Fermi acceleration revisited

Mappings that have been used to describe the Fermi acceleration mechanism are examined. It is shown that results which appear to be contradictory are due to differences in the mapping equations. For those mappings that can be locally approximated by the standard mapping, the value of the nonlinear parameter of the standard mapping, for which the last isolating KAM surface exists, can be used to predict the loss of KAM stability with action for the more general mappings. Previous results of the density distribution in the stochastic region of the phase space, averaged over phases, is shown to be consistent with the ergodic hypothesis. Fine scale structure of the mappings is found to be model dependent. The standerd mapping is a member of a class of mappings which retains some KAM trajectories at arbitrarily large nonlinearity.This feature is not generic to a wider class of mappings discussed in this paper. The stability of two-iteration fixed points are discussed in detail, including the bifurcation sequence for one type of mapping. (orig.)