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): x + omega(0)(2)x + epsilon(2) lambda g(x) + epsilon(2)f(x) p(omega(0)t) = epsilon(2)z( t). The method of stochastic averaging is used to derive a Fokker-Plan ck-Kolmogorov equation describing the probability density for the ampl itude of the solutions. In the case g(x) = x, an odd polynomial f(x) = k(3)x(3) + k(5)x(5) + ... and p(omega(0)t) = cos omega(0)t, we obtain the exact stationary probability density function and the first and s econd moments for the amplitude of the solutions. Numerical simulation shows very good agreement with the analytical results of this study.