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A possible mechanism, which explains the diffusion in the phase space due to the ripples in the quadrupole currents, is studied on a simplified version of the SPS lattice used for experiments. We describe the diffusion driven by a single resonance in the s

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MODULATED DIFFUSION FOR A SIMPLE LATTICE MODELA. BAZZANIDept. of Mathematics and INFN, Sezione di Bologna, P.zza di Porta San Donato n.5, I-40126 Bologna, Italy.

F. BRINIDept. of Physics and INFN, Sezione di Bologna, v. Irnerio n.46, I-40126 Bologna, Italy.A possible mechanism, which explains the di usion in the phase space due to the ripples in the quadrupole currents, is studied on a simpli ed version of the SPS lattice used for experiments. We describe the di usion driven by a single resonance in the space of the adiabatic invariant (action variable), by using the results of the Neishtadt's theory. Under suitable hypothesis, it is possible to introduce a random walk for the adiabatic invariant, which gives a quantitative description of the di usion. The comparison with the numerical results turns out to be very e ective.

1 INTRODUCTION The study of the long term dynamics aperture in a particle accelerator is related to the stability of the orbits in a neighborhood of an elliptic xed point of a symplectic map . The multipolar components of the magnetic eld correspond to nonlinear terms in the Taylor expansion of the map. The presence of Arnold di usion and overlapping of resonances allows the possibility of nding unstable orbits, near the elliptic xed point. However the numerical simulations show the following scenario: there is a threshold in the phase space after which a fast escape to in nity is observed (short term dynamics aperture); there is a neighborhood of the xed point, where no di usion can be detected; there is a region where some particular orbits escape to in nity after a big number of iterations, but usually the measure of the initial conditions of the unstable orbits is very small. This situation can be explained by taking into account both the extreme slowness and sensitivity to the initial condition of Arnold di usion and the limitation of the region of overlapping of resonances, which is a generic feature for polynomial symplectic maps. As a consequence to explain the slow di usion of particles1,2 3 4

Work partially supported by EC Human Capital and Mobility contract Nr. ERBCHRXCT940480 1