Percus–Yevick approximation

The direct correlation function represents the direct correlation between two particles in a system containing N − 2 other particles. It can be represented by

${\displaystyle c(r)=g_{\rm {total}}(r)-g_{\rm {indirect}}(r)\,}$

where ${\displaystyle g_{\rm {total}}(r)}$ is the radial distribution function, i.e. ${\displaystyle g(r)=\exp[-\beta w(r)]}$ (with w(r) the potential of mean force) and ${\displaystyle g_{\rm {indirect}}(r)}$ is the radial distribution function without the direct interaction between pairs ${\displaystyle u(r)}$ included; i.e. we write ${\displaystyle g_{\rm {indirect}}(r)=\exp[-\beta (w(r)-u(r))]}$. Thus we approximate c(r) by

${\displaystyle c(r)=e^{-\beta w(r)}-e^{-\beta [w(r)-u(r)]}.\,}$

If we introduce the function ${\displaystyle y(r)=e^{\beta u(r)}g(r)}$ into the approximation for c(r) one obtains

${\displaystyle c(r)=g(r)-y(r)=e^{-\beta u}y(r)-y(r)=f(r)y(r).\,}$

This is the essence of the Percus–Yevick approximation for if we substitute this result in the Ornstein–Zernike equation, one obtains the Percus–Yevick equation:

${\displaystyle y(r_{12})=1+\rho \int f(r_{13})y(r_{13})h(r_{23})d\mathbf {r_{3}} .\,}$

The approximation was defined by Percus and Yevick in 1958. For hard spheres, the equation has an analytical solution.[2]

• Hypernetted chain equation — another closure relation
• Ornstein–Zernike equation

• Exact solution of the Percus Yevick integral equation for hard spheres