On a simplified description of waves in non-collision plasmas

Aim of this methodic note is to collate approaches of A.A. Vlasov and L.D. Landau to the propagation of electromagnetic waves in hot rarified plasmas. Over half a century ago, A.A. Vlasov and L.D. Landau used the kinetic equation to show that – in accordance to the causality principle – electromagnetic waves propagating in equilibrium plasmas should decay even if the binary interaction between particles is negligibly weak. However, for a long time, the pioneer theories of A.A. Vlasov and L.D. Landau were regarded as not quite congenial. To reduce misconceptions in approaches to the kinetic effects at the wave propagation in non-collision plasmas, the paper submitted proposes to duplicate the method of kinetic equation with a simpler method – based on using elementary electron motion equations. The theoretical model represents a homogeneous plasma where the primary distribution of electron velocities is axis-symmetric; the longitudinal electric wave is propagating along this axis. The electron motion equations are used to derive an integral related to the plasma dielectric permittivity which is included into the wave dispersion equation. In particular, if the electron velocity distribution function is sufficiently smooth, the increment or the decrement of the wave is determined with derivative of the primary distribution function at the point of Cherenkov synchronism between electrons and the wave (the asymptotic solution of L.D. Landau). The simplified approach is illustrated with a wave propagation in a plasma where the electron velocity distribution is approximated with a Lorenz function. In this case, the wave decrement coincides with one obtained in the old paper of A.A. Vlasov; and at the Cherenkov synchronism at the «tail» of the function the wave decrement corresponds to the asymptotic theory of L.D. Landau. Thus, the simplified analysis has confirmed that the theories of A.A. Vlasov and L.D. Landau are mutually consistent.