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. The approximation is named after Jerome K. Percus and George J. Yevick.

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
where is the radial distribution function, i.e. and is the radial distribution function without the direct interaction between pairs included; i.e. we write. Thus we approximate c by
If we introduce the function into the approximation for c one obtains
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:
The approximation was defined by Percus and Yevick in 1958. For hard spheres, the equation has an analytical solution.