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.

Introduction
Resonant Interactions between Waves and Particles in the Inner Magnetosphere (or Radiation Belts)
Quasi-Linear Diffusion Theory
Test Particle Simulation of Wave-Particle Interactions
Fokker–Planck Diffusion Equation
Final Remarks
Acknowledgments
References

• Kyung-Chan Kim(Department of Astronomy and Space Science, College of Natural Sciences, Chungbuk National University, Cheongju, Chungbuk, 28644, Republic of Korea) Corresponding author
• Dae-Young Lee(Department of Astronomy and Space Science, College of Natural Sciences, Chungbuk National University, Cheongju, Chungbuk, 28644, Republic of Korea)
• Miroslav Hanzelka(GFZ German Research Centre for Geosciences, Potsdam, Germany)