Geodesic
Simulation of collisionless ultrarelativistic electron-proton plasma dynamics in a self-consistent electromagnetic field
Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 9, pp. 1635-1644

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The evolution of a collisionless electron-proton plasma in the self-consistent approximation is investigated. The plasma is assumed to move initially as a whole in a vacuum with the Lorentz factor. The behavior of the dynamical system is analyzed by applying a three-dimensional model based on the Vlasov–Maxwell equations with allowance for retarded potentials. It is shown that the analysis of the solution to the problem is not valid in the “center-of-mass frame” of the plasmoid (since it cannot be correctly defined for a relativistic plasma interacting via an electromagnetic field) and the transition to a laboratory frame of reference is required. In the course of problem solving, a chaotic electromagnetic field is generated by the plasma particles. As a result, the particle distribution functions in the phase space change substantially and differ from their Maxwell–Juttner form. Computations show that the kinetic energies of the electron and proton components and the energy of the self-consistent electromagnetic field become identical. A tendency to the isotropization of the particle momentum distribution in the direction of the initial plasmoid motion is observed. 
@article{ZVMMF_2016_56_9_a9,
     author = {S. L. Ginzburg and V. F. Dyachenko and Yu. N. Orlov and N. N. Fimin and V. M. Chechetkin},
     title = {Simulation of collisionless ultrarelativistic electron-proton plasma dynamics in a self-consistent electromagnetic field},
     journal = {\v{Z}urnal vy\v{c}islitelʹnoj matematiki i matemati\v{c}eskoj fiziki},
     pages = {1635--1644},
     publisher = {mathdoc},
     volume = {56},
     number = {9},
     year = {2016},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_9_a9/}
}
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S. L. Ginzburg; V. F. Dyachenko; Yu. N. Orlov; N. N. Fimin; V. M. Chechetkin. Simulation of collisionless ultrarelativistic electron-proton plasma dynamics in a self-consistent electromagnetic field. Žurnal vyčislitelʹnoj matematiki i matematičeskoj fiziki, Tome 56 (2016) no. 9, pp. 1635-1644. http://geodesic.mathdoc.fr/item/ZVMMF_2016_56_9_a9/