Towards the development of thermodynamics of nonextensive systems based on kappa-entropy Кaniadakis

In the framework of non-extensive statistical mechanics of Kaniadakis based on parametric $\kappa$-entropy, it is shown how to obtain the deformed statistical thermodynamics of complex anomalous systems and determine its properties. The paper presents the basic mathematical properties of the $\kappa$-logarithm and $\kappa$-exponent, as well as other related functions that arise during the development of the statistical mechanics of Kaniadakis. As a result, a generalization is obtained for the case under consideration of the zero law of thermodynamics for two independent subsystems with their thermal contact and the so-called physical temperature is introduced, which differs from the inversion of the Lagrange multiplier. Using the generalized first law of thermodynamics and the Legendre transformation, and based on the introduced Clausius entropy, new thermodynamic relations are obtained that are different from the relationships that were previously unsatisfactory from the point of view of deformed thermodynamics, which were traditionally used for non-extensive statistics. Based on the property of convexity of Bergman divergence, spontaneous transitions between stationary states of a complex $\beta$-system are studied and the Gibbs theorem and the Boltzmann $H$-theorem are proved.