This thesis describes theoretical studies of adiabatic motion of relativistic charged particles in the radiation belts and numerical modeling of multi-dimensional diffusion due to interactions between electrons and plasma waves.;A general Hamiltonian theory for the adiabatic motion of relativistic charged particles confined by slowly-varying background electromagnetic fields is presented based on a unified Lie-transform perturbation analysis in extended phase space (which includes energy and time as independent coordinates) for all three adiabatic invariants. First, the guiding-center equations of motion for a relativistic particle are derived from the particle Lagrangian. Covariant aspects of the resulting relativistic guiding-center equations of motion are discussed and contrasted with previous works. Next, the second and third invariants for the bounce motion and drift motion, respectively, are obtained by successively removing the bounce phase and the drift phase from the guiding-center Lagrangian. First-order corrections to the second and third adiabatic invariants for a relativistic particle are derived. These results simplify and generalize previous works to all three adiabatic motions of relativistic magnetically-trapped particles.;Interactions with small amplitude plasma waves are described using quasi-linear diffusion theory, and we note that in previous work numerical problems arise when solving the resulting multi-dimensional diffusion equations using standard finite difference methods. In this thesis we introduce two new methods based on stochastic differential equation theory to solve multi-dimensional radiation belt diffusion equations. We use our new codes to assess the importance of cross diffusion, which is often ignored in previous work, and effects of ignoring oblique waves, which are omitted in the parallel-propagation approximation of calculating diffusion coefficients. Using established wave models we show that ignoring cross diffusion or oblique waves may produce large errors at high energies. Results of this work are useful for understanding radiation belt dynamics, which is crucial for predictability of radiation in space.
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