Derivation of the Ornstein-Zernike differential equation from the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy

The theory of inhomogeneous fluids is applied to a $d$-dimensional system near its critical point to derive the probability of finding a particle at a distance $r$ from a pair separated by a distance $s$, given that $r\ensuremath{\gg}\ensuremath{\xi}\ensuremath{\gg}s$, where $\ensuremath{\xi}$ is the correlation length. When this result is used in the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy, an approximation-free equation is obtained, from which it follows that the pair correlations for $r\ensuremath{\gg}\ensuremath{\xi}$ satisfy the Ornstein-Zernike differential equation.