Diffusion rates in a 4-dimensional mapping model of accelerator dynamics

We study slow diffusion processes of orbits through thin chaotic layers of 2m (m ≥ 2)-dimensional symplectic maps (often referred to as Arnol'd diffusion) on a 4-dimensional mapping model of accelerator dynamics. Using a method proposed by Chirikov, we compute diffusion rates of vertical displacements, |y_n|, near the flat beam case, when the horizontal motion occurs within a thin chaotic layer 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 magnitude for N = 10⁸ iterations. For diffusion, |y_n| grows on the average with n (n ≤ 10⁸, with a rate D(R) ∝ exp(aR) (a > 0, R² = y²₀ + y²₁), which gives accurate estimates of escape times N_esc ∝ D^-1. Modelling synchrotron oscillations by a periodic modulation on the "betratron" frequencies, we find that diffusion rates increase significantly and rapid escape occurs above certain thresholds in the ε, Ω parameter values, ε and Ω being the amplitude and frequency of the modulation respectively.