Accurately modelling and forecasting of the dynamics of the Earth’s radiation belts with the available computer resources represents an important challenge that still requires significant advances in the theoretical plasma physics field of wave–particle resonant interaction. Energetic electron acceleration or scattering into the Earth’s atmosphere are essentially controlled by their resonances with electromagnetic whistler mode waves. The quasi-linear diffusion equation describes well this resonant interaction for low intensity waves. During the last decade, however, spacecraft observations in the radiation belts have revealed a large number of whistler mode waves with sufficiently high intensity to interact with electrons in the nonlinear regime. A kinetic equation including such nonlinear wave–particle interactions and describing the long-term evolution of the electron distribution is the focus of the present paper. Using the Hamiltonian theory of resonant phenomena, we describe individual electron resonance with an intense coherent whistler mode wave. The derived characteristics of such a resonance are incorporated into a generalized kinetic equation which includes non-local transport in energy space. This transport is produced by resonant electron trapping and nonlinear acceleration. We describe the methods allowing the construction of nonlinear resonant terms in the kinetic equation and discuss possible applications of this equation.

In the present investigation, radial diffusion of equatorially trapped electrons in the magnetospheres of Jupiter and Rotating Radio Transients (RRATs) are examined and compared. It is assumed that electrons lose energy through synchrotron radiation and the wave-particle interaction. The phase space density of the electrons, which go through gradB drift in Jupiter's and RRATs magnetospheres and thus resonate with the plasma waves, changes and this change predicted by the model seems to be consistent with the Pioneer 10 and Pioneer 11 data for Jupiter's case and a similar result obtained for RRATs.