Construction of the Formalism of Statistical Thermodynamics of Nonextensive Systems Based on the Kaniadakis Kappa-Entropy

Within the framework of the nonextensive Kaniadakis statistics based on parametric kappa-entropy, it has been shown how to obtain deformed thermodynamics of complex anomalous systems and determine its properties. The main mathematical properties of the κ-logarithm and κ-exponent, as well as other related functions arising in the development of the Kaniadakis statistical mechanics, have been presented. As a result, a generalization has been obtained for the nonextensive case of the zero law of thermodynamics for two independent subsystems in thermal contact and the so-called physical temperature has been introduced, which is different from the inversion of the Lagrange multiplier $$\beta $$ . With the involvement of the generalized first law of thermodynamics and the Legendre transformation and on the basis of the introduced Clausius entropy, new thermodynamic relations have been obtained, which differ from the relations previously derived by the traditional method for nonextensive statistics, which are unsatisfactory from the point of view of macroscopic thermodynamics. Based on the convexity property of the Bergman divergence, spontaneous transitions between stationary states of a complex $$\kappa $$ -system have been studied and the Gibbs theorem and the Boltzmann H-theorem have been proven.