ON THE EXISTENCE OF FREE-ENERGY DENSITIES IN NONUNIFORM FLUIDS

The perception that unambiguous constructions of local thermodynamic quantities cannot be given in equilibrium but non-uniform fluids has grown out of the recognition that many distinct thermodynamic fields satisfy the existing prescriptions for these quantities. That is, the existing prescriptions for thermodynamic fields underdetermine these quantities. It is shown that in many important cases. however, the requirement that thermodynamic consistency - the hallmark of classical thermodynamics - hold on all spatial scales does lead to a complete specification of free energy densities. The development presented requires - as in thermodynamics in general - a clear view of what fields actually fix the state of a non-uniform system. In the absence of a clear experimental foundation. the present development is built upon the statistical mechanical framework supporting density functional methods. It is noted that the identification of the free energy density as a stress tensor component is clouded by the fact that the traditional stress tensor formulation actually underdetermines a stress tensor. While this confusion should have no thermodynamic consequences it is also noted that the boundary conditions which play an essential role in constructing a thermodynamically meaningful stress tensor are not explicitly stated in general.