Density functional formalism in the canonical ensemble

Density functional theory (DFT), when applied to systems with T not equal 0, is based on the grand canonical extension of the Hohenberg-Kohn-Sham theorem due to Mermin (HKSM theorem). While a straightforward canonical ensemble (CE) generalization fails, work in nanopore systems could certainly benefit from a mesoscopic DFT in the CE. We show that, if the asymptotic behaviour of the canonical distribution functions is taken into account, the HKSM theorem can be extended to the CE. We generate N-modified correlation and distribution functions hierarchies, show that their functional relationship is equivalent to the one holding between the more conventional ones and prove that, if they are employed, either a modified external field or the density profiles can be indistinctly used as independent variables. We also write down the N-modified free energy functional and prove that its minimum is reached when the equilibrium values of the new hierarchy are used. This completes the extension of the HKSM theorem.