  Understanding the transport of energetic cosmic rays belongs to the most
challenging topics in astrophysics. Diffusion due to scattering by
electromagnetic fluctuations is a key process in cosmic-ray transport. The
transition from a ballistic to a diffusive-propagation regime is presented in
direct numerical calculations of diffusion coefficients for homogeneous
magnetic field lines subject to turbulent perturbations. Simulation results are
compared with theoretical derivations of the parallel diffusion coefficient's
dependencies on the energy and the fluctuation amplitudes in the limit of weak
turbulence. The present study shows that the widely-used extrapolation of the
energy scaling for the parallel diffusion coefficient to high turbulence levels
predicted by quasi-linear theory does not provide a universally accurate
description in the resonant-scattering regime. It is highlighted here that the
numerically calculated diffusion coefficients can be polluted for low energies
due to missing resonant interaction possibilities of the particles with the
turbulence. Five reduced-rigidity regimes are established, which are separated
by analytical boundaries derived in the present work. Consequently, a proper
description of cosmic-ray propagation can only be achieved by using a
turbulence-level-dependent diffusion coefficient and can contribute to solving
the Galactic cosmic-ray gradient problem.
