Nonlinear f Method for Beam-Beam Simulation

Abstract

We havedeveloped an e cacious algorithm for simulation of the beam-beam interaction in synchrotron colliders based on the nonlinear f method, where f is the much smaller deviation of the beam distribution from the slowly evolving main distribution f0. In the presence of damping and quantum uctuations of synchrotron radiation it has been shown that the slowly evolving part of the distribution function satis es a Fokker-Planck equation. Its solution has been obtained in terms of a beam envelope function and an amplitude of the distribution, which satisfy a coupled system of ordinary di erential equations. A numerical algorithm suited for direct code implementation of the evolving distributions for both f and f 0 has been developed. Explicit expressions for the dynamical weights of macro-particles for f as well as an expression for the slowly changing f 0 have been obtained