Long Time Algebraic Relaxation in Chaotic Billiards

The long-time algebraic relaxation process in spatially periodic billiards with infinite horizon is shown to display a self-similar time-asymptotic form. This form is identical for a class of such billiards, but can be different in an important special case.

Figure 1

Infinite horizon billiards. (a) Circular scatterers of radius $R$ whose centers are located on a square grid of periodicity length $L>2R$. This billiard is also referred to as the Sinai billiard [2]. (b) The circular scatterers of (a) are each given a random displacement $\overrightarrow{\Delta }$ that is uniform in the disc $\left|\overrightarrow{\Delta }\right|\le {\Delta }_0<(L-2R)/2$. (c) Randomly oriented squares of edge length $d$ where the centers of the squares are located on a square grid of periodicity length $L>\sqrt{2}d$. (d) A channel whose walls are circular arcs each subtending the same angle which is less than $\pi$ radians. (e) Similar to (d) but the circular arcs subtend $\pi$ radians (i.e., they are semicircles); orbits in this billiard can be mapped to orbits in the stadium billiard of Bunimovich [3] by reflection at the horizontal dashed lines in the figure.

Figure 2

(a) The time evolution of an initially uniform angular distribution of particles in the $\theta$ interval $[0.3,\pi /2]$ for the billiard of Fig. 1a with $L/R=3$. The distribution is shown at four times; + at $t=9W/v$, $\times$ at $27W/v$, $*$ at $81W/v$, and $\square$ at $243W/v$.(b) Distribution function for the billiard of Fig. 1e with $W/R=1$ at the same four times as in (a) and with the same initial $\theta$ distribution as in (a). The solid line in (b) is obtained from the theory, Eqs. (5, 6).

Figure 3

The evolution of ${\widehat{P}}_c(\theta ,t)$ and $\widehat{P}(\theta ,t)$.

Figure 4

(a) A one-bounce encounter with the wall resulting in the reversal of the particle’s vertical velocity component. (b) A two-bounce encounter with the wall resulting in an unchanged direction of the particle’s vertical velocity component. (c) $M\left(\epsilon \right)$ verses $\epsilon$ for ${\epsilon }_{\mathrm{m}\mathrm{a}\mathrm{x}}=2R\tan {\theta }^{\prime }\ll 1$.