Diffusive particle acceleration by an ensemble of shock waves

The theory of diffusive acceleration of cosmic rays by shock waves (first order Fermi acceleration) is generalized to the case that the particles propagate through a medium in which many shocks are present simultaneously. The basic assumption underlying the theory presented here is that the typical time between two shock transitions a particle undergoes is much longer than the time of spectral formation at an individual shock. This allows the separation of the process of shock acceleration from other propagation effects. We consider as such particle escape from the system (catastrophic losses) and continuous energy losses. The interplay of shock acceleration and these other propagation effects leads to the formation of particle spectra with spectral features such as breaks at characteristic momenta, or humps