The relativistic statistical theory and Kaniadakis entropy: an approach through a molecular chaos hypothesis

TL;DR

This paper extends relativistic statistical theory using Kaniadakis entropy, proving an H theorem and deriving a generalized Juttner distribution with applications to charged gases in electromagnetic fields.

Contribution

It introduces a covariant proof of the H theorem within Kaniadakis formalism and derives a relativistic power-law distribution generalizing the Juttner distribution.

Findings

Null entropy source states are described by a κ-generalized exponential distribution.

The formalism recovers standard results in the limit κ→0.

Application to a charged gas in electromagnetic fields demonstrates the model's validity.

Abstract

We have investigated the proof of the $H$ theorem within a manifestly covariant approach by considering the relativistic statistical theory developed in [G. Kaniadakis, Phy. Rev. E {\bf 66}, 056125, 2002; {\it ibid.} {\bf 72}, 036108, 2005]. As it happens in the nonrelativistic limit, the molecular chaos hypothesis is slightly extended within the Kaniadakis formalism. 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 $\kappa\to 0$.