# Percus–Yevick approximation

In statistical mechanics the Percus–Yevick approximation[1] is a closure relation to solve the Ornstein–Zernike equation. It is also referred to as the Percus–Yevick equation. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function.

## Derivation

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]

## References

1. ^ Percus, Jerome K. and Yevick, George J. Analysis of Classical Statistical Mechanics by Means of Collective Coordinates. Phys. Rev. 1958, 110, 1, DOI: 10.1103/PhysRev.110.1
2. ^ Wertheim, M. S. Exact Solution of the Percus-Yevick Integral Equation for Hard Spheres. Phys. Rev. Lett. 1963, 10, 321-323, DOI: 10.1103/PhysRevLett.10.321