Simpler free-energy functional of the Debye-Hückel model of fluids and the nonuniqueness of free-energy functionals in the theory of fluids

In previous publications [Piron and Blenski, Phys. Rev. E 94, 062128 (2016); Blenski and Piron, High Energy Density Phys. 24, 28 (2017)], the authors have proposed Debye-Hückel-approximate free-energy functionals of the pair distribution functions for one-component fluids and two-component plasmas. These functionals yield the corresponding Debye-Hückel integral equations when they are minimized with respect to the pair distribution functions, lead to correct thermodynamic relations, and fulfill the virial theorem. In the present paper, we update our results by providing simpler functionals that have the same properties. We relate these functionals to the approaches of Lado [Phys. Rev. A 8, 2548 (1973)] and of Olivares and McQuarrie [J. Chem. Phys. 65, 3604 (1976)]. We also discuss briefly the nonuniqueness issue that is raised by these results.