Particle‐in‐cell simulation studies of the non‐linear evolution of ultrarelativistic two‐stream instabilities

Gamma‐ray bursts are associated with relativistic plasma flow and intense X‐ray and soft gamma‐ray emissions. We perform particle‐in‐cell simulations to explore the growth and saturation of waves driven by the electrostatic two‐stream instability that may contribute to the thermalization of the relativistic plasma flows and to the electromagnetic emissions. We evolve self‐consistently the instability driven by two charge‐neutral and cool interpenetrating beams of electrons and protons that move at a relative Lorentz factor of 100. We perform three simulations with the beam density ratios of 1, 2 and 10. The simulations show that the electrostatic waves saturate by trapping the electrons of only one beam and that the saturated electrostatic wave fields spatially modulate the mean momentum of the second beam, while retaining its temperature. Cavities form in the charge density of the latter beam which, in turn, compress the electrostatic waves to higher intensities. A runaway process develops that terminates with the collapse of the waves and the development of an exponential electron high‐energy tail. We bring forward evidence that this energetic tail interacts stochastically with the charge density fluctuations of the relativistic proton beam. In response, an electron momentum distribution develops that follows an inverse power law up to a spectral break at four times the beam Lorentz factor.