Generalized Boltzmann Equation in a Manifestly Covariant Relativistic Statistical Mechanics

TL;DR

This paper develops a covariant relativistic Boltzmann equation for nonequilibrium systems in space-time, extending classical kinetic theory to relativistic regimes with invariant interactions and proving an H-theorem.

Contribution

It introduces a manifestly covariant relativistic Boltzmann-Uehling-Uhlenbeck equation using invariant interactions and proves the H-theorem within this framework.

Findings

Derivation of a covariant relativistic Boltzmann equation.

Proof of the H-theorem for relativistic kinetic evolution.

Expressions for energy density and pressure consistent with nonrelativistic limits.

Abstract

We consider the relativistic statistical mechanics of an ensemble of $N$ events with motion in space-time parametrized by an invariant ``historical time'' $\tau .$ We generalize the approach of Yang and Yao, based on the Wigner distribution functions and the Bogoliubov hypotheses, to find the approximate dynamical equation for the kinetic state of any nonequilibrium system to the relativistic case, and obtain a manifestly covariant Boltzmann-type equation which is a relativistic generalization of the Boltzmann-Uehling-Uhlenbeck (BUU) equation for indistinguishable particles. This equation is then used to prove the $H$-theorem for evolution in $\tau .$ In the equilibrium limit, the covariant forms of the standard statistical mechanical distributions are obtained. We introduce two-body interactions by means of the direct action potential $V(q),$ where $q$ is an invariant distance in the Minkowski space-time. The two-body correlations are taken to have the support in a relative $O( 2,1)$-invariant subregion of the full spacelike region. The expressions for the energy density and pressure are obtained and shown to have the same forms (in terms of an invariant distance parameter) as those of the nonrelativistic theory and to provide the correct nonrelativistic limit.