First-principles derivation of density-functional formalism for quenched-annealed systems

Abstract

We derive from first principles (without resorting to the replica trick) a density functional theory for fluids in quenched disordered matrices (QA-DFT). We show that the disorder-averaged free energy of the fluid is a functional of the average density profile of the fluid as well as the pair correlation of the fluid and matrix particles. For practical reasons it is preferable to use another functional: the disorder-averaged free energy plus the fluid-matrix interaction energy, which, for fixed fluid-matrix interaction potential, is a functional only of the average density profile of the fluid. When the matrix is created as a quenched configuration of another fluid, the functional can be regarded as depending on the density profile of the matrix fluid as well. In this situation, the replica-Ornstein-Zernike equations which do not contain the blocking parts of the correlations can be obtained as functional identities in this formalism, provided the second derivative of this functional is interpreted as the connected part of the direct correlation function. The blocking correlations are totally absent from QA-DFT, but nevertheless the thermodynamics can be entirely obtained from the functional. We apply the formalism to obtain the exact functional for an ideal fluid in an arbitrary matrix, and discuss possible approximations for non-ideal fluids.

Capillary condensation in pores with rough walls: A density functional approach

“…(6) makes difficult to derive the set of Ornstein-Zernike equations for the system from functional relations. For example, at present it is not at all clear what the meaning of the second derivatives of F ex (ρ m (z), ρ(R)) is [34]. Nevertheless, we have decided to follow the ideas of Schmidt [30][31][32] because at the moment this approach is the only one that can be implemented to the considered case in a relatively easy manner.…”

Recent developments in classical density functional theory: Internal energy functional and diagrammatic structure of fundamental measure theory

“…Work along these lines is in progress [60]. It would also be interesting to investigate the implications of Levy's method for the statistical mechanics of quenched-annealed fluid mixtures, where DFT was obtained both via the replica trick [61][62][63][64], and via a first-principles derivation following the Mermin-Evans arguments [65]. Зроблено огляд декiлькох недавнiх праць з теорiї функцiоналу густини для класичних неоднорiдних рiдин.…”

Variational principle of classical density functional theory via Levy’s constrained search method

“…For the case of the canonical ensemble we point the reader to the very significant body of work that has been carried out to formulate a computational scheme that permits to capture the effects that arise due to the constraint of fixed number of particles [21][22][23][24][25][26]. While the generalization to equilibrium mixtures is straightforward, we expect the application of Levy's method to DFT for quenched-annealed mixtures [28][29][30][31] to constitute an interesting topic for future work.…”