Well-behaved coarse-grained model of density-functional theory

A coarse-grained version of quantum density-functional theory featuring a limited spatial resolution is shown to model formal density-functional theory (DFT). This means that all densities are ensemble-V-representable (that is, by mixed states), the intrinsic energy functional F is a continuous function of the density, and the representing external potential is the functional derivative of the intrinsic energy, in the sense of directional derivatives within the domain of F. The representing potential v[ρ] also has a quasicontinuity property, specifically, v[ρ]ρ is continuous as a function of ρ. Convergence of the intrinsic energy, coarse-grained densities, and representing potentials in the limit of coarse-graining scale going to zero are studied vis-à-vis Lieb's L1 L3 theory. The intrinsic energy converges monotonically to its fine-grained (continuum) value, and coarse-grainings of a density ρ converge strongly to ρ. If a sequence of coarse-grained densities converges strongly to ρ and their representing potentials converge weak-^*, the limit is the representing potential for ρ. Conversely, L3^/2+L representability implies the existence of such a coarse-grained sequence.