I have previously done a post on chaos theory, and today I wanted to further explore one particular example in which chaotic behaviour may arise: a game of billiards.

In mathematics, we study the ‘ideal’ game of billiards: you have a table and ball, where the ball is massless and there is no friction. However, unlike a real game of billiards, there are no holes around the edge of the table, and since it doesn’t experience any friction, the ball will continue to move forever.

What will the trajectory of the ball look like?

Regular behaviour would occur if you adjust the initial direction so that the ball bounces off the midpoint of all four edges in turn and then returns to where it started, hence travelling along the same four line segments forever. However, this regular behaviour is very rare.

Periodic trajectories
Regular Behaviour | Source:

In fact, in the 1980s, mathematicians proved that for the vast majority of initial directions, not only will the ball retrace its steps but it will eventually explore the whole of the table. Additionally, it will visit each part of the table in equal measure. This is due to the fact that billiards behaves ergodically.

The ergodicity of billiards means that it’s hard to predict where a ball will be after a given amount of time. This is because, if you get the initial conditions of the ball slightly wrong, then the error will snowball, resulting in an inaccurate prediction. This is known as the ‘butterfly’ effect, and is one of the main characteristics of chaotic behaviour.

How is this useful?

Variations of the game are models for dynamical systems that occur in nature, for example the molecules that make up a gas, bouncing around an ambient space or in a simplified model of the electrical conductivity of metals (if you think of the electron as a little ball, then it behaves just like a billiard ball bouncing around on a table that contains a lattice of obstacles).

Corina Ulcigrai, an Italian Mathematician, says:

“Ergodicity is a very important property of chaotic systems. It dates back to the 19th century when the physicist Ludwig Boltzmann suggested that many dynamical systems are ergodic: they are so chaotic that if you look at the trajectory of a typical point this will in some sense explore all of the possibilities that are theoretically allowed.”

Sorry today’s post is short, I’ve been ill for the last week and haven’t had the chance to prepare a blog post for today. Hope you still enjoy it!


2 thoughts on “Billiards”

  1. Very nice introduction to Ergodicity. Please don’t apologize (for short post) when you haven’t done anything wrong 🙂 [You apologized thrice in one month, March 11, March 28, April 11]

    Liked by 2 people

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