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., $f(x,p)propto (sqrt{1+ kappa^2theta^2}+kappatheta)^{1/kappa}equivexp_kappatheta$, with $theta=alpha(x)+beta_mu p^mu$, where $alpha(x)$ is a scalar, $beta_mu$ is a four-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 $F^{mu u}$. All standard results are readly recovered in the particular limit $kappato 0$.