Percus–Yevick approximation

Example

Consider a classical fluid in $\R^3$ with number density $\rho$, inverse temperature $\beta$ and a pair potential $u(r)$. Let $g$ be the radial distribution function, $h = g - 1$, and let $c$ be the direct correlation function of the Ornstein–Zernike equation. The Percus–Yevick approximation is the closure $$c(r) = g(r) \left(1 - e^{\beta u(r)}\right).$$ With the cavity function $y(r) = g(r) e^{\beta u(r)}$ and the Mayer function $f(r) = e^{-\beta u(r)} - 1$ it reads $c(r) = f(r) \, y(r)$. Inserted in the Ornstein–Zernike equation it gives $$y(r_{12}) = 1 + \rho \int_{\R^3} f(r_{13}) \, y(r_{13}) \, h(r_{23}) \, dx_3, \qquad h = (1 + f) \, y - 1,$$ where $r_{ij} = \abs{x_i - x_j}$; this is a nonlinear integral equation for $y$, quadratic in $y$.

Check of the second form: $f y = (e^{-\beta u} - 1) g e^{\beta u} = g (1 - e^{\beta u})$.