We develop weighted macro-particle tracking and the Perron-Frobenius operator technique for simulating the time evolution of two beams coupled via the beam-beam interaction. The /spl pi/- and /spl sigma/-modes, with and without a sextupole perturbation, are studied based on the Vlasov-Poisson system. Extending the standard averaging formalism to maps with a small collective force, we derive an approximation to the "kick-rotate" model of our simulation. This eliminates the delta function smoothing in the Vlasov equation, common in many theoretical approaches. Action densities are quasi-equilibria, consistent with simulation, and linearization leads to uncoupled Fourier modes and third-kind integral equations. Extensions to 2-d.o.f. and a flexible general purpose code are being developed.
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机译：我们开发了加权宏粒子跟踪和Perron-Frobenius算子技术，用于模拟通过束-束相互作用耦合的两个束的时间演化。基于Vlasov-Poisson系统研究了带有或不带有六极摄动的/ spl pi /-和/ spl sigma /-模式。将标准平均形式主义扩展到具有较小集体力的地图，我们得出了模拟的“踢转”模型的近似值。这消除了Vlasov方程中的三角函数平滑，这在许多理论方法中都是很常见的。动作密度是准平衡的，与模拟一致，线性化导致不耦合的傅里叶模式和第三类积分方程。扩展到2-d.o.f。并且正在开发一种灵活的通用代码。
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