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We perform a detailed derivation of the bounce-averaged relativistic Fokker-Planck diffusion equation applicable to arbitrary magnetic field at a constant Roederer <I>L</I>. The form of the bounce-averaged diffusion equation is found regardless of details of the mirror geometry, suggesting that the numerical schemes developed for solving the modified two-dimensional (2-D) Fokker-Planck equation in a magnetic dipole should be feasible for similar computation efforts on modeling wave-induced particle diffusion processes in any non-dipolar magnetic field. However, bounce period related terms and bounce-averaged diffusion coefficients are required to be computed in realistic magnetic fields. With the application to the Dungey magnetosphere that is controlled by the intensity of southward interplanetary magnetic field (IMF), we show that with enhanced southward IMF the normalized bounce period related term decreases accordingly, and bounce-averaged diffusion coefficients cover a broader range of electron energy and equatorial pitch angle with a tendency of increased magnitude and peaking at lower energies. The compression of the Dungey magnetosphere can generally produce scattering loss of plasma sheet electrons <~4 keV and radiation belt electrons >~100 keV on a timescale shorter than that in a dipolar field, and induce momentum diffusion at high pitch angles closer to 90°. Correspondingly, the strong diffusion rate drops considerably as a product of changes in both the equatorial loss cone and the bounce period. The extent of differences in all the parameters introduced by the southward IMF intensification also becomes larger for a field line with higher equatorial crossing. With the derived general formulism of bounce-averaged diffusion equation for arbitrary 2-D magnetic field, our results confirm the need for the adoption of realistic magnetic fields to perform accurate determination of electron resonant scattering rates and precise multi-dimensional diffusion simulations of magnetospheric electron dynamics

Bounce-averaged Fokker-Planck diffusion equation in non-dipolar magnetic fields with applications to the Dungey magnetosphere