### Abstract

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 [Formula Presented](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.

Original language | English (US) |
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Pages (from-to) | 4027-4036 |

Number of pages | 10 |

Journal | Physical Review E - Statistical Physics, Plasmas, Fluids, and Related Interdisciplinary Topics |

Volume | 54 |

Issue number | 4 |

DOIs | |

State | Published - Jan 1 1996 |

### All Science Journal Classification (ASJC) codes

- Statistical and Nonlinear Physics
- Statistics and Probability
- Condensed Matter Physics