DIFFUSION RATES IN A 4-DIMENSIONAL MAPPING MODEL OF ACCELERATOR DYNAMICS

We study slow diffusion processes of orbits through thin chaotic layer s of 2m (m greater-than-or-equal-to 2)-dimensional symplectic maps (of ten referred to as Arnol'd diffusion) on a 4-dimensional mapping model of accelerator dynamics. Using a method proposed by Chirikov, we comp ute diffusion rates of vertical displacements, \y(n)\, near the flat b eam case, when the horizontal motion occurs within a thin chaotic laye r of the corresponding 2-dimensional x(n), x(n+1) map. Our computation distinguishes between regions of diffusive and quasiperiodic motion, in which the respective rates are found to differ by 10-12 orders of m agnitude for N = 10(8) iterations. For diffusion, \y(n)\ grows on the average with n (n less-than-or-equal-to 10(8)), with a rate D(R) is-pr oportional-to exp(aR) (a > 0, R2 = y0(2) + y1(2)), which gives accurat e estimates of escape times N(esc) is-proportional-to D-1. Modelling s ynchrotron oscillations by a periodic modulation on the ''betratron'' frequencies, we find that diffusion rates increase significantly and r apid escape occurs above certain thresholds in the epsilon, OMEGA para meter values, epsilon and OMEGA being the amplitude and frequency of t he modulation respectively.