In statistical mechanics the hypernetted-chain equation is a closure relation to solve the Ornstein–Zernike equation which relates the direct correlation function to the total correlation function. It is commonly used in fluid theory to obtain e.g. expressions for the radial distribution function. It is given by:

is the number density is the radial distribution function being the Thermodynamic temperature the Boltzmann constant.

Derivation

The direct correlation function represents the direct correlation between two particles in a system containing N2 other particles. It can be represented by

the potential of mean force. Thus we approximate by

by writing:

.

This equation is the essence of the hypernetted chain equation. We can equivalently write

If we substitute this result in the Ornstein–Zernike equation

one obtains the hypernetted-chain equation:

See also

• Classical-map hypernetted-chain method

• Percus–Yevick approximation another closure relation

• Ornstein–Zernike equation