# What is the proper time used in relativistic non-equilibrium statistical physics?

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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 ?

This post imported from StackExchange Physics at 2016-10-30 15:19 (UTC), posted by SE-user Shaktyai

edited Oct 30, 2016
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.

This post imported from StackExchange Physics at 2016-10-30 15:19 (UTC), posted by SE-user John Rennie
Exactly what I was looking for. Thank's a lot

This post imported from StackExchange Physics at 2016-10-30 15:19 (UTC), posted by SE-user Shaktyai
@JohnRennie perhaps you could post that as an answer? (with a brief statement of what the article actually says that answers the question)

This post imported from StackExchange Physics at 2016-10-30 15:19 (UTC), posted by SE-user David Z
@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 :-)

This post imported from StackExchange Physics at 2016-10-30 15:19 (UTC), posted by SE-user John Rennie
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 ...

This post imported from StackExchange Physics at 2016-10-30 15:19 (UTC), posted by SE-user Shaktyai

There are specific works on electrons beams. One partially answers your question. This didactic and short document shows the tricks used to bypass the elementary time variable in this electrons analysis ( see page 4 )  but it ends with this paragraph :

The differences between the nonrelativistic and relativistic quantum theory of electrons are so substantial that they completely invalidate some conclusions based on the Schrödinger equation. The first difference is the nonexistence of relativistic electron wave packets with fixed orbital angular momentum. The second difference is the presence of the relativistic length parameter—Compton wavelength—which determines the behavior at large distances from the center. The third difference is lack of freedom to manipulate separately orbital angular momentum and spin. Finally, there is an open problem of vortex lines in the relativistic case which boils down to the question: Which current, total (no vortex lines) or orbital (with vortex lines), is observed in experiments?

Relativistic Electron Wave Packets Carrying Angular Momentum by Iwo Bialynicki-Birula and Zofia Bialynicka-Birula

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