Occurrence of normal and anomalous diffusion in polygonal billiard channels

From extensive numerical simulations, we find that periodic polygonal billiard channels with angles which are irrational multiples of $\pi$ generically exhibit normal diffusion (linear growth of the mean squared displacement) when they have a finite horizon, i.e., when no particle can travel arbitrarily far without colliding. For the infinite horizon case we present numerical tests showing that the mean squared displacement instead grows asymptotically as $t\ln t$. When the unit cell contains accessible parallel scatterers, however, we always find anomalous super-diffusion, i.e., power-law growth with an exponent larger than $1$. This behavior cannot be accounted for quantitatively by a simple continuous-time random walk model. Instead, we argue that anomalous diffusion correlates with the existence of families of propagating periodic orbits. Finally we show that when a configuration with parallel scatterers is approached there is a crossover from normal to anomalous diffusion, with the diffusion coefficient exhibiting a power-law divergence.

Figure 1

Unit cells of (a) the polygonal Lorentz model and (b) the zigzag model.

Figure 2

(Color online) $⟨\mathrm{\Delta }{x}^2{⟩}_t$ as a function of $t$ in the zigzag model with $h_2=0.77$ and $h_3=0.45$. There is a finite horizon only when $h_1⩽h_3=0.45$; curvature is barely visible in the infinite horizon cases. Statistical errors are smaller than the linewidth.

Figure 3

(Color online) $⟨\mathrm{\Delta }{x}^2{⟩}_t$ as a function of $t$ on a double logarithmic plot in the zigzag model with the same parameters as in Fig. 2.

Figure 4

(Color online) $⟨\mathrm{\Delta }{x}^2{⟩}_t∕t$ as a function of $\log t$ in the zigzag model with the same parameters as in Fig. 2.

Figure 5

(Color online) Integrated velocity autocorrelation function $R\left(t\right)$ as a function of $\ln t$ in the (non-parallel) zigzag model with the same parameters as in Fig. 2 ($h_2=0.77$ and $h_3=0.45$).

Figure 6

(Color online) Growth rate ${\gamma }_q$ of $q$th moment $⟨\mid x{\mid }^q{⟩}_t$ for finite horizon $(h_1=0.4)$ and infinite horizon $(h_1=0.5)$ in the zigzag model with $h_2=0.77$ and $h_3=0.45$ for ${10}^6$ initial conditions.

Figure 7

(Color online) Mean squared displacement $⟨\mathrm{\Delta }{x}^2{⟩}_t$ for parallel models versus $t$ on a double logarithmic plot. The upper and lower lines are proportional to $t$ and $t^2$ for comparison. The remaining curves show, from top to bottom, the parallel zigzag model with $h_1=0.1$ and $h_3=5.2,10.2$ (both shifted vertically for clarity), and the parallel polygonal Lorentz model with $h_1=0.1$, $w=0.15$, and $h_3=5.2,10.2$. Statistical errors are slightly greater than the width of the lines.

Figure 8

(Color online) Diffusion exponent $\alpha$ as a function of $h_3$ in the parallel zigzag model with $h_1=0.1$ and the parallel polygonal Lorentz model with $h_1=0.1$ and $w=0.15$, calculated using ${10}^5$ initial conditions evolved until the presumed asymptotic regime. (Note that at $h_3=1=d$ there is a rational angle in both models.) Representative error bars show the standard deviation of the exponent over $100$ independent simulations.

Figure 9

(Color online) Growth exponent ${\gamma }_q$ of moments for the parallel polygonal Lorentz and zigzag models with $h_1=0.51$, $h_2=h_3=10.0$, and $w=0.21$.

Figure 10

(Color online) Rescaled demodulated densities in the polygonal Lorentz model, compared to a Gaussian with the corresponding variance. The inset shows the original and demodulated (heavy line) densities for $t=100$. The model parameters are $h_1=0.1$, $h_2=h_3=0.45$, $w=0.2$, $d=1$.

Figure 11

(Color online) Scatter plot representing the joint density $\psi (x,t)$ of laminar stretches in the zigzag model with $h_1=0.1$, $h_3=0.3$ and $h_2=h_3+\mathrm{\Delta }h$. The straight lines with slope $1$ correspond to the maximum possible speed.

Figure 12

(Color online) Tail of $\Psi \left(t\right)$ for the parallel zigzag model. The top curve is for $h_1=0.1$ and $h_3=0.3$. The lower three curves are for $h_1=0.49$, $h_3=5.1$, and $\mathrm{\Delta }h=0$, $0.0001$, and $0.001$ (top to bottom). The inset is a semilogarithmic plot of the $\mathrm{\Delta }h=0.001$ case, showing exponential decay.

Figure 13

(Color online) Families of trapped and propagating periodic orbits in parallel systems.

Figure 14

(Color online) Double logarithmic plot of $⟨\mathrm{\Delta }{x}^2{⟩}_t$ as a function of time for the polygonal Lorentz model with $h_1=0.1$, $h_3=0.45$, $w=0.2$, and $h_2=h_3+\mathrm{\Delta }h$. Deviations from an asymptotic linear fit to the lowest curve are shown for $\mathrm{\Delta }h=0$ (uppermost) and $\mathrm{\Delta }h={10}^{-m}$, $m=2,\dots ,11$ (bottom to top). The inset shows a double logarithmic plot of the diffusion coefficient $D$ as a function of $\mathrm{\Delta }h$ for the polygonal Lorentz model with $h_1=0.1$ for $h_3=0.3,0.45$, and for $h_3=0.45$ with $\mathrm{\Delta }h<0$.

Figure 15

(Color online) Data collapse of $\mid \mathrm{\Delta }h{\mid }^{\nu \alpha }⟨x^2\left(\mathrm{\Delta }h\right){⟩}_t$ as a function of $u=\mid \mathrm{\Delta }h{\mid }^{\nu }t$ for the zigzag model with $h_1=0.1$, $h_3=0.3$, which has $\alpha \simeq 1.81$ (upper curve) and the polygonal Lorentz model with $h_1=0.1$, $h_3=0.45$, with $\alpha =1.40$. In both models the value $\nu =0.37$ was used, and curves are shown for $\mathrm{\Delta }h={10}^{-m}$, $m=4,\dots ,11$. The solid lines show the extreme behaviors of the scaling function (4.3).