Percus–Yevick approximation

In statistical mechanics the Percus-Yevick approximation 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

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

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

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

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

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

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:

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

Derivation

See also