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Concept
Percus–Yevick approximation
Résumé
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.
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 potential u(r) is either zero or infinite, and therefore the Boltzmann factor 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.
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+2eta, 2)/pow(1-eta, 4);
const double b = -6etapow(1+eta/2, 2)/pow(1-eta, 4);
const double c = eta/2pow(1+2eta, 2)/pow(1-eta, 4);
const double A = 2qr;
const double A2 = AA;
const double G = a/A2(sin(A)-Acos(A))
b/A/A2(2Asin(A)+(2-A2)cos(A)-2)
c/pow(A,5)(-pow(A,4)cos(A)+4((3A2-6)cos(A)+A(A2-6)sin(A)+6));
return 1/(1+24etaG/A);
}
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.
Source officielle
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