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## On application of stochastic differential equations for simulation of nonlinear wave–particle resonant interactions

Long-term simulations of energetic electron fluxes in many space plasma systems require accounting for two groups of processes with well separated time-scales: a microphysics of electron resonant scattering by electromagnetic waves and a macrophysics of electron adiabatic heating/transport by mesoscale plasma flows. Examples of such systems are Earth's radiation belts and Earth's bow shock, where ion-scale plasma injections and cross-shock electric fields determine a general electron energization, whereas electron scattering by waves relaxes anisotropy of electron distributions and produces small populations of high-energy electrons. The application of stochastic differential equations is a promising approach for including effects of resonant wave–particle interaction into codes tracing electrons in models of large-scale electromagnetic fields. This study proposes and verifies such equations for the system with non-diffusive wave–particle interactions, i.e., the system with nonlinear effects of phase trapping and bunching. We consider electron resonances with intense electrostatic whistler-mode waves often observed in the Earth's radiation belts. We demonstrate that nonlinear resonant effects can be described by stochastic differential equations with the non-Gaussian probability distribution of random variations of electron energies.