Direct simulation of beam density evolution in phase space: Preliminary modeling in one degree of freedom

The concept of phase space density provides a properly macroscopic description of a beam bunch of mutually non-interacting particles. Ideally, simulations of particle beam behavior ought to concentrate directly on the dynamics of the beam phase space density rather than on that of individual particles--an immediate benefit being the elimination of the statistical uncertainty inevitably arising from the sampling of such particles. Here a preliminary step is taken in the direction of this desirable goal--a ''toy'' one-degree-of-freedom (the transverse horizontal) beam phase space density is simulated for a highly simplified lattice consisting of a purely linear FODO ring which contains a single thin-element nonlinear kick. These meager ingredients produce a fascinating collection of beam density snapshots in the phase space--including such phenomena as the filamentation and ''fragmentation'' of a kicked beam, the process of formation of phase space islands under a variety of circumstances, and spiral twisting of the beam density within the dynamic aperture