Stochastic Particle Acceleration in Parallel Relativistic Shocks
  We present results of test-particle simulations on both the first- and the second-order Fermi acceleration for relativistic parallel shock waves. Our studies suggest that the role of the second-order mechanism in the turbulent downstream of a relativistic shock may have been underestimated in the past, and that the stochastic mechanism may have significant effects on the form of the particle spectra and its time evolution. 
Joni J.P. Virtanen and Rami Vainio