Classical relativistic statistical mechanics: The case of a hot dilute plasma

Starting from predictive relativistic mechanics we develop a classical relativistic statistical mechanics. For a system of N particles, the basic distribution function depends, in addition to the $6N$ coordinates and velocities, on N times, instead of a single one as in the usual statistical mechanics. This generalized distribution function obeys N (instead of 1) continuity equations, which give rise to N Liouville equations in the case of a dilute plasma (i.e., to lowest, nonzero order in the charges). Hence, the Bogoliubov-Born-Green-Kirkwood-Yvon hierarchy for the reduced generalized distribution functions is derived. A relativistic Vlasov equation is obtained in this way. Thermal equilibrium is then considered for a dilute plasma. The calculation is explicitly worked out for a weakly relativistic plasma, up to order 1c², and known results are recovered.