EXACT INTEGRAL-EQUATIONS FOR THE DISTRIBUTION-FUNCTIONS OF LIQUIDS AND LIQUID-MIXTURES

A known identity relating the functional density derivative of the pair distribution function g(12) of nonuniform liquids to the triplet distribution function g(123) is obtained within the grand canonical ensemble, but then generalized to the case of nonuniform liquid mixtures with either short- or long-range forces. The resulting identities are easily applied to the case of uniform one- or multicomponent liquids, and when applied to the case of a binary Coulomb mixtures, they yield the canonical sum rules relating the n-point to the (n + 1)-point distribution functions. Finally, when applied to the one-component plasma (OCP), taken as a particular limiting case, the expected result that the total compressibility of the OCP must vanish in the thermodynamic limit is obtained.