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arXiv:cond-mat/9701092 · Journal of Physics Condensed Matter

Correlation function algebra for inhomogeneous fluids

Andrew O. Parry, Peter S. Swain

Published Mar 17, 1997• 4 citations
arXivHTMLPDF

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 . We show that for local variational models there exist two non-trivial identities relating both the transverse Fourier transform and the zeroth moment at different positions and . These relations form an algebra which severely restricts the possible form of the function . For the common situations in which the equilibrium one-body (magnetization/number density) profile exhibits an odd or even reflection symmetry in the plane, the algebra simplifies considerably, and is used to relate the correlation function to the finite-size excess free energy . 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.

DOI
10.1088/0953-8984/9/11/006

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Ising models on the plane square lattice with an arbitrary variation of the bond strengths, ${\mathit{J}}^{\mathrm{\ensuremath{\parallel}}}$(z) and ${\mathit{J}}^{\mathrm{\ensuremath{\perp}}}$(z), with one of the two axial coordinates, z, are considered. The total entropy is exactly represented as a functional of contributions ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z) to the local energy density arising from the Onsager fermions with wave vector q parallel to the layer axis, y. The resulting explicit local expression provides an effective variational principle for the free energy and energy-density profiles. In the scaling limit the problem reduces to a set of independent second-order differential equations for each ${\mathrm{\ensuremath{\varepsilon}}}_{\mathit{q}}$(z). The power of the method is demonstrated by application to an interface between two uniform but distinct regions; this includes the problem of a wall with a surface field, h1{\mathit{h}}_{1}h1​, as a special case. Previous results for the bulk and surface exponents and for the energy-energy correlation function are easily recovered. Near criticality the method yields, in addition, universal, scaled energy-density profiles, \ensuremath{\varepsilon}(z;T), which exhibit rich crossovers and nonmonotonic variation with z.
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A quest for microcanonical functionals describing near-critical behavior has led to an exact local variational functional for square lattice Ising models in zero field with varying bond strengths along the z axis. In the continuum limit near TC{\mathit{T}}_{\mathit{C}}TC​ this yields a second-order differential equation for the transverse Fourier coefficients of the energy density. Applications to a ``grain boundary'' between differing regions and to a wall with a surface field h1{\mathit{h}}_{1}h1​ give bulk and surface exponents and scaling-limit energy-density profiles which exhibit rich crossovers and nonmonotonic variation.
Equilibrium quantum statistical methods are used to study the pair correlation function for a magnetized free-electron gas in the presence of a hard wall that is parallel to the field. With the help of a path-integral technique and a Green function representation, the modifications in the correlation function caused by the wall are determined both for a non-degenerate and for a completely degenerate gas. In the latter case the asymptotic behaviour of the correlation function for large position differences in the direction parallel to the wall and perpendicular to the field, is found to change from Gaussian in the bulk to algebraic near the wall.
Abstract We consider near-critical two-dimensional statistical systems at phase coexistence on the half plane with boundary conditions leading to the formation of a droplet separating coexisting phases. General low-energy properties of two-dimensional field theories are used in order to find exact analytic results for one- and two-point correlation functions of both the energy density and order parameter fields. The subleading finite-size corrections are also computed and interpreted within an exact probabilistic picture in which interfacial fluctuations are characterized by the probability density of a Brownian excursion. The explicit analysis of the closed-form expression for order parameter correlations reveals the long-ranged character of interfacial correlations and their confinement within the interfacial region. The analysis of correlations is then carried out in momentum space through the notion of interface structure factor, which we extend to the case of systems bounded by a flat wall. The presence of the wall and its associated entropic repulsion leads to a specific term in the interface structure factor which we identify.