Phase space interpretation of exponential Fermi acceleration

Recently, the occurrence of exponential Fermi acceleration (FA) has been reported in a rectangular billiard with an oscillating bar inside (Shah et al 2010 Phys. Rev. E 81 056205). In this paper, we analyze the underlying physical mechanism and show that the phenomenon can be understood as a sequence of highly correlated motions, consisting of alternating phases of free propagation and motion along the invariant spanning curves of the well-known one-dimensional Fermi-Ulam model. The key mechanism for the occurrence of exponential FA can be captured in a random walk model in velocity space with step width proportional to the velocity itself. The model reproduces the occurrence of exponential FA and provides a good ab initio prediction of the value of the growth rate, including its full parameter dependence. Our analysis clearly points out the requirements for exponential FA, thereby opening the prospect of finding other systems exhibiting this unusual behavior.