On the ultrarelativistictwo-stream instability, electrostatic turbulence and Brownian motion

Experimental evidence indicates that bulk plasma flow at ultrarelativistic speeds is common in astrophysical settings, e. g. the collimated jets of active galactic nuclei and gamma ray bursts. The low-plasma density of such flows implies their collisionless relaxation by means of wave-particle interactions. Such processes are not well understood in the ultrarelativistic regime. The thermalization of two interpenetrating equally dense electron-proton (e(-)p) beams in the absence of a magnetic field is examined here by means of 1.5D particle-in-cell simulations. The relative beam speeds correspond to Lorentz factors in the range 200-1000. The constraint to one spatial simulation dimension, which is aligned with the beam velocity vectors, implies that only the two-stream (TS) instability and the Weibel-type instability can grow, while filamentation instabilities are excluded. With this constraint and for our plasma parameters, the TS instability dominates. The electrostatic waves grow, saturate by the trapping of electrons, and collapse. The interaction of the electrons with the electric fields after the wave collapse represents a relativistic Wiener process. In response, the electrons are rapidly thermalized. The final electron distribution can be interpreted as a relativistic Maxwellian distribution with a high-energy tail arising from ultrarelativistic phase space holes.