Pressure, sum rules and approximate density-functionals for fluid mixtures

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Abstract
The statistical mechanics of an inhomogeneous fluid comprising several species of molecule is investigated. It is shown that the density of each species, θ vw, at a hard wall is related to the pressure due to the species, Pv , at far distances through the equation and a virial expression for Pv is obtained. An alternative expression for Pv is obtained from the density-functional for the fluid mixture. These results are used to derive conditions which must be satisfied by an approximate density-functional to ensure that it yields the correct densities of the different species at a hard wall.