Statistical mechanics in the context of special relativity

In Ref. [Physica A $296,$ 405 (2001)], starting from the one parameter deformation of the exponential function ${\exp }_{\{\kappa \}}\mathrm{}\left(x\right)=(\sqrt{1+{\kappa }^2{x}^2}+\kappa {x)}^{1/\kappa },$ a statistical mechanics has been constructed which reduces to the ordinary Boltzmann-Gibbs statistical mechanics as the deformation parameter $\kappa$ approaches to zero. The distribution $f={\exp }_{\{\kappa \}}(-\beta E+\beta \mu )$ obtained within this statistical mechanics shows a power law tail and depends on the nonspecified parameter $\beta ,$ containing all the information about the temperature of the system. On the other hand, the entropic form $S_{\kappa }=\int d^3{p(c}_{\kappa }{f}^{1+\kappa }{+c}_{-\kappa }{f}^{1-\kappa }),$ which after maximization produces the distribution f and reduces to the standard Boltzmann-Shannon entropy $S_0$ as $\overrightarrow{\kappa }0,$ contains the coefficient $c_{\kappa }$ whose expression involves, beside the Boltzmann constant, another nonspecified parameter $\alpha .$ In the present effort we show that $S_{\kappa }$ is the unique existing entropy obtained by a continuous deformation of $S_0$ and preserving unaltered its fundamental properties of concavity, additivity, and extensivity. These properties of $S_{\kappa }$ permit to determine unequivocally the values of the above mentioned parameters $\beta$ and $\alpha .$ Subsequently, we explain the origin of the deformation mechanism introduced by $\kappa$ and show that this deformation emerges naturally within the Einstein special relativity. Furthermore, we extend the theory in order to treat statistical systems in a time dependent and relativistic context. Then, we show that it is possible to determine in a self consistent scheme within the special relativity the values of the free parameter $\kappa$ which results to depend on the light speed c and reduces to zero as $\overrightarrow{c}\infty$ recovering in this way the ordinary statistical mechanics and thermodynamics. The statistical mechanics here presented, does not contain free parameters, preserves unaltered the mathematical and epistemological structure of the ordinary statistical mechanics and is suitable to describe a very large class of experimentally observed phenomena in low and high energy physics and in natural, economic, and social sciences. Finally, in order to test the correctness and predictability of the theory, as working example we consider the cosmic rays spectrum, which spans 13 decades in energy and 33 decades in flux, finding a high quality agreement between our predictions and observed data.