Randomness in a Galton board from the viewpoint of predictability: Sensitivity and statistical bias of output states

The Galton board is a classic example of the appearance of randomness and stochasticity. In the dynamical model of the Galton board, the macroscopic motion is governed by deterministic equations of motion, and predictability depends on uncertainty in the initial conditions and its evolution by the dynamics. In this sense the Galton board is similar to coin tossing. In this paper, we analyze a simple dynamical model which is inspired by the Galton board. Especially, we focus on the predictability, considering the relation between the uncertainty of initial states and the structure of basins of initial states that result in the same exit state. The model has basins with fractal basin structure, unlike the basins in coin tossing models which have only finite structure. Arbitrarily small uncertainty of initial conditions can cause unpredictability of final states if the initial conditions are chosen in fractal regions. In this sense, our model is in a different category from the coin tossing model. We examine the predictability of a small Galton board model from the viewpoint of the sensitivity and the statistical bias of final states. We show that it is possible to determine the radii of scatterers corresponding to a given predictability criterion, specified as a statistical bias, and a given uncertainty of initial conditions.

Figure 1

(a) Schematic of the Galton board apparatus. (b) Model of the Galton board in the orthogonal coordinate system. The radii of circular disks are $r=0.60$, and the distance between the centers of the nearest circular disks is 2. An example of a trajectory of a particle falling and bouncing off circular disks is shown.

Figure 2

Small Galton board model for $r=0.80$. Two particles are released from the entrance and bounce off the circular elastic surfaces many times. One passes out through the right exit, and the other gets out through the left exit. The bold lines show reflecting surfaces in the model.

Figure 3

Basin structures for radii $r=0.50$, $0.60$, $0.70$, $0.80$, $0.90$, $0.95$, and $0.99$ (from above). The horizontal and vertical axes indicate the initial position $x_0$ and the initial velocity angle ${\theta }_0$, respectively. The white and black regions show initial states of particles that pass out through the right ($S_f=1$) and left ($S_f=-1$) exits, respectively.

Figure 4

Semilogarithmic plot of stripe widths as a function of the radii of the circular disks.

Figure 5

(b), (d), (f), (h) Enlargements of the basin structures near the origin. (a), (c), (e), (f) Plot of the residence time corresponding to the enlarged basins.

Figure 6

Enlargements of the basin structures near $0.3$ for $r=0.50$. The ranges of $x_0$ are $[0.25,0.35]$, $[0.291,0.292]$, $[0.29196,0.29197]$, $[0.29196185,0.29196195]$, and $[0.291961935,0.291961936]$.

Figure 7

Enlargements of the basin structures near $0.2$ for $r=0.50$.

Figure 8

Enlargements of the basin structures near origin for $r=0.99$.

Figure 9

Probabilities for exiting through the right side ($S_f=1$) for various initial uncertainties $\epsilon$. Two cases are shown, one with $r=0.50$, $x_0=0$ and one with $r=0.99$, $x_0\sim 0.3$.

Figure 10

Plot of the number $N\left(l\right)$ of segments covering all basin boundaries as a function of segment lengths $l$. The lines are calculated with the least mean square error, and their slopes indicate the Hausdorff dimension of basin boundaries. The slopes are $0.507$, $0.567$, $0.742$, $0.853$, and $0.991$ for $r=0.50$, $0.60$, $0.80$, $0.90$, and $0.99$, respectively.

Figure 11

Logarithmic plot of $f\left(\epsilon \right)$. The lines are calculated with the least mean square error, and their slopes indicate the exponents of the final state sensitivity $\gamma$. The slopes are $0.438$, $0.387$, $0.288$, $0.158$, and $0.007$ for $r=0.50$, $0.60$, $0.80$, $0.90$, and $0.99$, respectively.

Figure 12

Histograms of the averaged final states $\langle S_f\rangle$ for various sizes of the uniform uncertainty in the case of $r=0.50$. The vertical axes indicate relative frequency.

Figure 13

Enlarged basin structures (top left) and histograms of the averaged final states $\langle S_f\rangle$ for various sizes of the uniform uncertainty when the initial conditions are restricted to the region around the origin $[-0.05,0.05]$ in case of $r=0.50$.

Figure 14

Enlarged basin structure (top left) and histograms of the averaged final states ${\langle S_f\rangle }_{\epsilon }$ for various sizes of the uniform uncertainty when the initial conditions are restricted to the region $[0.29,0.3]$ in the case of $r=0.50$.

Figure 15

Histograms of the averaged final states ${\langle S_f\rangle }_{\epsilon }$ for various sizes of uniform uncertainty in the case of $r=0.99$. The vertical axes indicate relative frequency.

Figure 16

Variance $v_{Sf}$ of the final states as a function of the initial uncertainty for $r=0.50$ (top left), $0.90$ (top right), $0.99$ (bottom left). Critical uncertainty length for radii where the variance $v_{Sf}$ of the final states is $0.01$ (bottom right).

Figure 17

Small Galton board model for $r=0.80$. The bold lines show reflecting surfaces in the model, and there is an additional straight reflecting surface at the entrance in the case of positive initial momentum.

Figure 18

Basin structures. The horizontal axis shows the initial particle position $x_0$ ($\in [r-1,1-r]$), and the vertical axis shows the angle ${\theta }_0$ ($\in [0,\pi ]$) of the initial velocity. White shows the basin of the coarse-grained exit state ($S_f=1$), and black shows the basin of the coarse-grained exit state ($S_f=-1$). Left: $v_0=0.25$, middle: $v_0=0.50$, and right: $v_0=1.0$. (a)–(c) $r=0.50$, (d)–(f) $r=0.90$, and (g)–(i) $r=0.99$.

Figure 19

Enlarged basin structures for $r=0.50$: (a) $0.1\le x\le 0.2$ and $0.5\le \theta \le 1.5$, (b) $0.1\le x\le 0.105$ and $1.1\le \theta \le 1.14$, (c) $0.1\le x\le 0.1001$ and $1.112\le \theta \le 1.1135$, and (d) $0.1\le x\le 0.100001$ and $1.11324\le \theta \le 1.11326$.

Figure 20

Enlarged basin structures for $r=0.99$: (a) $0.05\le x\le 0.006$ and $1\le \theta \le 1.1$, (b) $0.05\le x\le 0.0051$ and $1\le \theta \le 1.01$, (c) $0.05\le x\le 0.005001$ and $1\le \theta \le 1.0001$, and (d) $0.05\le x\le 0.00500001$ and $1\le \theta \le 1.000001$.

Figure 21

Distribution of the averaged final state ${\langle S_f\rangle }_{\epsilon }$ for $r=0.90$ and $v_0=1.0$. The top four plots show histograms for various uncertainties ${\epsilon }_{\theta }$ of the initial velocity angles when uncertainty ${\epsilon }_x$ of the initial positions is fixed, and the bottom four plots are for various ${\epsilon }_x$ when ${\epsilon }_{\theta }$ is fixed. The vertical axes indicate relative frequency.

Figure 22

Distribution of the averaged final states ${\langle S_f\rangle }_{\epsilon }$ for $r=0.50$ (top four plots) and $0.99$ (bottom four plots). The uncertainty ${\epsilon }_{\theta }$ is fixed at $0.154$. The vertical axes indicate relative frequency.

Figure 23

Variance of ${\langle S_f\rangle }_{\epsilon }$ as a function of the uncertainty for $r=0.50$, $0.60$, $0.70$, $0.80$, $0.90$, $0.95$, and $0.99$: (top) ${\epsilon }_{\theta }$ is fixed at $0.154$ and ${\epsilon }_x$ varies; (bottom) ${\epsilon }_x$ is fixed at $0.050$ and ${\epsilon }_{\theta }$ varies.