  We propose a new approach, based on statistical mechanics, to predict the
saturated state of a single-pass, high-gain free-electron laser (FEL). In
analogy with the violent relaxation process in self-gravitating systems and in
the Euler equation of 2D turbulence, the initial relaxation of the laser can be
described by the statistical mechanics of an associated Vlasov equation. The
Laser field intensity and the electron bunching parameter reach
quasi-stationary values that are well fitted by a Vlasov stationary state if
the number of electrons $N$ is sufficiently large. Finite $N$ effects
(granularity) finally drive the system to Boltzmann-Gibbs statistical
equilibrium, but this occurs on times that are unphysical (i.e. excessively
long undulators). All theoretical predictions are successfully tested in finite
$N$ numerical experiments. Our results may provide important hints to the
optimization of the length of the FEL undulator.
