Manifestly covariant classical correlation dynamics I. General theory

TL;DR

This paper develops a covariant mathematical framework for classical many-body systems in special relativity, deriving a single-time Liouville equation and a closed nonlinear equation for the reduced distribution function, with extensions to general relativity.

Contribution

It introduces a manifestly covariant approach to correlation dynamics, deriving a single-time Liouville equation and a closed nonlinear equation for the reduced distribution function.

Findings

Derived a covariant Liouville equation for many-body systems.

Established that the reduced distribution function is fully determined by the one-body distribution.

Discussed extensions to general relativity and self-gravitating systems.

Abstract

n this series of papers we substantially extend investigations of Israel and Kandrup on nonequilibrium statistical mechanics in the framework of special relativity. This is the first one devoted to the general mathematical structure. Basing on the action-at-a-distance formalism we obtain a single-time Liouville equation. This equation describes the manifestly covariant evolution of the distribution function of full classical many-body systems. For such global evolution the Bogoliubov functional assumption is justified. In particular, using the Balescu-Wallenborn projection operator approach we find that the distribution function of full many-body systems is completely determined by the reduced one-body distribution function. A manifestly covariant closed nonlinear equation satisfied by the reduced one-body distribution function is rigorously derived. We also discuss extensively the generalization to the general relativity especially an application to self-gravitating systems.