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Exact solution of the Percus Yevick integral equation for hard spheres

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The exact solution for the Percus Yevick integral equation for the hard sphere model was derived by M. S. Wertheim in 1963 [1] (see also [2]), and for mixtures by Joel Lebowitz in 1964 [3].

The direct correlation function is given by (Eq. 6 of [1] )

C(r/R) = - \frac{(1+2\eta)^2 - 6\eta(1+ \frac{1}{2} \eta)^2(r/R) + \eta(1+2\eta)^2\frac{(r/R)^3}{2}}{(1-\eta)^4}


\eta = \frac{1}{6} \pi R^3 \rho

and R is the hard sphere diameter. The equation of state is given by (Eq. 7 of [1])

\frac{\beta P}{\rho} = \frac{(1+\eta+\eta^2)}{(1-\eta)^3}

where \beta is the inverse temperature. Everett Thiele also studied this system [4], resulting in (Eq. 23)

\left.h_0(r)\right. = ar+ br^2 + cr^4

where (Eq. 24)

a = \frac{(2x+1)^2}{(x-1)^4}


b= - \frac{12x + 12x^2 + 3x^3}{2(x-1)^4}


c= \frac{x(2x+1)^2}{2(x-1)^4}

and where x=\rho/4.

The pressure via the pressure route (Eq.s 32 and 33) is


and the compressibility route is