Conditional pair distributions in many-body systems: Exact results for   Poisson ensembles
  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. 
Ren\'e D. Rohrmann, Ernesto Zurbriggen
cond-mat.stat-mech astro-ph.CO cond-mat.soft