Thermodynamic Derivation of Classical Density Functional Theory

Classical density functional theory has evolved into a major branch of statistical and condensed matter physics. The fundamental equation of the equilibrium theory is [equation] where [equation] is the thermal Helmholtz free energy of the system as a functional of its non-uniform local number density [equation], is the external potential, μ is the uniform chemical potential, and [equation] denotes an isothermal functional derivative. This equation implicitly determines [equation] and is ordinarily derived from the grand canonical ensemble (GCE) of statistical mechanics. Here we show that it can also be simply derived from thermodynamics alone, and is therefore not inherently statistical in character or specific to the GCE. This derivation further shows that the isothermal functional derivative of A tis identically equal to the isentropic functional derivative of the internal energy E t, which is not immediately obvious in the statistical theory. This development shows that certain aspects of density functional theory are essentially macroscopic rather than statistical in character, thereby clarifying the physical content of the theory and making it more accessible to students and nonspecialists