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{(2\eta+1)^2}{(\eta-1)^4}


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


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

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


and the compressibility route is


A derivation of the Carnahan-Starling equation of state[edit]

It is interesting to note (Ref [5] Eq. 6) that one can arrive at the Carnahan-Starling equation of state by adding two thirds of the exact solution via the compressibility route, to one third via the pressure route, i.e.

Z = \frac{ p V}{N k_B T} =  \frac{2}{3} \left[   \frac{(1+\eta+\eta^2)}{(1-\eta)^3}  \right] +  \frac{1}{3} \left[     \frac{(1+2\eta+3\eta^2)}{(1-\eta)^2}  \right] = \frac{ 1 + \eta + \eta^2 - \eta^3 }{(1-\eta)^3 }

The reason for this seems to be a slight mystery (see discussion in Ref. [6] ).