  Recently, the occurrence of exponential Fermi acceleration has been reported
in a rectangular billiard with an oscillating bar inside [K. Shah, D. Turaev,
and V. Rom-Kedar, Phys. Rev. E {\bf 81}, 056205 (2010)]. In the present work,
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 Fermi acceleration 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 Fermi
acceleration and provides a good ab initio prediction of the value of the
growth rate including its full parameter-dependency. Our analysis clearly
points out the requirements for exponential Fermi acceleration, thereby opening
the perspective of finding other systems exhibiting this unusual behaviour.
