Ornstein–Zernike equation

Example

Consider a homogeneous and isotropic classical fluid in $\R^3$ with number density $\rho > 0$. Let $g(r)$ be its radial distribution function and $h(r) = g(r) - 1$ the total correlation function, where $r$ is the distance between two particles. The Ornstein–Zernike equation defines the direct correlation function $c$ by $$h(\abs{x - y}) = c(\abs{x - y}) + \rho \int_{\R^3} c(\abs{x - z}) \, h(\abs{z - y}) \, dz, \quad x, y \in \R^3.$$ For known $h$ it is a convolution integral equation of the second kind for $c$ on $\R^3$, and for known $c$ one for $h$.

Check by the Fourier transform: the integral is the convolution $c * h$, so the equation becomes $\hat h(q) = \hat c(q) + \rho \, \hat c(q) \hat h(q)$. Hence $\hat h = \frac{\hat c}{1 - \rho \hat c}$ and $\hat c = \frac{\hat h}{1 + \rho \hat h}$, wherever the denominators do not vanish.