Statistical mechanics and Vlasov equation allow for a simplified hamiltonian description of single pass free electron laser saturated dynamics

A reduced Hamiltonian formulation to reproduce the saturated regime of a single pass free electron laser, around perfect tuning, is here discussed. Asymptotically, $N_m$ particles are found to organize in a dense cluster, that evolves as an individual massive unit. The remaining particles fill the surrounding uniform sea, spanning a finite portion of phase space, approximately delimited by the average momenta $\omega_+$ and $\omega_-$. These quantities enter the model as external parameters, which can be self-consistently determined within the proposed theoretical framework. To this aim, we make use of a statistical mechanics treatment of the Vlasov equation, that governs the initial amplification process. Simulations of the reduced dynamics are shown to successfully capture the oscillating regime observed within the original $N$-body picture