Continuum percolation expressed in terms of density distributions

We present an 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-range interactions and we illustrate how the formalism can applied to higher-dimensional systems using the three-dimensional ideal gas as an example.