Island formation in collisionless kinetic plasmas with perturbed orbits

In the vinicity of phase-space resonances for a given species of plasma, the particle distribution function is flattened as free energy is exchanged between the plasma and resonant electromagnetic waves. Here, we present action-angle variables which explicitly separate the adiabatic invariant from the contribution that arises from perturbation of the Hamiltonian over the course of a single orbit. Then, we perform similar analysis for the orbit period, allowing one to identify a generating function $\psi$ which modifies the zeroth order contribution to the orbit period (unperturbed orbit) in the case where the particle energy is not time-invariant. We posit that a population of `quasi-trapped' particles cross the separatrix, directly allowing for island growth and decay. Then, we employ $\psi$ in the ensemble case by considering how this generating function allows for solutions of the 6+1D electrostatic Vlasov equation where finite growth and frequency sweeping of modes occur. Finally, we derive an approximate form for $\psi$ outside of the separatrix, allowing for qualitative observation of phase-space shear and island formation.