Beam-beam interaction models with a small stochastic perturbation

In this work, we study a class of differential equations, which may be used to model the beam-beam interaction in particle accelerators, in the presence of a small stochastic perturbation z(t): ẍ + ω02x + ϵ2λg(dotx) + ϵ2 f(x)p(ω0t) = ϵ2z(t). The method of stochastic averaging is used to derive a Fokker-Planck-Kolmogorov equation describing the probability density for the amplitude of the solutions. In the case g(dotx) = dotx, an odd polynomial f(x) = k3x3 + k5x5 + ⋯ and p(ω0t) = cosω0t, we obtain the exact stationary probability density function and the first and second moments for the amplitude of the solutions. Numerical simulation shows very good agreement with the analytical results of this study.