On the stability of spatially uniform Langmuir oscillations of electronic plasmas

We investigate the stability of spatially uniform, time-periodic solutions of the one-dimensional Vlasov-Maxwell system describing the longitudinal oscillations of an electronic plasma in an uniform neutralizing ion background. We show that such a stability problem can be trivially solved since the zero wave number mode of the electric field, i.e. its space average, performs pure Langmuir oscillations independently of the other modes. We however point out that such oscillations do affect on time average the evolution of the velocity distribution function in the frame at rest.