ELSEVIER Physica D 76 (1994) 8-21 PHYSICA Diffusion in Hamiltonian systems with a small stochastic perturbation A. Bazzani, S. Siboni, G. Turchetti Dipartirnento di Fisica della Universit~ di Bologna, 1-40136 Bologna, Italy and INFN, Sezione di Bologna, Italy 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 a 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 angle and in the action, obtained by solving the diffusion equation. 1. Introduction The diffusion in Hamiltonian systems is rele- vant to many areas of physics such as celestial mechanics (inclination of the axis of a planet [ 1 ] and stability of the asteroids [2] ), accelerator physics (slow dispersion of particles in a beam [3,4] ), plasma physics (diffusion of magnetic field lines in toroidal devices [5] ) and fluid dy- namics (Lagrangian description of the diffusion of particles in a two dimensional velocity field [6] ). Two distinct limiting cases can be con- sidered: strongly chaotic Hamiltonian systems and integrable Hamiltonian systems in presence of a weak stochastic perturbation. In the first case if a single ergodic component exists, with a large measure in any subset of phase space and positive Lyapunov exponents, the quasilinear theory provides a satisfactory description of the diffusion due to the ubiquity of chaotic orbits [7]. In the second case the unperturbed phase space is foliated by invariant manifolds and the stochastic perturbation induces the diffu- sion in the space of the integrals of motion [8]. The description of the process with a Fokker- Planck equation is not trivial because one has to take into account the interaction between the nonlinearity of the unperturbed system and the stochastic perturbation. When the system is al- most completely foliated with invariant curves, 0167-2789/94/$07.00 Q 1994 Elsevier Science B.V. All rights reserved SSDI 01 67-27 89 (94)00036-P