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In this work we discuss how and to what extent the thermodynamic concepts and the thermodynamic formalism can be extended to the description of high temperature states of the plasma not necessarily associated with a Boltzmann distribution and with thermal equilibrium. The discussion is based on the "magnetic or electrostatic entropy concept", an interpretative and predictive tool based on probability and information, defined in a suitably coarse-grained possibility space of all current density or of all electric charge density distributions under testable constraints, and whose variation properties are proven to be related under certain conditions to the equilibrium and the stability of the system. In the case of magnetic equilibrium the potentiality of the magnetic entropy concept is illustrated by comparing the predictions of the current density and pressure profiles with the observations in different tokamak machines and different tokamak regimes, as well as by showing how the equilibrium and the stability in devices as different as the reversed field pinch or the magnetic well are described by the variation properties of the same entropy functional applied to the different situations. In fact it emerges that the maximum of the entropy can be seen in these different cases as an optimization constraint for the minimum of the magnetic energy. The application of the entropy concept to the electrostatic processes shows in particular that the so-called reactive instabilities (non-dissipative, non-resonant instabilities with a marginal point) admit a neighboring state with higher entropy and are therefore of special relevance from the point of view of the physical evolution of the system. In this case the thermodynamic formalism allows the introduction of the concept of "thermodynamic fluctuations" of the macroscopic charge density and provides a method for the calculation of the "thermodynamic" fluctuation levels both on the stable as well as on the linearly unstable side of the marginal point. The paper discusses the relation between the variations of the entropy functional defined on statistical grounds and the motion of the underlying system of particles. It is found that the vanishing of the first variation of the entropy is connected, under certain assumptions, with the Hamilton's principle, while the second variation is not directly related to the dynamics but is an expression of the fact that the entropy is a predictive tool based on probability and information. [ABSTRACT FROM AUTHOR]