DIFFUSION IN HAMILTONIAN-SYSTEMS WITH A SMALL STOCHASTIC PERTURBATION

We study some examples of Hamiltonian systems perturbed by a small ran dom noise, which are relevant in accelerator physics; generalization t o other Hamiltonian systems is briefly sketched. Starting from the Lio uville equation, we derive a Fokker-Planck equation for the distributi on 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; howeve r there is evidence that such an equation governs the distribution ave raged on the angle even before the relaxation occurs, suggesting that an averaging principle does apply. We compare the solution of this equ ation with the numerical simulation obtained by using a symplectic int egrator of the stochastic Hamilton's equations. We consider also a sto chastically perturbed isochronous Hamiltonian and the corresponding ar ea preserving map. For the map we write the action diffusion coefficie nts and compare, with the numerical simulations, the averaged distribu tion functions both in the angle and in the action, obtained by solvin g the diffusion equation.