Similarities and differences between the Gaussian and random phase approximation for the partition function

Analogies and differences between the random phase approximation (RPA) and the Gaussian approximation at finite temperature are discussed. A comparison is made for Hartree and Hartree-Fock temperature dependent mean fields. Some ambiguities, in the definition of the Hartree mean field and in the relative correction, are analyzed. In a solvable model with a repulsive potential, the Gaussian approximation and the RPA, both built from the Hartree mean field, are found more reliable than the RPA built from a Hartree-Fock mean field.