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 ?
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$\begingroup$ icmp.lviv.ua/journal/zbirnyk.25/001/art01.pdf "Classical relativistic system of N charges. Hamiltonian description, forms of dynamics, and partition function" looks as if it answers exactly your question. $\endgroup$– John RennieJul 18, 2012 at 12:08
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$\begingroup$ @JohnRennie perhaps you could post that as an answer? (with a brief statement of what the article actually says that answers the question) $\endgroup$– David ZJul 19, 2012 at 6:40
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1$\begingroup$ @DavidZaslavsky a quick glance at the article convinced me that a brief description would be hard! The fact I found it is more a testament to my Google skills than my deep knowledge of relativistic statistical thermodynamics :-) $\endgroup$– John RennieJul 19, 2012 at 6:55
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$\begingroup$ The paper is quite complex, so far my researches to solve the problem has only brought back this paper: cft.edu.pl/~laturski/Physica/… I am not sure I understand how they have avoided the retarded time for each particle ... $\endgroup$– ShaktyaiJul 20, 2012 at 11:18
1 Answer
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).
