Hypernetted-chain equation

Example

Consider a classical fluid in $\R^3$ with number density $\rho$, inverse temperature $\beta$ and pair potential $u(r)$. Let $g$ be the radial distribution function, $h = g - 1$, and $c$ the direct correlation function of the Ornstein–Zernike equation. The hypernetted-chain closure is $$g(r) = \exp\big(-\beta u(r) + h(r) - c(r)\big), \quad \text{equivalently} \quad c(r) = h(r) - \ln g(r) - \beta u(r).$$ Together with the Ornstein–Zernike equation $$h(r_{12}) = c(r_{12}) + \rho \int_{\R^3} c(r_{13}) \, h(r_{23}) \, dx_3,$$ where $r_{ij} = \abs{x_i - x_j}$, it forms a closed nonlinear integral equation for $h$: substitute $c = h - \ln(1 + h) - \beta u$.

Check at low density: as $\rho \to 0$ the Ornstein–Zernike equation gives $c = h$ to leading order. Then the closure gives $g = e^{-\beta u}$, the exact low-density limit of the radial distribution function.