Nonlinear {delta}f Method for Beam-Beam Simulation

We have developed an efficacious algorithm for simulation of the beam-beam interaction in synchrotron colliders based on the nonlinear {delta}f method, where {delta}f is the much smaller deviation of the beam distribution from the slowly evolving main distribution f{sub 0}. In the presence of damping and quantum fluctuations of synchrotron radiation it has been shown that the slowly evolving part of the distribution function satisfies 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 differential equations. A numerical algorithm suited for direct code implementation of the evolving distributions for both {delta}f and f{sub 0} has been developed. Explicit expressions for the dynamical weights of macro-particles for {delta}f as well as an expression for the slowly changing f{sub 0} have been obtained.

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