The kinetic energy, or thermodynamic potential, of a non-interacting Fermion system, as a functional of the non-uniform density pattern evoked by an external potential, serves as a crucial tool in the analysis of interacting system ground states. Examples from classical lattice gas and continuum fluid theory are cited to show that analytic tractability can be dramatically enhanced by the introduction of auxiliary density fields with respect to which a suitable thermodynamic potential is stationary. This strategy is developed for the prototype of one-dimensional spinless Fermions, resulting in an exact highly overcomplete representation. A semi-classical variational ansatz recovers the familiar Thomas-Fermi plus renormalized Weiszacker expression, as well as leading corrections, and a simple extension creates a simple modification of the former. The corresponding 3-dimensional format is set up, and evaluated by extrapolation from the 1-dimensional case.