Do variational formulations for inhomogeneous density functions lead to unique solutions?

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Abstract
In principle, the equilibrium density in an inhomogeneous system is that density field which extremalizes the free energy and all the system’s equilibrium properties can be deduced from this. A simple, but qualitatively realistic model free energy is presented which shows that approximate free energy functionals can easily possess a large number of extremalizing solutions. The usual interpretation when multiple solutions are found is that the correct solution is the one associated with the lowest value of the free energy. This rule is not very reassuring when, as the model exhibits for some range of parameter values, a continuum of solutions can be found. A more careful analysis of the variational problem shows that a variational formulation only provides a complete characterization of an equilibrium system when the variational problem possesses a unique solution. A multiplicity of solutions actually corresponds to the existence of a multiplicity of Hamiltonians which could give rise to the postulated free energy functional. There is no variational basis for comparing different Hamiltonians, however, and hence choosing from among a multiplicity of solutions on the basis of the value of the free energy is an additional extrathermodynamic rule.