Robust Exponential Acceleration in Time-Dependent Billiards

A class of nonrelativistic particle accelerators in which the majority of particles gain energy at an exponential rate is constructed. The class includes ergodic billiards with a piston that moves adiabatically and is removed adiabatically in a periodic fashion. The phenomenon is robust: deformations that keep the chaotic character of the billiard retain the exponential energy growth. The growth rate is found analytically and is, thus, controllable. Numerical simulations corroborate the analytic predictions with good precision. The acceleration mechanism has a natural thermodynamical interpretation and is applied to a hot dilute gas of repelling particles.

Figure 1

General robust accelerator (a), divided stadium at the beginning of the first (b) and second (c) stages of the bar motion, (d) trapezium, (e) “double” Sinai billiard. Here $L=2h=2$, the radius of the disks in (e) is $\frac{1}{6}$, the distance between the centers is $\frac{4}{3}$, and the inclination angle of the trapezium is 10°. At the compression stage the bar displacement is 0.2 for trapezium, 0.28 for stadium, 0.18 for Sinai billiard, so the growth rate of Eq. (4) stays the same: $\mathcal{R}\approx 0.08$.

Figure 2

Ensemble energy growth (the energy is in the logarithmic scale). (a) Exponential growth for the three geometries of Figs. 1b, 1c, 1d, 1e, where the bar is introduced and removed abruptly. The energy is averaged over ensembles of 2000 initial conditions. (b) The same as (a) with slow and smooth bar motion (the bar velocity is continuous, piecewise linear in time). The graphs are smoother than in (a) since larger ensembles (${10}^4$ initial conditions) are taken. (c) When the bar is 90% of the billiard’s height and does not divide the billiard, the energy growth slows down significantly: $E\sim t^{1.55}$ for the stadium and Sinai billiards and $E\sim t^{1.95}$ for the trapezium (5000 initial conditions). The apparent crossover to faster growth rates at higher energies is in agreement with 11.