Conditional pair distributions in many-body systems: Exact results for Poisson ensembles

TL;DR

This paper introduces an exact conditional pair distribution function (CPDF) for many-body systems, providing detailed insights into four-body configurations and related structures in Poisson ensembles across different dimensions.

Contribution

It derives exact expressions for the CPDF in Poisson ensembles, enhancing the understanding of spatial configurations in many-body systems.

Findings

Exact CPDF expressions for 1D, 2D, and 3D systems.

Characterization of four-body, two-body, and three-body configurations.

Potential applications across physics, biology, and social sciences.

Abstract

We introduce a conditional pair distribution function (CPDF) which characterizes the probability density of finding an object (e.g., a particle in a fluid) to certain distance of other, with each of these two having a nearest neighbor to a fixed but otherwise arbitrary distance. This function describes special four-body configurations, but also contains contributions due to the so-called mutual nearest neighbor (two-body) and shared neighbor (three-body) configurations. The CPDF is introduced to improve a Helmholtz free energy method based on space partitions. We derive exact expressions of the CPDF and various associated quantities for randomly distributed, non-interacting points at Euclidean spaces of one, two and three dimensions. Results may be of interest in many diverse scientific fields, from fluid physics to social and biological sciences.