  It is shown, that under very general conditions, a generic time-dependent
billiard, for which a phase-space of corresponding static (frozen) billiards is
of the mixed type, exhibits the exponential Fermi acceleration in the adiabatic
limit. The velocity dynamics in the adiabatic regime is represented as an
integral over a path through the abstract space of invariant components of
corresponding static billiards, where the paths are generated probabilistically
in terms of transition-probability matrices. We study the statistical
properties of possible paths and deduce the conditions for the exponential
Fermi acceleration. The exponential Fermi acceleration and theoretical concepts
presented in the paper are demonstrated numerically in four different
time-dependent billiards.
