The Explicit Density Functional, and its Connection with Entropy Maximisation

The intrinsic Helmholtz free energy, commonly used as a basis for density functional theories, is here given explicitly as a cluster diagram expansion with density field points. Also given are explicit variational procedures for determining the chemical potential for a given density, the pair potential for a given pair correlation function, and the pair correlation function for a given pair potential. The physical meaning of the density functional is established within the context of a new derivation of statistical mechanics based on entropy that supplies a variational principle for equilibrium by generalising the thermodynamic potential to non-equilibrium states. This shows that the conventional density functional determines not only the equilibrium density, but also the probability of fluctuations about that density.