Modified Vlasov Equation and Collective Oscillations of Relativistic Radiating Electron Plasma

A Covariant Statistical Mechanics is developed for a system of electrons within the limits of plasma physics. By using the Landau-Lifshitz equation of motion for point charged particles, the phase space reduces to be a seven dimensional space instead of the ten dimensional space proposed by Hakim and Mangeney [J. Math. Phys. 1968, 9, 1, 116]. A modified Vlasov equation is obtained which permits one to deduce a general expression for the dispersion relations. Unlike other works, neither the Special Relativity nor any of the properties of the gauge are used to discard some of the wave modes. Instead, the continuity equation is used as a constraint embedded in the definition of the 4 vector current. This permits to make a mathematical analysis for calculating and discarding the transverse and longitudinal modes for any temperature. An expression for the longitudinal wave modes is deduced for any temperature and wave number. Usually, certain approximations are made to obtain such an expression that matches our result. But in general, without making such approximations, the result differs by a factor. The Landau damping and the damping due to the radiation reaction force are obtained and contrasted at low and high temperature cases. The results are compared with the ones obtained within Magnetohydrodynamics. Moreover, a numerical method is applied in order to compare the dispersion relations obtained with the approximations made. Surprisingly, it is sampled numerically that the dispersion relations are bounded for each temperature and that there is a cut-off for each wave number.