Application of Scaling Properties of the Vlasov and Fokker-planck Equations to Improved Macroparticle Models

Here ψ(E; t) is the energy distribution of a coasting beam. It is often easier in practical applications to follow the evolution of phase space density by tracking a representative distribution of macroparticles using the equations of motion. There are scaling relations derived from the PDE’s which can make dramatic improvements in the efficiency or practical scope of macroparticle models. Scaling rules reported previously are recapitulated below. Their usefulness in speeding up macroparticle model calculations is discussed with most attention to the simpler case of time scaling in the Vlasov equation.