Cosmic-ray transport in inhomogeneous media

A theory of cosmic-ray transport in multi-phase diffusive media is developed, with the specific application to cases in which the cosmic-ray diffusion coefficient has large spatial fluctuations that may be inherently multi-scale. We demonstrate that the resulting transport of cosmic rays is diffusive in the long-time limit, with an average diffusion coefficient equal to the harmonic mean of the spatially varying diffusion coefficient. Thus, cosmic-ray transport is dominated by areas of low diffusion even if these areas occupy a relatively small, but not infinitesimal, fraction of the volume. On intermediate time scales, the cosmic rays experience transient effective sub-diffusion, as a result of low-diffusion regions interrupting long flights through high-diffusion regions. In the simplified case of a two-phase medium, we show that the extent and extremity of the sub-diffusivity of cosmic-ray transport is controlled by the spectral exponent of the distribution of patch sizes of each of the phases. We finally show that, despite strongly influencing the confinement times, the multi-phase medium is only capable of altering the energy dependence of cosmic-ray transport when there is a moderate (but not excessive) level of perpendicular diffusion across magnetic-field lines.