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

$c(r)=g_{\rm {total}}(r)-g_{\rm {indirect}}(r)\,$

where $g_{\rm {total}}(r)$ is the radial distribution function, i.e. $g(r)=\exp[-\beta w(r)]$ (with w(r) the potential of mean force) and $g_{\rm {indirect}}(r)$ is the radial distribution function without the direct interaction between pairs $u(r)$ included; i.e. we write $g_{\rm {indirect}}(r)=\exp[-\beta (w(r)-u(r))]$. Thus we approximate c(r) by

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

If we introduce the function $y(r)=e^{\beta u(r)}g(r)$ 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:

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

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 ${\text{e}}^{-u/k_{\text{B}}T}$ 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)
{
    const double a = pow(1+2*eta, 2)/pow(1-eta, 4);
    const double b = -6*eta*pow(1+eta/2, 2)/pow(1-eta, 4);
    const double c = eta/2*pow(1+2*eta, 2)/pow(1-eta, 4);
    const double A = 2*qr;
    const double A2 = A*A;
    const double G = a/A2*(sin(A)-A*cos(A))
        + b/A/A2*(2*A*sin(A)+(2-A2)*cos(A)-2)
        + c/pow(A,5)*(-pow(A,4)*cos(A)+4*((3*A2-6)*cos(A)+A*(A2-6)*sin(A)+6));

    return 1/(1+24*eta*G/A);
}

See also

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