Abstract. The analytical expressions for the ultrarelativistic transverse momentum distributions of the Tsallis and the Tsallis-2 statistics were obtained. We found that the transverse momentum distribution of the Tsallis-factorized statistics, which is now largely used to describe the experimental transverse momentum spectra of hadrons measured in pp collisions at LHC and RHIC energies, in the ultrarelativistic case is not equivalent to the transverse momentum distributions of the Tsallis and the Tsallis-2 statistics. However, we revealed that this distribution exactly coincides with the transverse momentum distribution of the Tsallis-2 statistics in the zeroth term approximation and is transformed to the transverse momentum distribution of the Tsallis statistics in the zeroth term approximation by changing the parameter q to 1/qc. We demonstrated analytically on the basis of the ultrarelativistic ideal gas that the Tsallis-factorized statistics is not equivalent to the Tsallis and the Tsallis-2 statistics. In the present paper the Tsallis statistics corresponds to the standard expectation values.PACS. 13.85.-t Hadron-induced high-and super-high-energy interactions -13.85.Hd Inelastic scattering: many-particle final states -24.60.-k Statistical theory and fluctuations

The microscopic foundation of the generalized equilibrium statistical mechanics based on the Tsallis entropy is given by using the Gibbs idea of statistical ensembles of the classical and quantum mechanics. The equilibrium distribution functions are derived by the thermodynamic method based upon the use of the fundamental equation of thermodynamics and the statistical definition of the functions of the state of the system. It is shown that if the entropic index ξ = 1/(q − 1) in the microcanonical ensemble is an extensive variable of the state of the system, then in the thermodynamic limitz = 1/(q − 1)N = const the principle of additivity and the zero law of thermodynamics are satisfied. In particular, the Tsallis entropy of the system is extensive and the temperature is intensive. Thus, the Tsallis statistics completely satisfies all the postulates of the equilibrium thermodynamics. Moreover, evaluation of the thermodynamic identities in the microcanonical ensemble is provided by the Euler theorem. The principle of additivity and the Euler theorem are explicitly proved by using the illustration of the classical microcanonical ideal gas in the thermodynamic limit.

We show that starting with either the non-extensive Tsallis entropy in Wang's formalism or the extensive Rényi entropy, it is possible to construct the equilibrium statistical mechanics with non-Gibbs canonical distribution functions. The statistical mechanics with Tsallis entropy does not satisfy the zeroth law of thermodynamics at dynamical and statistical independence request, whereas the extensive Rényi statistics fulfills all requirements of equilibrium thermodynamics. The transformation formulas between Tsallis statistics in Wang representation and Rényi statistics are presented. The one-particle distribution function in Rényi statistics for classical ideal gas and finite particle number has a power-law tail for large momenta.