Distribution Function of Classical Fluids of Hard Spheres. I

The Born–Green–Yvon (BGY) hierarchy is truncated by introducing a superposition approximation $g_{1234}{\mathrm{\simeq g}}_{123}{g}_{124}{g}_{134}{g}_{234}{\mathrm{\hspace{0.17em}/\hspace{0.17em}[g}}_{12}{g}_{13}{g}_{14}{g}_{23}{g}_{24}{g}_{34}]$ to the quadruplet correlation function $g_{1234}$ . The resulting pair of two simultaneous integral equations including the triplet‐ and pair‐ correlation functions $g_{123}$ and $g_{12}$ , 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 $B_5$ for the BGY2 theory is ${\text{(0.1090\hspace{0.17em}±\hspace{0.17em}0.0008) (B}}_2{)}^4$ or ${\text{(0.1112\hspace{0.17em}±\hspace{0.17em}0.0005) (B}}_2{)}^4$ depending on whether the virial theorem or the Ornstein–Zernike compressibility relation is used in the calculations of $B_5$ ; the exact Monte Carlo value, $B_5{\text{\hspace{0.17em}=\hspace{0.17em}0.1103\hspace{0.17em}(B}}_2{)}^4$ , lies between the two values. (b) The third coefficient $g_{12}^3$ in the density series of $g_{12}$ agrees exactly with the exact value given in the literature if $\text{r\hspace{0.17em}/\hspace{0.17em}σ\hspace{0.17em}≥\hspace{0.17em}2}$ ($\sigma$ is the sphere diameter), and it differs only slightly from the exact value if $\text{0\hspace{0.17em}≤\hspace{0.17em}r/σ\hspace{0.17em}≤\hspace{0.17em}2}$ , i.e., the deviation lies within 3%. (c) The data for the second coefficient $g_{123}^2$ of the density series of $g_{123}$ 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 $\mathrm{g(r)}$ is exact for one‐dimensional hard rods, and is independent of any approximation used to describe the $g_{123}$ . A proof that the BGY2 $\mathrm{g(r)}$ is exact for the hard‐rod system is also given. These results indicate that the BGY2 theory compares favorably with any other theories for $\mathrm{g(r)}$ and that it satisfactorily describes low and moderately dense fluids.