Studies in Clustering Theory
STATE UNIV OF NEW YORK AT STONY BROOK DEPT OF CHEMISTRY
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In recent years the properties of percolation models have been studied intensively. The purpose of our project was to develop a general theory of percolation and clustering between particles of arbitrary size and shape, with arbitrary correlations between them. The goal of such a theory includes the treatment of continuum percolation as well as a novel treatment of lattice percolation. We made substantial progress toward this goal. The quantities basic to a description of clustering, the mean cluster size, mean number of clusters, etc. were developed. Concise formulas were given for the terms in such series, and proved, at least for sufficiently low densities, that the series are absolutely convergent. These series can now be used to construct Pade approximants that will allow one to probe the percolation transition. A scaled- particle theory of percolation was developed which gives analytic approximants for the mean number of clusters in a large class of two and three dimensional percolation models. Although this quantity is essential in many applications, e. g., explaining colligative properties, and interpreting low-angle light- scattering data, no systematic studies of it have been done before this work. Recently carried out detailed computer simulations show that the mean number of clusters is given to high accuracy by several of there approximations. Extensions of this work will allow calculation of the complete cluster size distribution. This work should be of practical importance in studying systems ranging from colloidal dispersions to nano-scale metal clusters.
- Fluid Mechanics