  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.
