Correlation function algebra for inhomogeneous fluids

We consider variational (density functional) models of fluids confined in parallelplate geometries (with walls situated in the planes z = 0 and z = L respectively) and focus on the structure of the pair correlation function G(r1, r2). We show that for local variational models there exist two non-trivial identities relating both the transverse Fourier transform G(zμ, zν ; q) and the zeroth moment G0(zμ, zν) at different positions z1, z2 and z3. These relations form an algebra which severely restricts the possible form of the function G0(zμ, zν). For the common situations in which the equilibrium one-body (magnetization/number density) profile m0(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 γ (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.