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

## DerivationEdit

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. (with *w*(*r*) the potential of mean force) and is the radial distribution function without the direct interaction between pairs included; i.e. we write . Thus we *approximate* *c*(*r*) by

If we introduce the function into the approximation for *c*(*r*) 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.^{[2]}

## See alsoEdit

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

## ReferencesEdit

**^**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**^**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