Distribution Function of Classical Fluids of Hard Spheres. I

The Born–Green–Yvon (BGY) hierarchy is truncated by introducing a superposition approximation g1234≃g123g124g134g234 / g12g13g14g23g24g34 to the quadruplet correlation function g1234. The resulting pair of two simultaneous integral equations including the triplet- and pair- correlation functions g123 and g12, which hereafter will be called the BGY2 equations, is reformulated to give mathematically simpler forms. The BGY2 theory is checked for its internal consistency by using the hard-sphere potential. The following results have been obtained: (a) The fifth virial coefficients B5 for the BGY2 theory is (0.1090 ± 0.0008) (B2)4 or (0.1112 ± 0.0005) (B2)4 depending on whether the virial theorem or the Ornstein–Zernike compressibility relation is used in the calculations of B5; the exact Monte Carlo value, B5 = 0.1103 (B2)4, lies between the two values. (b) The third coefficient g123 in the density series of g12 agrees exactly with the exact value given in the literature if r / σ ≥ 2 (σ is the sphere diameter), and it differs only slightly from the exact value if 0 ≤ r/σ ≤ 2, i.e., the deviation lies within 3%. (c) The data for the second coefficient g1232 of the density series of g123 show that the Kirkwood superposition approximation is better for a symmetric configuration than an asymmetric one when particles 1, 2, and 3 lie close to each other. (d) It is shown that the pressure calculated from the virial theorem using the BGY g(r) is exact for one-dimensional hard rods, and is independent of any approximation used to describe the g123. A proof that the BGY2 g(r) is exact for the hard-rod system is also given. These results indicate that the BGY2 theory compares favorably with any other theories for g(r) and that it satisfactorily describes low and moderately dense fluids.