  By combining the upper and lower bounds to the free energy as given by the
Gibbs inequality for two systems with the same intermolecular interactions but
with external fields differing from each other only in a finite region of space
Gamma, we show that the corresponding equilibrium densities must also differ
from each other somewhere in Gamma. We note that the basic equations of density
functional theory arise naturally from a simple rearrangement and
reinterpretation of the terms in the upper bound Gibbs inequality for such
systems and briefly discuss some of the complications that occur when the
intermolecular interactions of the two systems also differ.
