Hypernetted-chain equation

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:

${\displaystyle \ln y(r_{12})=\ln g(r_{12})+\beta u(r_{12})=\rho \int \left[h(r_{13})-\ln g(r_{13})-\beta u(r_{13})\right]h(r_{23})\,d\mathbf {r_{3}} ,\,}$

where ${\displaystyle \rho ={\frac {N}{V}}}$ is the number density of molecules, ${\displaystyle h(r)=g(r)-1}$, ${\displaystyle g(r)}$ is the radial distribution function, ${\displaystyle u(r)}$ is the direct interaction between pairs. ${\displaystyle \beta ={\frac {1}{k_{\rm {B}}T}}}$ with ${\displaystyle T}$ being the Thermodynamic temperature and ${\displaystyle k_{\rm {B}}}$ the Boltzmann constant.