Liouville theory for fully analytic studies of transverse beam dynamics in laser-plasma ion accelerators

The exact solution of the Liouville equation expressed in terms of exponential operators can describe the phase space evolution of particle beams in transport lines. In this paper, we generalize the solution of the above equation for the case of beam losses induced by apertures and for particle beams with large spreads in the momentum space. We discuss the applicability of such approach to ion beams produced by high-intensity lasers interacting with critical plasmas, based on the comparison between theoretical findings and measurements.