Copyright information to be inserted by the Publishers MODUL(3)

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

MODULATED DIFFUSION FOR A SIMPLE LATTICE MODEL

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that the perpetual adiabatic invariance (no limits on t) holds for a large set of initial conditions. As a consequence the instabilities due to a slow modulation are driven by the presence of resonances and in particular of separatrices in the phase space. The adiabatic description of the dynamics near to a resonance is based on the slow pulsation of the separatrix in the phase space due to the slow time dependence of the Hamiltonian (show g. 1 left); starting from an initial condition out of the separatrix the actual trajectory will follow the unperturbed orbits of H (q; p; ) in order to preserve the value of the a.i. (i.e. the area of the unperturbed orbit for a 2D system); but since the separatrix is moving in the phase space, at a certain time it could happen that the orbit crosses the separatrix and starts to be trapped in the resonance region. In such a case the inequality (1) does not hold any more and it is necessary to compute the

changing of the a.i. due to a crossing of a separatrix . The main result of Neishtadt's theory is that if one considers an orbit in the region spanned by the separatrix, after one period of the slow modulation, the value of the a.i. will be changed by a quantity8,9

I1

? I= f (I0

0

;;

0; log )

(2)

where and 0 are two variables 2 0; 1] related to the value of the phase variable at the crossing times (we have two crossings of the separatrix in one period); the leading term in eq. (2) is of order log . In order to understand a possible mechanism for the di usion in the phase space, we consider the following scenario: the unperturbed system H (q; p; ) has a single resonance in the phase space for each value of and the region spanned by the separatrix when= t varies, is free from other resonances (this condition will be not true for quasi integrable systems, but in this case we suppose that the amplitude of other resonances is of order ); the amplitude of the region spanned by the separatrix is much larger than . Let us consider a cluster of initial conditions in the region A spanned by the separatrix; the size of the cluster has to be larger than . The distribution function of the increments for the a.i. can be computed by using eq. (2) and considering; 0 as random variables uniformly distributed in 0; 1] . In spite of the complicated expression of the r.h.s. of eq. (2), we make the operative hypothesis that the increment I? I could be approximated by10 1 0

I1

? I= f^(I; log )0 0

(3)

where is an universal random variable whose distribution can be determined numerically, whereas the function f^ takes into account the dependence on the action of the di usion coe cient and it is zero at the boundary of the region A. The main idea of our approach is to consider successive crossings of the separatrix as a random walk for the a.i. with independent increments; the probability space is de ned by the set of the initial conditions. This approach is correct if we show that the values of phase variable after one period T of the modulation are uncorrelated. According to the results in, the relation between two successive phase values11,12