Two-parameter Sharma–Taneja–Mittal entropy as the basis of family of equilibrium thermodynamics of nonextensive systems

In the framework of statistical mechanics based on the two-parameter Sharma–Taneja–Mittal entropy, it is shown how one can obtain the equilibrium thermodynamics of nonextensive systems and determine its properties. The basic mathematical properties of the doubly deformed logarithm and exponent, as well as other related functions that are necessary for constructing non-extensive thermostatics, are presented. A generalization is obtained for the non-extensive case of the zero law of thermodynamics and the so-called physical temperature is introduced, which differs from the inversion of the Lagrange multiplier $\beta$. Based on the Clausius macroscopic entropy and using the generalized first law of thermodynamics and the Legendre transformation, new thermodynamic equations for nonextensive systems are obtained that are satisfactory from the point of view of macroscopic thermodynamics. In addition, taking into account the convexity property of the Bragmann divergence, it was shown that for $(k,\zeta)$-systems the principle of maximum the Sharma–Taneja–Mittal equilibrium entropy is preserved, the Legendre structure of the theory and the $H$-theorem describing the randomization of the system during spontaneous transitions.