Two-Parameter Sharma–Mittal Entropy Functional as the Foundation of the Family of Generalized Thermodynamics of Nonextensive Systems

The properties of the family of generalized entropies defined by the Sharma–Mittal measure have been investigated. This family includes the Tsallis entropy, the Rényi entropy, the Landsberg–Vedral entropy, the Gaussian entropy, and the classical Boltzmann–Gibbs–Shannon entropy. A two-parameter thermodynamics of nonextensive systems has been constructed on the basis of the Sharma–Mittal statistics and its relationship with generalized one-parameter thermodynamics based on the indicated deformed entropies of the family has been shown. A generalization of the zeroth law of thermodynamics for two independent nonextensive systems in thermal contact has been obtained, introducing into consideration the so-called physical temperature, which differs from the inversion of the Lagrange multiplier $$\beta $$ . Based on this and taking into account the generalized first law of thermodynamics and the Legendre transformation, a redefinition of the thermodynamic relationships obtained within the framework of the Sharma–Mittal statistics has been given. Finally, based on the two-parameter information of the Sharma–Mittal difference, the Gibbs theorem and the H‑theorem on the change of these measures during evolution over time have been formulated and proven.