Triangle distribution and equation of state for classical rigid disks

The triangle distribution function f((3)) for three mutual near neighbors i n the plane describes basic aspects of short-range order and statistical th ermodynamics 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 rudimenta ry closure suggested by scaled-particle theory is introduced. In conjunctio n 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 qu alitatively 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 .