Sinai Billiard (Chaotic)

A billiard particle moves in a straight line at constant speed inside a closed boundary and undergoes specular reflection, \(\mathbf{v}' = \mathbf{v} - 2(\mathbf{v}\cdot\mathbf{n})\,\mathbf{n}\), whenever it strikes the boundary. The Sinai billiard’s boundary is a square cell, \(|x| \le L/2,\ |y| \le L/2\), with a circular scatterer of radius \(r_s < L/2\) removed from its center, \(x^2 + y^2 = r_s^2\). It is a classic example of a defocusing chaotic billiard: because the scatterer is convex as seen from inside the cell, it disperses nearby trajectories exponentially fast (unlike the flat walls of a Rectangle billiard, which neither focus nor defocus). This example plots a single trajectory and the resulting Poincare section, which fills densely with points rather than tracing out smooth curves.

import matplotlib.pyplot as plt
import numpy as np

from physicskit.chaos.systems.billiards import SinaiBilliard
from physicskit.chaos.visualizers.dynamic_plots import animate_billiard_trajectory
from physicskit.chaos.visualizers.phase_space import plot_billiard_trajectory, plot_poincare_section

billiard = SinaiBilliard(cell_size=2.0, scatterer_radius=0.5)

Animation

Watch the scatterer disperse the ray: notice how quickly it starts exploring the whole cell, and how the Poincare section fills in densely rather than tracing a smooth curve.

anim = animate_billiard_trajectory(billiard, pos=billiard.sample_interior_point(), vel=(0.4, 0.9), n_bounces=100, interval=50)

plt.show()

To save the animation to a file instead of (or in addition to) displaying it interactively, use e.g.:

anim.save("sinai_billiard_animation.gif", writer="pillow", fps=25)

Trajectory

Notice how quickly the ray explores the whole cell after just a few bounces off the central scatterer.

fig, ax = plot_billiard_trajectory(billiard, pos=billiard.sample_interior_point(), vel=np.array([0.4, 0.9]), n_bounces=100)

Poincare section

A single ray, given enough bounces, would already fill the section densely (this is the hallmark of ergodicity); here we pool several shorter rays for a quick illustration.

fig, ax = plot_poincare_section(billiard, n_rays=40, n_bounces=200)

plt.show()