The Earth’s radiation belts, which extend from near the Earth to approximately geosynchronous orbit, contain highly energetic particles that actively interact with various plasma waves. This study reviews two numerical approaches to studying waveparticle interactions in the Earth’s radiation belts and discusses their respective advantages and limitations. The first approach involves diffusion simulations based on quasi-linear theory, which is well-suited for describing the collective dynamics of many particles from a statistical perspective. The second approach, test particle simulation, focuses on the detailed motion of individual particles, revealing nonlinear phenomena such as phase trapping and bunching. Both methods allow for the derivation of diffusion coefficients, which quantify the timescale of wave-particle interactions and help explain how particles either precipitate into the atmosphere or accelerate to higher energies in the Earth’s radiation belts. Additionally, these methodologies can be adapted to study the dynamics of planetary radiation belts, such as those around Jupiter and Saturn, by adjusting for the specific environmental parameters of each planet.

Nonthermal radiation from supernova remnants (SNRs) provides observational evidence and constraints on the diffusive shock acceleration (DSA) hypothesis for the origins of Galactic cosmic rays (CRs). Recently it has been recognized that a variety of plasma wave-particle interactions operate at astrophysical shocks and the detailed outcomes of DSA are governed by their complex and nonlinear interrelationships. Here we calculate the energy spectra of CR protons and electrons accelerated at Type Ia SNRs, using time-dependent, DSA simulations with phenomenological models for magnetic field amplification due to CR streaming instabilities, Alfv´enic drift, and free escape boundary. We show that, if scattering centers drift with the Alfv´en speed in the amplified magnetic fields, the CR energy spectrum is steepened and the acceleration efficiency is significantly reduced at strong CR modified SNR shocks. Even with fast Afv´enic drift, DSA can still be efficient enough to develop a substantial shock precursor due to CR pressure feedback and convert about 20-30% of the SN explosion energy into CRs. Since the high energy end of the CR proton spectrum is composed of the particles that are injected in the early stages, in order to predict nonthermal emissions, especially in X-ray and -ray bands, it is important to follow the time dependent evolution of the shock dynamics, CR injection process, magnetic field amplification, and particle escape. Thus it is crucial to understand the details of these plasma interactions associated with collisionless shocks in successful modeling of nonlinear DSA.