Method of projection operator for the study of angle-averaged distribution function of beam particles in hadron storage rings

Abstract

Starting from the Vlasov equation for the single-particle phase-space distribution of beam particles in hadron storage rings, the angle-averaged distribution function of the action variables is studied by means of projection operator. It is shown that the angle-averaged distribution function is governed by a kinetic equation similar to the Fokker-Planck equation but with a memory integral. For localized nonlinear perturbations, this kinetic equation can be further reduced to a moment map which can be iterated numerically for studying the beam-size growth in hadron storage rings. To examine the validity of this treatment, the evolution of beam size is studied with examples of integrable and nonintegrable systems. It is found that the result from the moment map agrees very well with the exact solution for the integrable example or that from multiparticle tracking for the nonintegrable example when the system is not very close to low-order resonances. The angle-averaged distribution function is a promising tool to study the effect of the space charge and beam-beam interaction on high-intensity hadron beams. © 1998 The American Physical Society.