Analytic solution of the Ornstein-Zernike relation for inhomogeneous liquids

The properties of a classical simple liquid are strongly affected by the application of an external potential that supports inhomogeneity. To understand the nature of these property changes, the equilibrium particle distribution functions of the liquid have, typically, been calculated directly using either integral equation or density functional based analyses. In this study, we develop a different approach with a focus on two distribution functions that characterize the inhomogeneous liquid: the pair direct correlation function c(r₁,r₂) and the pair correlation function g(r₁,r₂). With g(r₁,r₂) considered to be an experimental observable, we solve the Ornstein-Zernike equation for the inhomogeneous liquid to obtain c(r₁,r₂), using information about the well studied and resolved g⁽⁰⁾(r₁,r₂) and c⁽⁰⁾(r₁,r₂) for the parent homogeneous (⁽⁰⁾) system. In practical cases, where g(r₁,r₂) is available from experimental data in a discrete form, the resulting c(r₁,r₂) is expressed as an explicit function of g(r₁,r₂) in a discrete form. A weaker continuous form of solution is also obtained, in the form of an integral equation with finite integration limits. The result obtained with our formulation is tested against the exact solutions for the correlation and distribution functions of a one-dimensional inhomogeneous hard rod liquid. Following the success of that test, the formalism is extended to higher dimensional systems with explicit consideration of the two-dimensional liquid.