Fermi acceleration and adiabatic invariants for non-autonomous billiards

Recent results concerned with the energy growth of particles inside a container with slowly moving walls are summarized, augmented, and discussed. For breathing bounded domains with smooth boundaries, it is proved that for all initial conditions the acceleration is at most exponential. Anosov-Kasuga averaging theory is reviewed in the application to the non-autonomous billiards, and the results are corroborated by numerical simulations. A stochastic description is proposed which implies that for periodically perturbed ergodic and mixing billiards averaged particle energy grows quadratically in time (e.g., exponential acceleration has zero probability). Then, a proof that in non-integrable breathing billiards some trajectories do accelerate exponentially is reviewed. Finally, a unified view on the recently constructed families of non-ergodic billiards that robustly admit a large set of exponentially accelerating particles is presented.