ICBM, Carl von Ossietzky Universität, Carl-von-Ossietzky Str. 9-11, 26111 Oldenburg, Germany.
Chaos. 2012 Jun;22(2):026121. doi: 10.1063/1.4722744.
In this study, we compare the dynamical properties of chaotic and nearly integrable time-dependent focusing billiards with elastic and dissipative boundaries. We show that in the system without dissipation the average velocity of particles scales with the number of collisions as ̅V∝n(α). In the fully chaotic case, this scaling corresponds to a diffusion process with α≈1/2, whereas in the nearly integrable case, this dependence has a crossover; slow particles accelerate in a slow subdiffusive manner with α<1/2, while acceleration of fast particles is much stronger and their average velocity grows super-diffusively, i.e., α>1/2. Assuming ̅V∝n(α) for a non-dissipative system, we obtain that in its dissipative counterpart the average velocity approaches to ̅V(fin)∝1/δ(α), where δ is the damping coefficient. So that ̅V(fin)∝√1/δ in the fully chaotic billiards, and the characteristics exponents α changes with δ from α(1)>1/2 to α(2)<1/2 in the nearly integrable systems. We conjecture that in the limit of moderate dissipation the chaotic time-depended billiards can accelerate the particles more efficiently. By contrast, in the limit of small dissipations, the nearly integrable billiards can become the most efficient accelerator. Furthermore, due to the presence of attractors in this system, the particles trajectories will be focused in narrow beams with a discrete velocity spectrum.
在这项研究中，我们比较了具有弹性和耗散边界的混沌和近可积时变聚焦台球的动力学特性。我们表明，在无耗散的系统中，粒子的平均速度与碰撞次数的标度关系为 ̅V∝n(α)。在完全混沌的情况下，这种标度对应于扩散过程，α≈1/2；而在近可积的情况下，这种依赖关系存在交叉；慢粒子以亚扩散的方式缓慢加速，α<1/2，而快粒子的加速则更强，它们的平均速度呈超扩散增长，即 α>1/2。假设非耗散系统中 ̅V∝n(α)，我们得到在其耗散对应系统中，平均速度趋近于 ̅V(fin)∝1/δ(α)，其中 δ 是阻尼系数。因此，在完全混沌的台球中， ̅V(fin)∝√1/δ，在近可积系统中，特征指数 α 随 δ 从 α(1)>1/2 变化到 α(2)<1/2。我们推测，在中等耗散的极限下，混沌时变台球可以更有效地加速粒子。相比之下，在小耗散的极限下，近可积台球可以成为最有效的加速器。此外，由于该系统中存在吸引子，粒子轨迹将在离散速度谱中聚焦在狭窄的射束中。
