Scaling, bounds, and inequalities for the noninteracting density functionals at finite temperature

Finite-temperature density functional theory requires representations for the internal energy, entropy, and free energy as functionals of the local density field. A central formal difficulty for an orbital-free representation is the construction of the corresponding functionals for noninteracting particles in an arbitrary external potential. That problem is posed here in the context of the equilibrium statistical mechanics of an inhomogeneous system. The density functionals are defined and shown to be equal to the extremal state for a functional of the reduced one-particle statistical operators. Convexity of the latter functionals implies a class of general inequalities. First, it is shown that the familiar von Weizs\"acker lower bound for zero temperature functionals applies at finite temperature as well. An upper bound is obtained in terms of a single-particle statistical operator corresponding to the Thomas-Fermi approximation. Next, the behavior of the density functionals under coordinate scaling is obtained. The inequalities are exploited to obtain a class of upper and lower bounds at constant temperature, and a complementary class at constant density. The utility of such constraints and their relationship to corresponding results at zero temperature are discussed.