The storage-ring FOKKER–PLANCK equation: inclusion of the free electron laser amplifier contributions

We analyze the longitudinal phase-space dynamics of e-beam circulating in storage-rings, by exploiting analytical and semianalytical solutions of the relevant FOKKER–PLANCK equation. Particular attention is devoted to the modification induced by the inclusion of the free electron laser amplifier contributions. We study time-dependent solutions and discuss the evolution towards the equilibrium configuration, we also analyze the effect of the laser parameter on the equilibrium conditions and on the longitudinal damping time.