Phase Space Dynamics in Theory and Simulation

Particle motion in accelerators is usually assumed in 6D phase space, under the action of external forces for acceleration and focusing, and with space charge interaction resulting from the Coulomb interaction between particles. The by far dominating contribution to direct Coulomb interaction in accelerator beams is via the so-called mean field or self-consistent space charge potential, which is calculated from an assumed smoothed charge density n(x, y, z, t), and ignoring the discreteness of charges. Assuming that collisional effects can be neglected, this chapter introduces the Vlasov-Poisson equations and the important concepts of emittance, rms equivalence as well as beam anisotropy. In the context of computer simulation, however, some aspects of artificial collision and noise effects due to numerical discretisation are an important subject, which is also briefly discussed.