Density-Functional Theory of Inhomogeneous Fluids in the Canonical Ensemble

Abstract

We present a density-functional approach for dealing with inhomogeneous fluids in the canonical ensemble. A general relation is proposed between the free-energy functionals in the canonical and the grand canonical ensembles. The minimization of the canonical-ensemble free-energy functional gives rise to Euler-Lagrange equations which involve averaged Ornstein-Zernike equations of second and third order. The theory is especially appropriate for systems with a small, fixed number of particles. As an example of application we obtain accurate results for the density profile of a hard-sphere fluid in a closed spherical cavity that contains only a few particles.

Equivalence of two approaches for the inhomogeneous density in the canonical ensemble

“…The proof of the equivalence gives additional support to the saddle point approximation for the CE free-energy functional introduced in [10]. This approximation allows for a CE-DFT treatment of fluids confined in a closed cavity with excellent agreement with simulation data.…”

“…On one hand the DFT could be formulated in the canonical ensemble [9], with a minimum free-energy principle with fixed T and N , and an appropriate CE functional. Very recently, this approach has been explicitly realized [10] by considering an approximate expression for the CE functional. On the other hand, one can perform a conventional DFT study in the GCE and then relate the obtained properties to those of the CE.…”

“…Lett. 84, 1220(2000], is equivalent to first order to the result of the series expansion of the CE inhomogeneous density introduced by González et al [Phys. Rev.…”

“…On the basis of the standard saddle point relation between the CE Helmholtz free energy and the GCE grand potential [11] the following approximation for the CE free-energy functional F c was proposed in Ref. [10]:…”

Density functional theory in the canonical ensemble: I. General formalism

“…In fact, years ago, Ramshaw [14] explicitly worked out the two-body OZ equation for a one component system, showing how the stripping of the correlation function h(r) off its asymptotic behaviour solves the problems found in a more conventional approach. Another approach to a DFT in the CE, equivalent to the one given in [9], is discussed in [15]. In [16] the existence of an extension of the OZ equation for the CE is discussed in the framework of a DFT theory.…”