Two-parameter functional of entropy Sharma–Mittal as the basis of the family of generalized thermodynamices of non-extensive systems

The properties of the family of generalized entropies given by the Sharm–Mittal entropy $S_{qr}^{SM}k[1-(\Sigma_j p_j^q)^{(r-1)(q-1)}]/(r-1)$, which includes the entropy of Tsallis $S_q^{Ts}$ ($r=q$), the Renyi entropy $S_r^R$ ($r\to 1$), the Landsberg–Vedral entropy $S_q^{LV}$ ($r=2-q$), the Gauss entropy $S_r^G$ ($q\to 1$), and the classical Boltzmann–Gibbs–Shannon entropy $S^{BGS}$ ($r,q\to 1$) are investigated. Based on the Sharm–Mittal statistics, the two-parameter thermodynamics of non-extensive systems is constructed and its interrelation with generalized one-parameter thermodynamices based on the named deformed entropies of the family is shown. A generalization of the zero law of thermodynamics is obtained for two independent non-extensive systems at their thermal contact, introduce into consideration a so-called physical temperature different from the inversion of the Lagrange multiplier $\beta$. 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 is given. On the basis of the two-parametric information of Sharm–Mittal's difference, Gibbs's theorem and the $H$-theorem on the change of these measures in the course of time evolution are formulated and proved.