The Proton Distribution Function in Weakly Magnetized Turbulent Plasmas
  We calculate the proton distribution function due to heating by subsonic (Alfvenic) turbulence in a weakly magnetized collisionless plasma. The distribution function is nonthermal. For non-relativistic energies, it is an exponential of the magnitude of the proton velocity. For ultra-relativistic energies, it can be characterized as a power law with a momentum-dependent slope. 
Andrei Gruzinov and Eliot Quataert