Relativistic statistical theory and generalized stosszahlansatz

Abstract

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 $κ$-formalism, and the second law of thermodynamics implies that the $κ$ parameter lies on the interval [-1,1]. It is shown that the collisional equilibrium states (null entropy source term) are described by a $κ$ power law generalization of the exponential Juttner distribution, e.g., $f(x,p)\propto (\sqrt{1+ κ^2θ^2}+κθ)^{1/κ}\equiv\exp_κθ$, with $θ=α(x)+β_μp^μ$, where $α(x)$ is a scalar, $β_μ$ is a four-vector, and $p^μ$ is the four-momentum. As a simple example, we calculate the relativistic $κ$ power law for a dilute charged gas under the action of an electromagnetic field $F^{μν}$. All standard results are readly recovered in the particular limit $κ\to 0$.