Tunable Fermi Acceleration in the Driven Elliptical Billiard

We explore the dynamical evolution of an ensemble of noninteracting particles propagating freely in an elliptical billiard with harmonically driven boundaries. The existence of Fermi acceleration is shown thereby refuting the established assumption that smoothly driven billiards whose static counterparts are integrable do not exhibit acceleration dynamics. The underlying mechanism based on intermittent phases of laminar and stochastic behavior of the strongly correlated angular momentum and velocity motion is identified and studied with varying parameters. The diffusion process in velocity space is shown to be anomalous and we find that the corresponding characteristic exponent depends monotonically on the breathing amplitude of the billiard boundaries. Thus it is possible to tune the acceleration law in a straightforwardly controllable manner.

Figure 1

Schematic picture of the setup of the elliptical billiard.

Figure 2

The functions $F(n)$ (upper curve, light gray) and $|\overrightarrow{v}|(n)$ (lower curve, light gray) for a single trajectory for $C=0.2$ in arbitrary units. The solid black lines indicate the cumulative mean in each case. The two successive magnifications demonstrate that the corresponding fluctuations occur on many scales.

Figure 3

Power spectrum $S(k)$ of a typical laminar (a) and turbulent phase (b) in arbitrary units.

Figure 4

The function $⟨|\overrightarrow{v}|⟩(n)$ (arbitrary units) for an ensemble of ${10}^3$ trajectories for different values of the driving amplitude $C$, together with the best linear fit (between the two crosses).