# Continuum percolation expressed in terms of density distributions

## TL;DR

This paper introduces a new integral equation approach to analyze the connectivity in pairwise interacting systems, linking pair connectedness with density distributions and pair correlation functions, applicable across various dimensions and interaction types.

## Contribution

It develops a novel formalism relating pair connectedness to density distributions, extending analytical solutions to higher dimensions and complex interactions.

## Key findings

- Analytical solutions for pair connectedness in one-dimensional systems with specific potentials.
- Extension of the formalism to higher-dimensional systems like the 3D ideal gas.
- Framework accommodating external fields and long-range interactions.

## Abstract

We present a new approach to derive the connectivity properties of pairwise interacting n-body systems in thermal equilibrium. We formulate an integral equation that relates the pair connectedness to the distribution of nearest neighbors. For one-dimensional systems with nearest-neighbor interactions, the nearest-neighbor distribution is, in turn, related to the pair correlation function g through a simple integral equation. As a consequence, for those systems, we arrive at an integral equation relating g to the pair connectedness, which is readily solved even analytically if g is specified analytically. We demonstrate the procedure for a variety of pair-potentials including fully penetrable spheres as well as impenetrable spheres, the only two systems for which analytical results for the pair connectedness exist. However, the approach is not limited to nearest-neighbor interactions in one dimension. Hence, we also outline the treatment of external fields and long-ranged interactions, and we illustrate how the formalism can applied to higher-dimensional systems using the three-dimensional ideal gas as an example.