Maximum Particle Energies by Fermi Acceleration and the Origin of Cosmic Rays above the Knee

We derive the maximum accelerated particle energy from first-order and second-order Fermi acceleration at nonrelativistic and relativistic shocks for explosions taking place in a uniform surrounding medium. Second-order stochastic processes in relativistic flows are shown to be capable of accelerating cosmic rays to ultra-high energies. Cosmic rays above the knee of the cosmic ray spectrum can be accelerated by the second-order Fermi mechanism in relativistic flows, such as those occuring in gamma-ray bursts and unusual supernovae like SN 1998bw.