Aspects of diffusion in the stadium billiard

We perform a detailed numerical study of diffusion in the $\varepsilon$ stadium of Bunimovich, and propose an empirical model of the local and global diffusion for various values of $\varepsilon$ with the following conclusions: (i) the diffusion is normal for all values of $\varepsilon$ ($\le 0.3$) and all initial conditions, (ii) the diffusion constant is a parabolic function of the momentum (i.e., we have inhomogeneous diffusion), (iii) the model describes the diffusion very well including the boundary effects, (iv) the approach to the asymptotic equilibrium steady state is exponential, (v) the so-called random model (Robnik et al., 1997) is confirmed to apply very well, (vi) the diffusion constant extracted from the distribution function in momentum space and the one derived from the second moment agree very well. The classical transport time, an important parameter in quantum chaos, is thus determined.

Figure 1

The geometry and notation of the stadium billiard of Bunimovich.

Figure 2

The phase space of the stadium for $\varepsilon =1$, with ${10}^4$ bounces along an orbit emanating from $(s=\pi /4,p=0)$, showing the avoidance of the bouncing ball areas.

Figure 3

The distribution function $\rho (p,N)$ after $N=100$ collisions (a), $N=300$ collisions (b), and $N=700$ collisions (c). The ${10}^5$ initial conditions at $p_0=0$ are as described in the text. $\varepsilon =0.1$. The blue full line is the theoretical prediction for the inhomogeneous diffusion (9), the dashed red curve corresponds to the homogeneous diffusion (6).

Figure 4

The distribution function $\rho (p,N)$ after $N=100$ collisions (a), $N=300$ collisions (b), and $N=700$ collisions (c). The ${10}^5$ initial conditions at $p_0=0.25$ are uniformly distributed over $s\in [0,\mathcal{L})$, and are taken at $N=100$ collisions. $\varepsilon =0.1$. The blue full line is the theoretical prediction for the inhomogeneous diffusion (9), the long-dashed red curve corresponds to the homogeneous diffusion (6), while the short-dashed black line is the theoretical prediction of the inhomogeneous diffusion starting from the initial $\delta$ spike $\rho (p,t=0)=\delta (p-p_0)$ rather than from the delayed histogram of (a).

Figure 5

As in Fig. 4 but with $p_0=0.50$.

Figure 6

As in Fig. 4 but with $p_0=0.75$.

Figure 7

The variance of $p$ as a function of the number of collisions $N$ for four different values of $\varepsilon$, starting at the initial condition $p_0=0$. From top to bottom we show the numerical results for $\varepsilon =0.2$ (magenta), $\varepsilon =0.12$ (red), $\varepsilon =0.1$ (green), $\varepsilon =0.08$ (blue) calculated with ${10}^5$ initial conditions. The theoretical fitting curves according to Eq. (19) are shown with black dashed lines. The inset is a magnification of the first 100 bounces and shows the initial nondiffusive phase.

Figure 8

The variance as a function of the number of collisions $N$ for $\varepsilon =0.1$, starting at four initial conditions $p_0=0,0.25,0.50,0.75$. Each curve is shifted upwards by 0.1 in order to avoid the overlapping of curves. They all converge to $1/3$ when $N\to \infty$. The dotted black curve shows the theoretical prediction for the variance (18), while the dashed curve shows the empirical model (19).

Figure 9

$D_{\mathrm{dis}}=D_0\overline{l}=1/\left(6N_T\right)$ and $C$ as functions of $\varepsilon$, at $p_0=0$, using the Eq. (19). The $log$ is decadic. Ideally, $C$ should be unity according to Eqs. (13) and (14).

Figure 10

$D_{\mathrm{dis}}=D_0\overline{l}$ from (18), and $N_T$ with $C$ from (19) as functions of $p_0$ for $\varepsilon =0.1$.

Figure 11

The filling of the cells $\chi \left(N\right)$ for the stadium billiard $\varepsilon =0.1$ in the lin-lin plot (a), the decadic log-lin plot (b) and the distribution of cells with the occupancy number $M$ (c). The black dashed curve in (a) and (b) is the random model Eq. (21). There are $25\times {10}^6$ collisions and $N_c={10}^6$ cells, so that the mean occupancy number $\mu =\langle M\rangle =25$. The chaotic orbit has the initial conditions $(s=\pi /4,p=0)$. The black full curve is the best fitting Gaussian, while the red dashed curve is the best fitting Poissonian distribution. The theoretical values $\mu$ and ${\sigma }^2=\mu b=\mu (1-a)$ and their numerical values agree very well.

Figure 12

As in Fig. 11 but $\varepsilon =0.01$. Some deviations from the random model are seen, which are always negative due to the sticky objects in the phase space which delay the diffusion.

Figure 13

As in Fig. 11 but for the mixed type billiard [27] with $\lambda =0.15$, for an orbit with ${10}^8$ collisions. The asymptotic value of $\chi$ is ${\chi }_A=0.80$, as we see in (a). Large deviations from the random model are seen also in (b), which are always negative due to the sticky objects in the phase space which delay the diffusion. In (c) we see the distribution of cells at time $N=5\times {10}^7$ according to the occupancy $M$, which now consists of the the Gaussian bulge corresponding to the cells of the largest chaotic region, and the $\delta$ peak at $M=0$ corresponding to the regular and other smaller chaotic regions. The initial condition (using the standard Poincaré-Birkhoff coordinates) is $(s=4.0,p=0.75)$.

Figure 14

As in Fig. 13 but now in decadic log-log plot to show that the approach to the asymptotic value ${\chi }_A$ is neither an exponential function nor a power law. The curves correspond to three different initial conditions, $(s,p)=(4.0,0.75),(2.5,0.75),(2.5,0.15)$, in green, blue, and red correspondingly.