Anomalous diffusion equation: Application to cosmic ray transport

Abstract

We consider the propagation of galactic cosmic rays under assumption that the interstellar medium is a fractal one. An anomalous diffusion (AD) equation in terms of fractional derivatives is used to describe cosmic ray propagation. The anomaly in the model results from large free paths (“Lévy flights”) of particles between galactic inhomogeneities, and from the fact that a particle stays in a trap for a long time. Asymptotical solutions of the AD equation for point instantaneous and for impulse sources with inverse power spectrum relating to supernova bursts are found. They cover both subdiffusive and superdiffusive regimes, and are expressed in terms of the stable distributions. The energy dependence of spectral exponent of observed particles in the different regimes is discussed.