Lineal measures of clustering in overlapping particle systems

The lineal-path function L(z) gives the probability of finding a line segment of length z entirely in one of the phases of a disordered multiphase medium. We develop an exact methodology to determine L(z) for the particle phase of systems of overlapping particles, thus providing a measure of particle clustering in this prototypical model of continuum percolation. We describe this procedure for systems of overlapping disks and spheres with a polydispersivity of sizes and for randomly aligned equal-sized overlapping squares. We also study the effect of polydispersivity on the range of the lineal-path function. We note that the lineal-path function L(z) is a rigorous lower bound on the two-point cluster function C₂(z), which is not available analytically for overlapping particle models for spatial dimension d{}2. By evaluating the second derivative of L(z), we then evaluate the chord-length distribution function for the particle phase. Computer simulations that we perform are in excellent agreement with our theoretical results. {} 1996 The American Physical Society.