Application of scaling properties of the Vlasov and the Fokker-Planck equations to improved macroparticle models

Numerical simulations of cooling processes over minutes or hours of real time are usually carried out using direct solution of the Fokker-Planck equation. However, by using scaling rules derived from that equation, it is possible to use macroparticle representations of the beam distribution. Besides having applications for cooling alone, the macroparticle approach allows combining the cooling process with other dynamical processes which are represented by area-preserving maps. A time-scaling rule derived from the Vlasov equation can be used to adjust the time step of a map-based dynamics calculation to one more suitable for combining with a macroparticle Fokker-Planck calculation. The time scaling for the Vlasov equation is also useful for substantially more rapid calculations when a macroparticle model of a conservative multiparticle system requires a large number of macroparticles to faithfully produce the collective potential or when the model must simulate a long time period