Ring approximation at equilibrium: The hard sphere pair correlation function

The ring approximation has been used in the kinetic theory of hard sphere fluids to investigate non-equilibrium phenomena depending on dynamic correlations, not included in the Boltzmann or Enskog equations. We show that when applied to equilibrium the ring approximation reproduces correctly the main qualitative features of the pair correlation function. We find a maximum at contact followed by exponentially damped oscillations. The equation of state comes out exactly up to the third virial coefficient whereas the detailed structure of pair correlations is quantitatively correct at low volume fraction not exceeding 0.1.