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  "body": "\n## Example\n\nConsider a classical fluid in $\\R^3$ with number density $\\rho$, inverse temperature $\\beta$ and pair potential $u(r)$.\nLet $g$ be the radial distribution function, $h = g - 1$, and $c$ the direct correlation function of the [[Ornstein–Zernike equation|29T4NC87AR]].\nThe hypernetted-chain closure is\n$$g(r) = \\exp\\big(-\\beta u(r) + h(r) - c(r)\\big), \\quad \\text{equivalently} \\quad c(r) = h(r) - \\ln g(r) - \\beta u(r).$$\nTogether with the Ornstein–Zernike equation\n$$h(r_{12}) = c(r_{12}) + \\rho \\int_{\\R^3} c(r_{13}) \\, h(r_{23}) \\, dx_3,$$\nwhere $r_{ij} = \\abs{x_i - x_j}$, it forms a closed [[nonlinear integral equation|BSCYBP6AS6]] for $h$: substitute $c = h - \\ln(1 + h) - \\beta u$.\n\nCheck at low density: as $\\rho \\to 0$ the Ornstein–Zernike equation gives $c = h$ to leading order.\nThen the closure gives $g = e^{-\\beta u}$, the exact low-density limit of the radial distribution function.\n\n## Remarks\n\nThe closure is usually solved numerically by iteration, with the convolution done by the fast Fourier transform.\nThe [[Percus–Yevick approximation|K3E536PSRJ]] is usually more accurate for hard spheres, while the hypernetted-chain equation is usually preferred for long-range potentials such as the Coulomb potential.\n\n## Explanations\n\n### L1 human\n\nThe equation describes how particles in a liquid arrange around each other.\nIt combines the exact Ornstein–Zernike relation with an approximate rule that leaves out a certain class of terms, the so-called bridge diagrams, in an expansion of the correlations.\nThe resulting equation is nonlinear because the rule involves a logarithm.\n",
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