Based on the perturbation theory with multiple scales, we have developed a technique to study the evolution of particle distribution as a function of oscillation amplitudes in a hadron storage ring. With a renormalization scheme for the zeroth-order term, a uniformly valid perturbation expansion for the distribution function is obtained. For localized nonlinear perturbations such as the beam-beam interaction at colliding points, this renormalization scheme results in a functional mapping for the particle distribution and the diffusion processes of particles in the beam can be studied numerically without resorting to the tracking of individual particles. A case involving a single nonlinear kick in the ring is presented to illustrate the method in detail.

• Received 15 January 1993

DOI:https://doi.org/10.1103/PhysRevE.47.4405