Percus–Yevick approximation

In statistical mechanics the Percus–Yevick approximation[1] is a closure relation to solve the Ornstein–Zernike 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.

is the radial distribution function without the direct interaction between pairs

For hard spheres, the potential u(r) is either zero or infinite, and therefore the Boltzmann factor

This leaves just two parameters: the hard-core radius R (which can be eliminated by rescaling distances or wavenumbers), and the packing fraction η (which has a maximum value of 0.64 for random close packing).

Under these conditions, the Percus–Yevick equation has an analytical solution, obtained by Wertheim in 1963.

[2][3][4] The static structure factor of the hard-spheres fluid in Percus–Yevick approximation can be computed using the following C function: For hard spheres in shear flow, the function u(r) arises from the solution to the steady-state two-body Smoluchowski convection–diffusion equation or two-body Smoluchowski equation with shear flow.

An approximate analytical solution to the Smoluchowski convection-diffusion equation was found using the method of matched asymptotic expansions by Banetta and Zaccone in Ref.

Approximate solutions for the pair distribution function in the extensional and compressional sectors of shear flow and hence the angular-averaged radial distribution function can be obtained, as shown in Ref.,[6] which are in good parameter-free agreement with numerical data up to packing fractions