The model of ensembles for the beam dynamics simulation

It is proposed to use the “Ensembles” of particles for modeling the beams in accelerators. Each ensemble describes the longitudinal and transverse dynamics of the nonmonochromatic bunch in the six dimensional phase space. taking into account all coupling effects, coming from the relativistic relation between variation of the velocity projection and energy spread. Ensemble parameters include average values of coordinate and momentum, bunch dimensions, noncorrelated momentum spread and all second order correlation parameters. Self-consistent equations for the Ensemble are derived from the Vlasov equation. The presentation by vector and matrix gives the possibility to define an implicit algorithm for computer simulations. Examples of application of this model for the calculation of the beam dynamics in the accelerators are presented