Triangle distribution and equation of state for classical rigid disks

The triangle distribution function f((3)) for three mutual near neighbors in the plane describes basic aspects of short-range order and statistical thermodynamics in two-dimensional many-particle systems. This paper examines prospects for constructing a self-consistent calculation for the rigid-disk-system f((3)). Wr present several identities obeyed by f((3)). A rudimentary closure suggested by scaled-particle theory is introduced. In conjunction with three of the basic identities, this closure leads to an unique f((3)) over the entire density range, The pressure equation of state exhibits qualitatively correct behaviors in both the low-density and the close-packed limits, but no intervening phase transition appears. We discuss extensions to improved disk closures. and to the three-dimensional rigid-sphere system.