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 formula_2 is the radial distribution function, i.e. formula_3 (with "w"("r") the potential of mean force ) and formula_4 is the radial distribution function without the direct interaction between pairs formula_5 included; i.e. we write formula_6. Thus we "approximate" "c"("r") by If we introduce the function formula_8 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. Hard spheres. For hard spheres, the potential "u(r)" is either zero or infinite, and therefore the Boltzmann factor formula_11 is either one or zero, regardless of temperature  T. Therefore structure of a hard-spheres fluid is temperature independent. 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. Solution as C code. The static structure factor of the hard-spheres fluid in Percus-Yevick approximation can be computed using the following C function: double py(double qr, double eta)