Abstract
The intent of the present chapter is to derive and discuss some basic results and specific developments in continuum percolation theory. We will begin with a discussion of exact results for cluster statistics and other percolation descriptors for a prototypical model of continuum percolation, namely, identical overlapping spheres in d dimensions. Subsequently, we will describe an Ornstein-Zernike formalism to find the pair-connectedness function P 2(r) for general isotropic models of continuum percolation. The reader should note the beautiful correspondence of this theory to the Ornstein-Zernike formalism for the total correlation function h(r) of equilibrium (or thermal) systems discussed in Chapter 3. This will be followed by a discussion of various approximation schemes to close the resulting integral equation, including the Percus-Yevick approximation. The next topic will be the two-point cluster function C 2(r). First we will present an exact series representation of C 2(r) for dispersions and then discuss its analytical evaluation for certain models. The chapter will conclude with a presentation of percolation thresholds for overlapping sphere systems, overlapping particles of nonspherical shape, and interacting particle systems. The reader is referred to Meester and Roy (1996) for a more mathematical treatment of continuum percolation.
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© 2002 Springer Science+Business Media New York
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Torquato, S. (2002). Some Continuum Percolation Results. In: Random Heterogeneous Materials. Interdisciplinary Applied Mathematics, vol 16. Springer, New York, NY. https://doi.org/10.1007/978-1-4757-6355-3_10
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DOI: https://doi.org/10.1007/978-1-4757-6355-3_10
Publisher Name: Springer, New York, NY
Print ISBN: 978-1-4757-6357-7
Online ISBN: 978-1-4757-6355-3
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