In the literature one often finds covariant relativistic generalizations of classical non equilibrium statistical equations (Boltzmann, Vlasov, Landau, Fokker-Planck, etc...) but I wonder what is the meaning of the time which is used. As far as I know, one can only write the interaction between two relativistic charged particles by doing the computation in the proper space-time frame of one of the particles. With three relativistic charged particles I am already wondering about how to tackle the problem of proper time, so for N close to a mole...I am lost. Since non-equilibrium statistical mechanics is derived from Hamiltonian mechanics, I can reformulate my question as follows. What is the Hamiltonian of N relativistic interacting charged particles ?
Relativistic non-equilibrium statistical physics is done in a field theoretic setting, not in a multiparticle setting. There everything is naturally covariant in space-time coordinates, and the question of proper time does not arise. (Instead one has the preferred time of a coordinate systen comoving with the fluid.)
Note that there is no consistent classical relativistic multiparticle setting; see Currie, Jordan and Sudarshan, Reviews of Modern Physics 35 (1963), 350. Relativistic quantum multiparticle theory is possible in a pure particle framework; see, e.g.,
B.D. Keister and W.N. Polyzou, Relativistic Hamiltonian Dynamics in Nuclear and Particle Physics, in: Advances in Nuclear Physics, Volume 20, (J. W. Negele and E.W. Vogt, eds.) Plenum Press 1991.
But it is somewhat awkward to use and I haven't seen any statistical mechanics based on it. The conventional approach to relativistic quantum multiparticle theory is through quantum field theory. For a treatment of relativistic statistical mechanics in these terms see, e.g., the book Calzetta,& Hu, Nonequilibrium quantum field theory. Cambridge University Press (2008).