Stochastic cyclotron acceleration in fluctuating transverse electric fields: a Fokker–Planck approach

Abstract

Particle acceleration in magnetized plasmas often occurs through resonant interactions with transverse electric field fluctuations. This study presents a detailed analytical derivation of the stochastic acceleration of a charged particle in a uniform magnetic field subjected to a time-varying, statistically isotropic Gaussian white-noise electric field. Using the method of variation of parameters, the equations of motion are transformed into a complex evolution equation for the gyration amplitude, effectively decoupling the guiding-center drift from the gyratory motion. We explicitly derive the Fokker–Planck diffusion coefficients D1 and D2 from the two-time correlation function of the stochastic electric field, demonstrating that the diffusion rate is proportional to the spectral density W(ωg) of the electric field at the cyclotron resonance. For a constant (white-noise) energy spectrum we obtain the exact analytical solution for the energy distribution function, F(w,t)=1D1texp[−w+w0D1t]I0(2ww0D1t), in terms of modified Bessel functions of the first kind I0. This result describes the transition from an initial mono-energetic state to a non-thermal distribution with a high-energy tail. The model is explicitly valid in the dilute, collisionless plasma regime where Coulomb collision mean free paths greatly exceed the system scale, as in the solar wind and magnetospheric environments. Our findings quantify the efficiency of cyclotron resonance as a stochastic heating process and offer a predictive tool for relating observed turbulence spectra to accelerated particle populations.