Relativistic statistical theory and generalized stosszahlansatz

We have investigated the proof of the $H$ theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [Phy. Rev. E {\bf 66}, 056125, 2002; {\it ibid.} {\bf 72}, 036108 2005]. In our analysis, however, we have not considered the so-called deformed mathematics as did in the above reference. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the $\kappa$-formalism, and the second law of thermodynamics implies that the $\kappa$ parameter lies on the interval [-1,1]. It is shown that the collisional equilibrium states (null entropy source term) are described by a $\kappa$ power law generalization of the exponential Juttner distribution, e.g., $f(x,p)\propto (\sqrt{1+ \kappa^2\theta^2}+\kappa\theta)^{1/\kappa}\equiv\exp_\kappa\theta$,with$\theta=\alpha(x)+\beta_\mu p^\mu$,where$\alpha(x)$isascalar,$\beta_\mu$isafour-vector,and$p^\mu$ is the four-momentum. As a simple example, we calculate the relativistic $\kappa$ power law for a dilute charged gas under the action of an electromagnetic field $F^{\mu \nu }$. All standard results are readly recovered in the particular limit $\kappa \to 0$.