  A general formulation of the equilibrium state of a many-electron system in
terms of a (mixed-state, ensemble) density matrix operator in the Fock space,
based on the maximum entropy principle, is introduced. Various characteristic
functions/functionals are defined and investigated: the basic Massieu function
for fully open thermodynamic system (ensemble), the effective action function
for the fully closed (isolated) system, and a series of Legendre transforms for
partially open/closed ones - the Massieu functions. Convexity and/or concavity
properties of these functions are determined, their first and second
derivatives with respect to all arguments are obtained. Other characteristic
functions - the Gibbs-Helmholtz functions - are obtained from previous ones as
their Massieu-Planck transforms, i.e. by specific transformation of arguments
(which involves the temperature) and by applying the temperature with the minus
as a prefactor. Such functions are closer to traditional (Gibbs, Helmholtz)
thermodynamic potentials. However, the first and second derivatives of these
functions represent more complicated expressions than derivatives of the
Massieu functions. All introduced functions are suitable for application to the
extensions of the density functional theory, both at finite and zero
temperature.
