Ultraslow diffusion and weak ergodicity breaking in right triangular billiards

We investigate the diffusion behavior of a right triangular billiard system by transforming its dynamics to a two-dimensional piecewise map. We find that the diffusion in the momentum space is ultraslow, i.e., the mean squared displacement grows asymptotically as the square of the logarithm of time. The mechanism of the ultraslow diffusion behavior is explained and numerical evidence corroborating our conclusion is provided. The weak ergodicity breaking of the system is also discussed.

Figure 1

The forward regions of the forward iteration Eq. (3). The three regions, $L, M$, and $R$, are partitioned by the two solid lines the slopes of which equal to 1.

Figure 2

The evolution diagram of the variable $\vartheta$ follows Eqs. (4) and (5a).

Figure 3

The dependence of ${\sigma }^2\left(\tilde{n}\right)$ on ${(lnt)}^2$. The average ensemble consists of ${10}^6$ initial conditions with identical ${\vartheta }_0$ but uniformly distributed random $x_0$. To explore the role of waiting time, another time scale, $t_S$, by omitting the time of all $\mathbf{M}$ operations, is also considered.

Figure 4

The first ${10}^{10}$ images of the initial point $(\sqrt{3},0.5)$ on ${\mathrm{\Theta }}^{+}({\vartheta }_0,\tilde{n}),0\le \tilde{n}\le 20$. The temporarily forbidden zones can be seen clearly even for $\tilde{n}=3$. The pink bar below is the corresponding ${\mathrm{\Gamma }}^{+}({\vartheta }_0,\tilde{n})$ plotted for reference.

Figure 5

The time dependence of $K$, the number of different $\tilde{n}$ values visited by a trajectory. The squares are for the averaged results over an ensemble of ${10}^6$ initial conditions uniformly distributed along the line ${\vartheta }_0=0.58$. The solid curve is for the result of a single trajectory with initial point $({\vartheta }_0,x_0)=(0.58,0.2)$, as an example. The dashed line is plotted for reference.

Figure 6

The dependence of $\eta$ on $\varepsilon$. The asterisks are for the averaged results over an ensemble of ${10}^6$ initial conditions uniformly distributed along the line ${\vartheta }_0=\sqrt{3}$. The dashed line is plotted for reference. In the inset we show the time dependence of ${\sigma }^2\left(\tilde{n}\right)$ for $\varepsilon =0.01$ as an example. The solid line gives the slope $\eta =0.6$.

Figure 7

A particle moves in a right triangular billiard the hypotenuse of which has unit length. The collisions between the particle and the sides are all elastic. The motion in the figure from leaving $x_i$ to colliding with the hypotenuse at $x_{i+1}$ is denoted by a letter $R$, the mapping formula of which for $\theta$ and $x$ can be derived using the law of sines.