Monte Carlo tests of scaling and universality for non-interacting discs. J. Phys. A: Math. Gen. 14, L291–L299.CrossRefGoogle Scholar
[5] Gilbert, E. N. (1961) Random plane networks. J. SIAM 9, 533–543.Google Scholar
[6] Given, J. A., Kim, I. C., Torquato, S. and Stell, G. (1990) Comparison of analytic and numerical results for the mean cluster density in continuum percolation. J. Chem. Phys. 93, 5128–5139.Google Scholar
[7] Given, J. A. and Stell, G. (1989) The continuum Potts model and continuum percolation. Physica A 161, 152–180.CrossRefGoogle Scholar
[8] Grimmett, G. R. (1989) Percolation. Springer, New York.Google Scholar
[9] Haan, S. W. and Zwanzig, R. (1977) Series expansions in a continuum percolation problem. J. Phys. A: Math. Gen. 10, 1547–1555.CrossRefGoogle Scholar
[10] Hall, P. (1988) Introduction to the Theory of Coverage Processes. Wiley, New York.Google Scholar
[11] Helte, A. (1994) Fourth-order bounds on the effective conductivity for a system of fully penetrable spheres. Proc. R. Soc. London A 445, 247–256.Google Scholar
[12] Kratky, K. W. (1978) The area of intersection of n equal circular disks. J. Phys. A: Math. Gen. 11, 1017–1024.CrossRefGoogle Scholar
[13] Kratky, K. W. (1981) Intersecting disks (and spheres) and statistical mechanics. I. Mathematical basis. J. Statist. Phys. 25, 619–634.CrossRefGoogle Scholar
[14] Lee, S. B. and Torquato, S. (1988) Pair-connectedness and mean cluster size for continuum-percolation models: computer-simulation results. J. Chem. Phys. 89, 6427–6433.Google Scholar
[15] Penrose, M. D. (1991) On a continuum percolation model. Adv. Appl. Prob. 23, 536–556.Google Scholar
[16] Penrose, M. D. (1996) Continuum percolation and Euclidean minimal spanning trees in high dimensions. Ann. Appl. Prob. 6, 528–544.Google Scholar
[17] Powell, M. J. D. (1964) The volume internal to three intersecting hard spheres. Mol. Phys. 7, 591–592.CrossRefGoogle Scholar
[18] Quintanilla, J. and Torquato, S. (1996) Clustering properties of d-dimensional overlapping. Phys. Rev. E 54, 5331–5339.Google Scholar
[19] Ree, F. H., Keller, R. N. and Mccarthy, S. L. (1966). Radial distribution function of hard spheres. J. Chem. Phys. 44, 3407–3425.CrossRefGoogle Scholar
[20] Roach, S. A. (1968) The Theory of Random Clumping. Methuen, London.Google Scholar
[21] Rowlinson, J. S. (1963) The triplet distribution function in a fluid of hard spheres. Mol. Phys. 6, 517–524.Google Scholar
[22] Sevick, E. M., Monson, P. E. and Ottino, J. M. (1988). Monte Carlo calculations of cluster statistics in continuum models of composite materials. J. Chem. Phys. 88, 1198–1206.Google Scholar
[23] Stell, G. (1984) Exact equation for the pair-connectedness function. J. Phys. A: Math. Gen. 17, L855–L858.Google Scholar
[24] Stell, G. (1995) Continuum theory of percolation and association. Physica A 231, 1–19.Google Scholar
[25] Stell, G. (1996) Continuum theory of percolation. J. Phys: Cond. Matter 8, A1–A17.Google Scholar
[26] Torquato, S. Beasley, J. D. and Chiew, Y. C. (1988) Two-point cluster function for continuum percolation. J. Chem. Phys. 88, 6540–6547.CrossRefGoogle Scholar
[27] Weissberg, H. L. and Prager, S. (1962) Viscous flow through porous media. II. Approximate three-point correlation function. Phys. Fluids 5, 1390–1392.Google Scholar