Thin structure of the transit time distributions of open billiards

It is known that typical open billiards distribution of transit times is an exponentially decaying function, possibly with a power-law tail. In the paper we show that on small scales some of such distributions change their appearance. These distributions contain a quasiperiodic thin structure, which carries a significant amount of information about the system. Origin and properties of this structure are discussed.

Figure 1

General view of (a) the Bunimovich stadium with the parameters $l=20, h=10$ and (b) the parabolic billiard with the parameters $a=40, a_x=20, a_y=10$. Also shown is (c) the phase portrait of the parabolic billiard with the same parameters.

Figure 2

Distribution of transit times for the Bunimovich stadium with the parameters $l=20, h=10, d=0.5$ (the mean free path $\overline{l}\approx 24.5$) on different scales. Positions of the hole are shown by the arrows in the insets. The initial data of $N_r={10}^9$ trajectories were distributed evenly along the hole and had the Lambert distribution of the initial directions of motion. Each point presents the sum of all escape times within the same time interval ${\mathrm{\Delta }}_t$. It has the following duration values: (a) ${\mathrm{\Delta }}_t=40$, (b) ${\mathrm{\Delta }}_t=1$, (c) ${\mathrm{\Delta }}_t=0.25$, (d) ${\mathrm{\Delta }}_t=1$, (e) ${\mathrm{\Delta }}_t=2$, (f) ${\mathrm{\Delta }}_t=0.05$, (g) ${\mathrm{\Delta }}_t=1$, and (h) ${\mathrm{\Delta }}_t=0.05$. The insets on figures (d) and (e) show the discrete Fourier transform spectrum of the presented distribution part.

Figure 3

Distributions of transit times for the parabolic billiard with parameters $a=40, a_x=20, a_y=10, d=0.5, \overline{t}\approx 29.4$ on different scales. The initial data of $N_r={10}^9$ trajectories were distributed evenly along the length of the hole and had Lambert distribution of the initial directions of motion. Each point on the graphs shows the sum of all escape times within the same time interval ${\mathrm{\Delta }}_t$. It has the following duration values: (a) ${\mathrm{\Delta }}_t=128$, (b) ${\mathrm{\Delta }}_t=1$, (c) ${\mathrm{\Delta }}_t=1$, (d) ${\mathrm{\Delta }}_t=1$, (e) ${\mathrm{\Delta }}_t=1$, (f) ${\mathrm{\Delta }}_t=0.05$, (g) ${\mathrm{\Delta }}_t=1$, and (h) ${\mathrm{\Delta }}_t=0.05$. The insets on figures (e) and (g) show the discrete Fourier transform spectrum of the presented distribution part.

Figure 4

Phase portraits and escape time distributions for the parabolic billiard (a),(c) with the parameters $a=180, a_x=50, a_y=30, d=0.5, {\mathrm{\Delta }}_t=10$ and (b),(d),(e) with the parameters $a=100, a_x=80, a_y=12, d=0.5, {\mathrm{\Delta }}_t=10$. (f) Initial part of the distribution of transit times for the Bunimovich stadium with the parameters $l=0.01, h=10, d=0.5, {\mathrm{\Delta }}_t=0.05$. The marks below show the multiples of the billiards perimeter. The inset on figure (e) shows the discrete Fourier transform spectrum of this part of distribution.

Figure 5

Two parts of the same distribution of escape times for the Bunimovich billiard with the parameters $l=20, h=10, d=0.5$. Each of these two parts of the distribution contains two peaks. The marks at the top show the peaks positions calculated according to the formula (1). Short mark—without the second term; long mark—according to the complete formula. On the right is shown the typical flattening of the trajectory belonging to the distribution peak.

Figure 6

Ratio of the average number of returns, contained in the distribution sites with a peak, to such for the sites without peak. The distribution, for which this value was calculated, is shown in Fig. 2. It is visible that the contribution of peaks in the total number of returns tends to a constant nonzero value.

Figure 7

Modification of the thin structure of distribution for Bunimovich stadium with the decrease of parameter $l$ to zero value, which corresponds to the integrable circle billiard. The distributions were built for the parameters $h=10, d=0.5$ and (a) $l=4, {\mathrm{\Delta }}_t=0.25$, (b) $l=1.2, {\mathrm{\Delta }}_t=0.25$, (c) $l=1.2, {\mathrm{\Delta }}_t=0.75$, (d) $l=0.01, {\mathrm{\Delta }}_t=1$, (e) $l=0.01, {\mathrm{\Delta }}_t=0.25$, and (f) $l=0.01, {\mathrm{\Delta }}_t=0.05$. The insets on figures (d) and (e) show the discrete Fourier transform spectrum of the presented distribution part.