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

Thermodynamic extension of density-functional theory. I. Basic Massieu  function, its Legendre and Massieu-Planck transforms for equilibrium state in  terms of density matrix