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  "body": "\n## Example\n\nConsider a classical fluid in $\\R^3$ with number density $\\rho$, inverse temperature $\\beta$ and a pair potential $u(r)$.\nLet $g$ be the radial distribution function, $h = g - 1$, and let $c$ be the direct correlation function of the [[Ornstein–Zernike equation|29T4NC87AR]].\nThe Percus–Yevick approximation is the closure\n$$c(r) = g(r) \\left(1 - e^{\\beta u(r)}\\right).$$\nWith the cavity function $y(r) = g(r) e^{\\beta u(r)}$ and the Mayer function $f(r) = e^{-\\beta u(r)} - 1$ it reads $c(r) = f(r) \\, y(r)$.\nInserted in the Ornstein–Zernike equation it gives\n$$y(r_{12}) = 1 + \\rho \\int_{\\R^3} f(r_{13}) \\, y(r_{13}) \\, h(r_{23}) \\, dx_3, \\qquad h = (1 + f) \\, y - 1,$$\nwhere $r_{ij} = \\abs{x_i - x_j}$; this is a [[nonlinear integral equation|BSCYBP6AS6]] for $y$, quadratic in $y$.\n\nCheck of the second form: $f y = (e^{-\\beta u} - 1) g e^{\\beta u} = g (1 - e^{\\beta u})$.\n\n## Remarks\n\nFor hard spheres of diameter $\\sigma$, $g(r) = 0$ for $r \u003c \\sigma$ and the closure gives $c(r) = 0$ for $r \u003e \\sigma$.\nIn this case the equation has a known closed-form solution, in which $c$ is a cubic polynomial in $r$ for $r \u003c \\sigma$.\nWith the packing fraction $\\eta = \\frac{\\pi}{6} \\rho \\sigma^3$, the compressibility route gives the equation of state $\\frac{\\beta P}{\\rho} = \\frac{1 + \\eta + \\eta^2}{(1 - \\eta)^3}$, and the virial route gives $\\frac{1 + 2\\eta + 3\\eta^2}{(1 - \\eta)^2}$.\nThe two routes differ because the closure is an approximation.\n\n## Explanations\n\n### L1 human\n\nThe Ornstein–Zernike equation alone has two unknown functions, so a second rule is needed to close it.\nThe Percus–Yevick rule says that the direct correlation is zero wherever the particles do not interact.\nFor hard balls this rule leads to a formula for the pressure that is close to computer simulations at moderate densities.\n",
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