Billiards with memory framework leads to mathematical questions

Physical Assessment Letters (2024). DOI: 10.1103/PhysRevLett.132.157101″>

Adding one simple rule to an idealized billiards game leads to a wealth of intriguing mathematical questions, as well as applications in the physics of living organisms. This week, researchers from the University of Amsterdam, including two master's students as first authors, did just that published a piece of paper in it Physical Assessment Letters about the fascinating dynamics of billiards with memory.

An idealized version of the game of billiards has fascinated mathematicians for decades. The basic question is simple: once a billiard ball is played, where does it go and where does it end up? Suppose the billiard table is perfect: the walls are perfectly resilient, there are no other objects on the table, the motion of the ball is frictionless, and so on. Then the ball won't really “end up” anywhere; it will go on forever.

But will it ever return to where it started? Does it eventually visit every part of the table? If we slightly change the direction of the ball, or its starting location, will the path it follows resemble the previous one?

All these questions prove to be mathematically very intriguing. Their answers are not always known, especially if the shape of the billiard table is not simple, such as a square or a rectangle. For example, in triangular billiards with angles less than 100 degrees, it is known that there are always periodic paths for the ball to follow that return on themselves.

This can be proven mathematically. Now change one of the angles to a slightly larger angle, and no mathematician will know the answer anymore.

Idealized billiards games are not just a favorite pastime of mathematicians. They also have a profound influence on physics and other sciences. Many of the questions about billiards can be formulated as questions about chaos: Do similar initial conditions of a dynamical system, whether a ball on a billiard table, a molecule in a gas, or a bird in a flock, always lead to similar end results?

A new rule

In research at the University of Amsterdam, a team of physicists has realized that by slightly changing the rules of the game of billiards, the number of applications in the real world increases even further.

Mazi Jalaal, co-author of the publication and head of the group in which the research was conducted, explains: “In nature, many living organisms have an external form of memory. For example, they leave traces to remember where they have been. They can then use that information to follow the same route again, or for example when looking for food so as not to explore the same region again.'

This last option led the researchers to an interesting idea: what if we added one rule to the game of billiards, namely that the ball should never cross its own previous path? The result is that the effective size of the billiard table becomes smaller and smaller. In fact, the ball eventually becomes trapped in its own trajectory.

Intriguing new questions

The trapping effect makes the system even more intriguing. Even simple questions now become extremely fascinating. How far does a ball travel before it gets stuck? The answer varies depending on the shape of the table as well as the starting point and direction of the ball.

Sometimes the ball travels a length only a few times the length of the table, sometimes it can travel 100 times that length before getting stuck. Where the ball ultimately ends up when caught is also a complicated question; repeating the experiment millions of times on a computer, each time with a slightly different starting position and speed, leads to beautiful patterns of final configurations.

The image at the top of this text shows some of these beautiful examples. Interestingly, the resulting dynamical systems can be chaotic. Changing the starting position or the speed of the self-avoiding ball only slightly could result in it getting stuck at a completely different point on the billiard table.

Moreover, unlike what happens on a regular billiard table, it is not very likely that the self-avoiding ball will just land anywhere. Some regions are more likely than others. To explain and prove all these features, mathematicians certainly have their work cut out for them.

Endless applications

An interesting feature of the publication is that both first authors are master's students. Jalaal adds: “The idea of ​​a 'billiards with memory' is simple enough and new enough that studying it doesn't require years of experience. Thijs and Stijn have done a great job of mastering the material and finding clever ways to to study all these things. new open problems. I'm very happy that they can already be lead authors of a publication.”

The results are just the first steps in what could be a whole new area of ​​research. Not only are there many interesting mathematical questions now waiting to be answered; the applications in physics, including biophysics, are also endless.

Jalaal says: “The concept of trapping requires research, including in real-life systems. For example, we know that single-celled slime molds use self-avoidant pathways. Do they get trapped too, and what happens if they do? Or Do they have smart mechanisms to prevent this? Do they use it to improve food search strategies?

“The results would help us better understand these biological systems, and perhaps even integrate the lessons we learn to optimize this form of memory billiards for use in robots.”