Percus-Yevick approximation

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

[edit] 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 $g total (r)$ is the radial distribution function, i.e. $g (r) = e x p [ - β w (r)]$ (with w(r) the potential of mean force) and $g indirect (r)$ is the radial distribution function without the direct interaction between pairs $u (r)$ included; i.e. we write $g indirect (r) = e x p - β[w (r) - u (r)]$ . Thus we approximate c(r) by

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

If we introduce the function $y (r) = e β u (r) g (r)$ 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:

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