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Nonlinear $δf$ Method for Beam-Beam Simulation

Yunhai Cai·Alexander W. Chao·Stephan I. Tzenov·Toshi Tajima
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Abstract

We have developed an efficacious algorithm for simulation of the beam-beam interaction in synchrotron colliders based on the nonlinear $δf$ method, where $δf$ is the much smaller deviation of the beam distribution from the slowly evolving main distribution $f_0$. 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 $δf$ and $f_0$ has been developed. Explicit expressions for the dynamical weights of macro-particles for $δf$ as well as an expression for the slowly changing $f_0$ have been obtained.

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Yunhai Cai, Alexander W. Chao, Stephan I. Tzenov, Toshi Tajima. 2000-10-23. Nonlinear $δf$ Method for Beam-Beam Simulation. https://arxiv.org/abs/physics/0010055

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