Velocity Correlation and the Spatial Diffusion Coefficients of Cosmic Rays: Compound Diffusion

We consider the transport of charged particles in a stochastic magnetic field using a method based on the velocity correlation function ⟨ v i (0) v j ( t )⟩ developed by R. Kubo. This can be used under very general conditions to evaluate the corresponding spatial diffusion coefficients, if the fluctuations are statistically homogeneous in space and time. Although Kubo's theory is quite general, it is not obvious how it can be applied to describe compound diffusion when particles are strictly tied to the magnetic field lines and perpendicular transport results solely from the random walk of the field lines. This motion is non-Markovian and leads to a slower ⟨Δ x 2 ⟩ ∝ t 1/2 diffusion in contrast to the ⟨Δ x 2 ⟩ ∝ t dependence of the standard diffusion. We demonstrate how compound diffusion fits into Kubo's formalism. As intuitively as can be anticipated, the non-Markovian nature of the motion results in a long-term anticorrelation in ⟨ v j (0) v i ( t )⟩, which causes the ordinary spatial diffusion coefficient to vanish identically. The ⟨Δ x 2 ⟩ ∝ t 1/2 dependence of the compound diffusion can also be recovered from the Laplace transform of the velocity correlation function. Some implications of the long-term anticorrelation are discussed.