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.