Scattering is a fundamental tool for studying many physical systems. In a scattering experiment, particles with different initial states interact with the system of interest and their characteristics are modified between before and after the scattering, i.e between input and output variables. It may happen that arbitrarily small changes in the input variable of scattering system can cause large changes in the output variable. This sensitive dependence on initial conditions signifies the appearance of chaos. We study here chaotic phenomena related to the Gaspard-Rice's scattering system. We will highlight the chaotic nature of this system and compute its fractal dimension.
Gaspard-Rice's system consists of points particles incident on three circular hard disks in a two-dimensional plane. We choose the radii of the disks to be $R_{1}=R_{2}=R_{3}=1$. The distances between the individual disks are $d=2.5$. The disks are located at $x=-d\sqrt{3}/6$, $y=d/2$ (disk1=up disk), $x=d\sqrt{3}/3$, $y=0$ (disk2= right disk) and $x=-d\sqrt{3}/6$, $y=-d/2$ (disk3=down disk). Initially, a particle coming from $x=-\infty$ bounces off the three disks when it collides with them and spends a finite amount of time in the region between the disks (the scattering region). Then, particles exits to infinity with an angle $\theta$. For convenience, we assume that the incident particle trajectory is parallel to the $Ox$ axis with an impact parameter $b$, i.e the initial ordinate of the particle. The output angle of exit depends of the impact parameter, so one can write $\theta=\theta(b)$. The time $T(b)$ (the delay time) that the particle spends in the scattering region before exiting also depends on $b$.
Firstly, We begin by bring out the chaotic nature of this system. Our Matlab graphical interface allows to compute the output angle as a function of $b$ input parameter. Below examples of trajectory computed for $b=0.21005$ et $b=0.21010$ :
Figure 1 : Matlab Interface - Trajectory for $b=0.21005$
Figure 2 : Matlab Interface - Trajectory for $b=0.21010$
We can see that a small difference in the impact parameter $b$ ($b_{1}=0.21005$ and $b_{2}=0.2101$) leads to small differences in the scattering angles (respectively $\theta_{1}=5.6351$ and $\theta_{2}=5.6317$). Intial conditions are then called "certain" against small perturbations. This behavior implies that the scattering function $\theta(b)$ and the delay time function $T(b)$ near $b_{1}$ and $b_{2}$ are smooth function of $b$.
More complicated dynamics can occur as figures below where two particle trajectories with impact parameters $b_{1}=0.33005$ and $b_{2}=0.33010$ are shown. The scattering angles for these two trajectoriess are $\theta(b_{1})=3.1246$ and $\theta(b_{2})=1.839$. Such a large difference in the scattering angles is also reflected in the large difference in the delay times. We call such initial conditions "uncertain" against small perturbations.
Figure 3 : Matlab Interface - Trajectory for $b=0.33005$
Figure 4 : Matlab Interface - Trajectory for $b=0.33010$
Hence, around the impact parameters $b\approx 0.33$, a small difference in the initial conditions leads to a big difference in the outcome of the trajectories. This sensitive dependance on initial condition is a clear signature of chaos. This dependance also occurs for a large number of impact parameters (actually an infinity). Chaotic behavior is illustrated on the following figure where $\theta(b)$ is computed with $100000$ points in $0 < b <0.5$ interval.
Figure 5 : Representation of output angle for $0 < b <0.5$
One can distinguish in the same time continuous and discontinuous regions. This double characteristic is repeated on smaller scales. The following figure shows this result with $\theta(b)$ computed on the interval $0.383 < b < 0.391$ : there is a similar aspect with the above figure.
Figure 6 : Representation of output angle for $0.383 < b <0.391$
To further quantify the nature of chaotic scattering, a parameter is often used : the uncertainty dimension. This parameter was introduced by Celso Grebogi to characterize fractal basin boundaries, which arise commonly in dissipative chaotic systems with multiple attractors. It has been conjectured that the uncertainty dimension is equal to the fractal dimension of typical chaotic sets. The procedure for computing the uncertainty dimension is as follows : for a given perturbation $\epsilon$, we can compute, by comparing the number of bounces in between the hard disks, the fraction of uncertain initial conditions $f(\epsilon)$ for many randomly chosen initial conditions.
In our numerical experiments, $f(\epsilon)$ is obtained by accumulating $10000$ initial conditions for each of the $200$ values for $\epsilon$ taken in the interval $[10^{-8},10^{−2}]$. As $\epsilon$ decreases, we expect $f(\epsilon)$ to decrease as well. The following figure shows our results :
In most physical situations, $f(\epsilon)$ scales with $\epsilon$ as : $f(\epsilon)\propto \epsilon^{\alpha}$ where $\alpha$ is the uncertainty exponent. The dimension of the fractal set is given by $D=1-\alpha$. Intuitively, obtaining the number of uncertain initial conditions for perturbations of different orders of magnitude is equivalent to counting the number of singularities in the scattering and delay time functions for different scales. Hence, we expect that the uncertainty dimension is a good approximation to the box-counting dimension. The uncertainty algorithm offers a great computational advantage because it requires relatively little memory in comparison to the box-counting procedure. Figure below represents the uncertainty fraction $f(\epsilon)$ versus the perturbation $\epsilon$ on a logarithmic scale, where $\epsilon$ varies by seven orders of magnitude. The plot can be robustly fitted by a straight line : the slope of which is $\alpha=0.412044\pm0.00401$. We conclude that the fractal dimension of the set of singularities in the scattering and delay-time functions is $D=1-\alpha=0.587956\pm0.00401$.
