Compound and perpendicular diffusion of cosmic rays and random walk of the field lines. I. Parallel particle transport models

A Chapman-Kolmogorov equation description of compound transport of cosmic rays due to random walk of the magnetic field lines, and for a range of models for particle transport along the field, is developed. The probability distribution, Pp, for the particle propagation along the field corresponds to either (1) a ballistic or scatter-free model, (2) a parallel diffusion model, or (3) a telegrapher equation model. The probability distribution function (pdf) describing the magnetic field statistics, P-FRW, is assumed to be Gaussian. These models are used to discuss features of the dropout events in the low-energy, solar cosmic-ray intensity observed by Mazur et al. We show that the Chuvilgin and Ptuskin transport equation for compound diffusion, at sufficiently late times, can be written as a fractional Fokker-Planck equation, involving ordinary diffusion parallel to the mean magnetic field and compound diffusion of the particles normal to the field. The Green's function solution of the equation and the corresponding spatial moments of the particle transport, both parallel and perpendicular to the field, are obtained. The two-dimensional pdf for compound diffusion across the field is obtained as an inverse Laplace transform, or as a real integral.