Particle focusing in oscillating dissipative billiards

We develop and analyze a scheme to achieve both spatial and energetic focusing of an ensemble of neutral particles which is based on an oscillating billiard with frictional forces. The interplay of two competing mechanisms, acceleration due to collisions with the oscillating billiard walls and deceleration caused by friction, leads to the emergence of attractors in phase space. Their specific properties, i.e., spatial localization and energy spread, can be controlled and tuned by varying, e.g., the frequency of the time-dependent billiard.

Figure 1

(Color online) Boundary of the elliptical billiard in coordinate space at two different phases $\xi$ together with a typical period four attractor $\left(a_0=1.5,b_0=1,C=0.25,\omega =1\right)$. At the single crossing point, the two fluxes ${\Phi }_1$ and ${\Phi }_3$ intersect.

Figure 2

(Color online) $⟨v⟩\left(n\right)$ (dashed) and its standard deviation ${\sigma }_v\left(n\right)$ (solid) (1000 particles with $v_0=0.1$, and ${\xi }_0,{\varphi }_0,{\alpha }_0$ distributed uniformly; $C=0.25$, $a_0=1.5$, $b_0=1$, $k={10}^{-3}$). The inset shows the distribution ${\rho }_n\left(v\right)$ at $n={10}^3$ (dashed) and at $n={10}^7$ (solid).

Figure 3

(Color online) Collision resolved evolution of the phase space density (logarithmic color map) ${\rho }_{n_1,n_2}\left(\varphi ,\alpha \right)$: (a) ${10}^4<n<2\times {10}^4$, (b) $4\times {10}^5<n<{10}^6$, and (c) $2\times {10}^7<n<{10}^8$. The ensemble gets focused with increasing number of collisions more and more onto attractors [bright dots and wavy lines in panel (c)]. (For better visibility, the extension of these bright dots has been magnified artificially.)