Overcomplete density functional description

Although formally exact expressions can be written for thermodynamic potentials of many-particle systems in external fields, Legendre transformation to particle density as the controlling field has many virtues, insensitivity in the face of singular states being notable. It is also true that nonuniform systems, classical or quantum, which lead to simple closed-form solutions, tend to do so principally in the density functional form. However, as gleaned from the heavily investigated cases of classical lattice gas networks, this may be achieved only if additional auxiliary densities are appended, with respect to which the associated thermodynamic potential is stationary. Here, we successfully apply the overcompleteness strategy to the prototypical case of free fermion ground states in one-dimensional space, extrapolate to three-dimensional space, and apply a basic one-functional variational ansatz. The qualitative properties of the resulting free energy and potential-density profile relations are in accord with those that any valid formulation must satisfy.