Radial Distribution Functions and Equation of State of the Hard-Disk Fluid

Abstract

We evaluated by a Monte Carlo method the hard‐disk radial distribution function at densities: ${\mathrm{\rho \hspace{0.17em}/\hspace{0.17em}\rho }}_0\text{\hspace{0.17em}=\hspace{0.17em}0.4, 0.5,}\mathrm{and}\text{0.6}$ (${\rho }_0$ is equivalent to the close‐packed density of hard disks). The results were used to find out the relative merits of solutions of the four integral equations (the Born–Green–Yvon, the modified Born–Green–Yvon, the Percus–Yevick, and the convolution–hypernetted‐chain equations) describing approximately the behavior of the radial distribution function. We also evaluated the first several density coefficients of the pressure and the radial distribution functions corresponding to these approximate equations. The Padé approximants formed by the calculated coefficients for the approximate equations were found to satisfactorily describe pressures for these equations over the range of low to medium density of the hard‐disk fluids. A simple analytic expression, Eq. (13), for the pressure which describes fairly well the dense hard‐disk fluid was also derived by using the modified Born–Green–Yvon equation. This expression yields better or comparable results for virial coefficients and pressures at low and medium fluid densities compared with the results obtained from the Percus–Yevick equation in combination with the virial theorem.