  Missing cascades from TeV blazar beams indicate that collective plasma
effects may play a significant role in their energy loss. It is possible to
mimic the evolution of such highly energetic pair beams in laboratory
experiments using modern accelerators. The fate of the beam is governed by two
different processes, energy loss through the unstable mode and energetic
broadening of the pair beam through diffusion in momentum space. We chalk out
this evolution using a Fokker-Planck approach in which the drift and the
diffusion terms respectively describe these phenomena in a compact form. We
present particle-in-cell simulations to trace the complete evolution of the
unstable beam-plasma system for a generic narrow Gaussian pair beam for which
the growth rate is reactive. We show that the instability leads to an energetic
broadening of the pair beam, slowing down the instability growth in the linear
phase, in line with the analytical and numerical solutions of the Fokker-Planck
equation. Whereas in a laboratory experiment the change in the momentum
distribution is an easily measured observable as a feedback of the instability,
the consequence of diffusive broadening in an astrophysical scenario can be
translated to an increase in the opening angle of the pair beam.
