Phase-space composition of driven elliptical billiards and its impact on Fermi acceleration

Abstract

We demonstrated very recently [Lenz, New J. Phys. 11, 083035 (2009)] that an ensemble of particles in the driven elliptical billiard shows a surprising crossover from subdiffusion to normal diffusion in momentum space. This crossover is not parameter induced, but rather occurs dynamically in the evolution of the ensemble. In this work, we consider three different driving modes of the elliptical billiard and perform a comprehensive analysis of the corresponding four-dimensional phase space. The composition of this phase space is different in the high-velocity regime compared to the low-velocity regime. We will show, among others, by investigating periodic orbits and probability distributions of laminar phases that the stickiness properties, which eventually determine the diffusion, are intimately connected with this change in the composition of the phase space with respect to velocity. In the course of the evolution, the accelerating ensemble thus explores regions of varying stickiness, leading to the mentioned crossover in the diffusion.

From billiards to thermodynamic laws: Stochastic energy exchange model

“…We remark that approximating a difficult chaotic billiard system by a more mathematically tractable stochastic process is a very generic strategy, which can be potentially applied to other highly chaotic billiard-like systems in physics. For example, it is known that Fermi acceleration can be found in many chaotic billiards [29,30,31,41]. And the rate of energy growth is found to be significantly larger in many chaotic billiards or stochastic acceleration models [25,26].…”

Exponential Fermi acceleration in general time-dependent billiards

“…From Eqs. (31) and (32) we find a normal component of a boundary velocity n · u = − 1 ∇h(r, t) ∂ h(r, t) ∂t .…”

“…A time-dependent billiard in which all corresponding static billiards are integrable [32]. In this case the adiabatic invariance of actions [33] ensures that F 1 → 0 for every ζ-trajectory.…”

Fermi acceleration in chaotic shape-preserving billiards

“…where I is the identity matrix and α is a constant. We can determine α by taking the determinant of both hand sides of (11). In 2D these gives α 2 = |J| 2 and the relation…”