Long-Range Correlations in a Closed System with Applications to Nonuniform Fluids

We investigate the corrections to the representation of the joint distribution of $q+l$ particles, $n_{q+l}$, by the product $n_q{n}_l$ for large separation between the sets of $q$ and $l$ particles. For a system in which there exists a "finite correlation length," we find explicitly the $\frac{1}{N}$ correction term to the simple product, where $N$ is the number of particles in our system. When $q+l$ is equal to two, this expression reduces to that familiar from the Ornstein-Zernike relations for scattering of light from a fluid. In a uniform gas, our derivation also yields the explicit $\frac{1}{N}$ dependence of equilibrium distributions. Our result on the asymptotic form is then used to determine the low-order distribution functions for an equilibrium system of varying density, as well as for a nonequilibrium system represented by a local-equilibrium ensemble. These distribution functions are shown to be governed by the temperature and density in the vicinity of the molecules considered. We find as expected that the two-body distribution function coincides, to within quadratic terms in the gradients, with its equilibrium value for a uniform system at the temperature and density of the midpoint. For the higher-order distributions, correction terms linear in the gradients are found.