Entropy and the spontaneous emission of plasma waves.

An entropy theorem applicable to the quasi-linear formulation of the Boltzmann-Vlasov equation for beam-plasma interactions with spontaneous emission is presented. The theorem elucidates the interplay between induced and spontaneous emission of plasma waves and their effects on streaming electrons (i.e., diffusion and 'friction' in velocity space). It is suggested that spontaneous emission terms are essential for a complete treatment of the quasi-linear Boltzmann-Vlasov formulation.