Diffusion in Hamiltonian systems driven by harmonic noise

Abstract

We study the dynamics of randomly perturbed integrable Hamiltonian systems. In the limit of small perturbations, we show that the distribution function of the action variable satisfies a Fokker-Planck equation whose diffusion coefficient depends on the correlation function of the stochastic process. By using an harmonic noise we show the effect of resonances between the spectral density of the noise and the proper frequencies of the system. We explicitly consider the dynamics of a pendulum whose potential is stochastically perturbed; this model is relevant for the study of the synchrotron motion in accelerator physics. The numerical results for the distribution function of the energy are in very good agreement with the solutions of the Fokker-Planck equation.