Diffusion in Hamiltonian systems with a small stochastic perturbation

Abstract

We study some examples of Hamiltonian systems perturbed by a small random noise, which are relevant in accelerator physics: generalization to other Hamiltonian systems is briefly sketched. Starting from the Liouville equation, we derive a Fokker-Planck equation for the distribution function in the unperturbed action angle variables, which is valid for a vanishingly small noise. When the angle distribution has relaxed we write a simple equation for the distribution in the action; however there is evidence that such an equation governs the distribution averaged on the angle even before the relaxation occurs, suggesting that an averaging principle does apply. We compare the solution of this equation with the numerical simulation obtained by using a symplectic integrator of the stochastic Hamilton's equations. We consider also stochastically perturbed isochronous Hamiltonian and the corresponding area preserving map. For the map we write the action diffusion coefficients and compare, with the numerical simulations, the averaged distribution functions both in the action, obtained by solving the diffusion equation.