On stochastic complex beam–beam interaction models with Gaussian colored noise

This paper is to continue our study on complex beam–beam interaction models in particle accelerators with random excitations Y. Xu, W. Xu, G.M. Mahmoud, On a complex beam–beam interaction model with random forcing [Physica A 336 (2004) 347–360]. The random noise is taken as the form of exponentially correlated Gaussian colored noise, and the transition probability density function is obtained in terms of a perturbation expansion of the parameter. Then the method of stochastic averaging based on perturbation technique is used to derive a Fokker–Planck equation for the transition probability density function. The solvability condition and the general transforms using the method of characteristics are proposed to obtain the approximate expressions of probability density function to order ε. Also the exact stationary probability density and the first and second moments of the amplitude are obtained, and one can find when the correlation time equals to zero, the result is identical to that derived from the Stratonovich–Khasminskii theorem for the same model under a broad-band excitation in our previous work.