Modified Vlasov equation and dispersion relations for a relativistic radiating electron plasma

A modified Vlasov equation is obtained by developing a covariant statistical mechanics for a system of electrons without considering the effects of the ions and including the Landau–Lifshitz equation of motion. General dispersion relations for the transverse and longitudinal modes for any temperature are expressed. The results are similar to those found by Hakim & Mangeney (Phys. Fluids, vol. 14, 1971, pp. 2751–2781) for both the modified Vlasov equation and the dispersion relations. However, for the longitudinal mode, unlike the development of Hakim and Mangeney, correct expansions are done in order to give a numerical approach to obtain the longitudinal relativistic dispersion relations for any value of the wavenumber. Accordingly, new loop solutions, with turning points, crossing the super-luminous region and the super-thermal region are found. Although the expressions for the Landau damping and the damping due to the radiation reaction force coincide with the Hakim and Mangeney results for some particular cases, in general they are different. A Landau anti-damping appears in the second branch of the loop in a small region between the cutoff point and the intersection with the super-thermal line. The analysis of this effect leads us to a kind of wave pulse. We will call them bipolar waves. The treatment contains the relativistic interactions between all the electrons in the system with retarded effects. This explain the differences with Zhang’s recent work (Phys. Plasmas, vol. 20, 2013, 092112–092132). It is shown that for low densities, the cutoff of the wave is due to the dispersion relations and not due to the radiation reaction force damping. While for both high densities and temperatures, the damping due to the radiation reaction force is important.