Discussion on applicability of the complex fractional moment method in Hamiltonian system

In this paper, the complex fractional moment (CFM) method is extended from the Langevin system to the Hamiltonian system for the first time, the applicability and advantages of the CFM method in Hamiltonian systems are investigated. According to the quasi-periodic property, the stochastic averaging technique is applied to the stochastic quasi-nonintegrable Hamiltonian system, and the one-dimensional Hamiltonian stochastic differential equation is obtained. By Mellin transformation, the associated Fokker Planck Kolmogorov (FPK) equation governing the Hamilton function is transformed into a set of Ordinary Differential Equations (ODEs) of CFM. The semi-analytical solutions of the FPK equation are obtained by solving the ODEs. Furthermore, by a polynomial approximation method for the Hamiltonian case proposed in this paper, the numerical results show the influence of parameters and initial values on the evolution of the transient probability density functions (PDF) of the system. The accuracy of CFM method in the Hamiltonian case is verified by error analysis.