On the derivation of thermodynamic equilibrium criteria

It is well established that, besides the two thermodynamic formulations based on the fundamental equations (i.e., entropic formulation S=S(U, V, N); Energetic Formulation U=U(S, V, N)), the equilibrium states of a system can be represented by equivalent formulations derived by making Legendre transforms of either of the fundamental equations [H.B. Callen, Thermodynamics and an Introduction to Thermostatics, 2nd edn., Wiley, NY, 1985]. The main purpose of this article is to present in a new and consistent way, the derivation of the equilibrium criteria for both thermodynamic potentials (i.e., Legendre transforms of internal energy) and Massieu functions (i.e., Legendre transforms of entropy) with the aid of concepts such as boundaries, reservoirs, and certain auxiliary devices known as equilibrium or intermittent connections.