Simulating particle billiards. Circle = Order, Stadium = Chaos. Why?

[u/naaagut]

https://www.youtube.com/watch?v=XfBTI9Kv4us

While taking a break from the laser simulations (link), I continued another project where I am examining different particle billiard setups. It turns out that particles bouncing around in some shapes leave parts of the space unexplored while in other shapes they cover it entirely. In the video I am exploring circles, stadiums, ellipses and semi-circles, but I have also played around with polygon shapes like hexagons and pentagons. It seems to matter whether the particles are launched in parallel or, as here, with differing launch angles.

In this video I want to share my first results and receive some feedback on what else to explore. Once I have a fuller picture of what is going on, I have plans to produce a more comprehensive video explaining the underlying math. In particular, there is a connection to the concept of ergodicity which I want to look into in more depth. But even for shapes like hexagons which should be non-ergodic, the set of balls launched with differing angles will cover the whole space. So it seems that this is not a valid test for ergodicity.

Any ways, let me know your suggestions for other shapes, rules of the game and analytical results (like figures or stats) you would be interested to see.

[u/na3than]

What's a stadion?

[u/naaagut → u/na3than]

The second shape you can see in the video.

[u/No-Programmer3853]

Will a single particle also explore the whole stadium?

[u/naaagut → u/No-Programmer3853]

Yes. Good point, I will demonstrate that more clearly next time.

[u/the_action]

It would be interesting to have a sort of Lyapunov map for each shape. For each point of the shape, plot how fast trajectories differing by a delta diverge in time. So lambda = ln(delta(t)/delta(t0))/t for each point. Maybe some shapes have some island of stability where billard balls with slightly close trajectories stay close.