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Application of the density functional formalism to the canonical ensemble isof practical interest in cases where there is a marked difference between, say,the canonical and the grand canonical ensemble (cavities or pores). Animportant role is played by the necessary modification of the famousOrnstein-Zernike relation between pair correlation and direct correlationfunction, as the former is no longer invertible in a strict sense in (finite)canonical ensembles. Here we approach the problem from a different directionwhich may complement the density functional approach. In particular, we developrigorous canonical ensemble versions of the LMBW equations, relating densitygradient and exterior potential in the presence of explicit (singular)containing potentials. This is accomplished with the help of integral operatorand Hilbert space methods, yielding among other things representations of thedirect correlation function on certain subspaces. The results are particularlynoteworthy and transparent if the segregating potential is a linear(gravitational) one. In that case the modifications in the LMBW equations canbe expressed as pure, seemingly non-local integrations over the containerboundaries.Comment: 24 pages, Latex, more detailed results concerning the direct  correlation function, more reference

Modifications of the Ornstein–Zernike relation and the LMBW equations in the canonical ensemble via Hilbert-space methods