Nonperturbative geometrodynamic calculation of chaotic mixing time in charged-particle beams

The time scale for irreversible mixing in a charged-particle bunch as a consequence of time-independent, nonlinear space-charge forces is estimated analytically to be a few plasma periods, much shorter than the two-body relaxation time. The basis for the estimate is a metric tensor inferred from Hamilton's least-action principle. Geodesics derived from the metric tensor correspond to particle trajectories. Their behavior reflects the properties of the curvilinear manifold in which they are embedded, among which irregularities associated with parametric resonances are of foremost importance. Exponential separation of nearby chaotic trajectories is thereby accessible to the geometrodynamic approach. The e-folding time associated with dispersing an initially localized perturbation throughout the bunch characterizes the process of irreversible mixing. It thereby constrains both the placement and size of hardware for emittance compensation that may be needed, for example, to undo phase-space degradation arising from coherent synchrotron radiation in magnetic bends. These constraints are estimated for linacs powering modern infrared and x-ray free-electron lasers