The velocity correlation function in cosmic-ray diffusion theory

Abstract

The concept of velocity correlation functions is introduced and applied to the calculation of cosmic ray spatial diffusion coefficients. It is assumed that the pitch angle scattering coefficient is already known from some other theory, and is reasonably well-behaved. Previous results for the coefficient for diffusion parallel to the mean field are recovered when the velocity-changing mechanism is artificially restricted to pitch angle scattering. The velocity correlation method is then applied to the more general case where there are fluctuations in the local mean field. It is found that the parallel diffusion coefficient is reduced in proportion to the amplitude of the field fluctuations, and that the ratio of the perpendicular to parallel diffusion coefficients cannot be greater than 〈δB x 2 〉/B 0 2 . It is shown in the appendix that the Liouville form of the scattering equation implies that the Fokker-Planck coefficients 〈Δμ2〉/Δt=2D μμ and 〈Δμ〉/Δt=∂D μμ /∂μ, and that all higher-order coefficients are identically zero.