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

4Z T0

A. BAZZANI, F. BRINI

after a period T, can be expressed in term of the integral@@I

(In?

1

; t In

)(? In? )dt1 1

(4)

where (I; t) is the amplitude dependent rotation number (tune) and In? and In the initial and nal values of the a.i.. As a consequence if the integral (4) is 1, then the increments will be independent. Taking into account that In? In?/ log and that T/ 1=, one can estimate that the integral (4) is 1 if is su ciently small and the derivative@=@I is not too small, which is in general true near a separatrix. We have numerically checked the independence of the increments for the a.i. and our result seems to be generic for symplectic maps; but for 2D symmetric Hamiltonian systems there are examples for which the condition (4) 1 fails and the phases are correlated for

more than one period T . Therefore we can describe the di usion of the a.i. due to the presence of an isolated single resonance by using the random walk^ In? In?= f (In?; log ) (5)1 13 14 1 1

where In is the value of the a.i. after n periods. The equation (5) will be a crude approximation of the di usion in the case that is not small or that the orbits are very close to boundary of the region A. Moreover the di usion described by the random walk (5) is a bounded di usion, since the orbits cannot overcome the region A spanned by the separatrix, so that the nal distribution function for the a.i. will be an uniform distribution in the region A. In order to describe the escape to in nity of an orbit in the phase space we have to consider the overlapping of regions spanned by the separatrices of di erent resonances and to introduce the transition probabilities for the a.i. to go from one region to another. This mechanism is indeed observed in the numerical simulations but up to now there is no analytical description.15

3 DESCRIPTION OF THE MODEL AND NUMERICAL RESULTS In the 1991 experiments at CERN to study the di usion of the beam due to the ripples in the quadrupole magnets, the SPS had a special con guration with eight sextupoles strongly exited to produce intense nonlinear elds . The betatronic motion for a at beam is well described by the composition of eight quadratic area-preserving maps, which take into account the e ect of the strong sextupoles. This is the model we have used for the numerical simulations; the same model was considered to study the di usion of the nonlinear invariant, when a stochastic noise is introduced in the linear tune . We cannot claim that our model is realistic, but we expect that it is su ciently generic to perform all the features of a realistic model in the at beam approximation.5 16