Relativistic statistical theory and generalized stosszahlansatz

TL;DR

This paper extends relativistic statistical theory by introducing a $ ext{kappa}$-generalization of the exponential distribution, proving an $H$ theorem without deformed mathematics, and analyzing equilibrium states under electromagnetic fields.

Contribution

It develops a covariant relativistic statistical framework with a $ ext{kappa}$-power law distribution, extending the molecular chaos hypothesis and deriving new equilibrium solutions.

Findings

The $ ext{kappa}$-distribution describes relativistic equilibrium states.

The $H$ theorem holds within the $ ext{kappa}$-formalism without deformed mathematics.

Standard results are recovered as $ ext{kappa} o 0$.

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 $\kappa$-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 $\alpha(x)$ 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 $F^{\mu\nu}$. All standard results are readly recovered in the particular limit $\kappa\to 0$.