  The change of the distribution function of electron-positron pair beams
determines whether GeV photons can be produced as secondary radiation from TeV
photons. We will discuss the instabilities driven by pair beams. The system of
a thermal proton-electron plasma and the electron-positron beam is collision
free. We have, therefore, used the Particle-in-Cell simulation approach. It was
necessary to alter the physical parameters, but the ordering of growth rates
has been retained. We were able to show that plasma instabilities can be
recovered in particle-in-cell simulations, but their effect on the pair
distribution function is negligible for beam-background energy density ratios
typically found in blazars.
