Diffusive propagation of ultra-high-energy cosmic rays and the propagation theorem

We present a detailed analytical study of the propagation of ultra - high- energy (UHE) particles in extragalactic magnetic fields. The crucial parameter that affects the diffuse spectrum is the separation between sources. In the case of a uniform distribution of sources with a separation between them much smaller than all characteristic propagation lengths, the diffuse spectrum of UHE particles has a universal form, independent of the mode of propagation. This statement has the status of theorem. The proof is obtained using the particle number conservation during propagation and also using the kinetic equation for the propagation of UHE particles. This theorem can be also proved with the help of the diffusion equation. In particular, it is shown numerically how the diffuse fluxes converge to this universal spectrum, when the separation between sources diminishes. We study also the analytic solution of the diffusion equation in weak and strong magnetic fields with energy losses taken into account. In the case of strong magnetic fields and for a separation between sources large enough, the GZK cutoff can practically disappear, as it has been found early in numerical simulations. In practice, however, the source luminosities required are too large for this possibility.