  The current article completes our investigation of the hard-particle
interaction by determining their distribution functionals. Beginning with a
short review of the perturbation expansion of the free-energy functional, we
derive two representations of the correlation functionals in rooted and
unrooted Mayer diagrams, which are related by a functional derivative. This map
allows to transfer the mathematical methods, developed previously for unrooted
diagrams, to the current representation in rooted graphs. Translating then the
Mayer to Ree-Hoover diagrams and determining their automorphism groups, yields
the generic functional for all r-particle distributions. From this we derive
the examples of 2- and 3-particle correlations up to four intersection centers
and show that already the leading order reproduces the Wertheim, Thiele, Baxter
solution for the contact probability of spheres. Another calculation shows the
failure of the Kirkwood superposition approximation for any r-particle
correlation.
