Correlation function algebra for inhomogeneous fluids

TL;DR

This paper develops an algebraic framework for the pair correlation function in inhomogeneous fluids confined between parallel plates, revealing new identities that connect correlation functions with free energy and scaling behaviors.

Contribution

It introduces a novel algebraic structure for the correlation function in variational fluid models, simplifying analysis and relating correlations to free energy in confined geometries.

Findings

Derived identities linking correlation functions at different positions.

Reproduced known scaling laws for free energy at criticality.

Extended the algebraic approach to non-planar geometries.

Abstract

We consider variational (density functional) models of fluids confined in parallel-plate geometries (with walls situated in the planes z=0 and z=L respectively) and focus on the structure of the pair correlation function G(r_1,r_2). We show that for local variational models there exist two non-trivial identities relating both the transverse Fourier transform G(z_\mu, z_\nu;q) and the zeroth moment G_0(z_\mu,z_\nu) at different positions z_1, z_2 and z_3. These relations form an algebra which severely restricts the possible form of the function G_0(z_\mu,z_\nu). For the common situations in which the equilibrium one-body (magnetization/number density) profile m_0(z) exhibits an odd or even reflection symmetry in the z=L/2 plane the algebra simplifies considerably and is used to relate the correlation function to the finite-size excess free-energy \gamma(L). We rederive non-trivial scaling expressions for the finite-size contribution to the free-energy at bulk criticality and for systems where large scale interfacial fluctuations are present. Extensions to non-planar geometries are also considered.