Billiard systems change character of dynamics with different billiard ball radius

In the transition from mathematical billiards to physical billiards, where a ball goes from being a point particle to having a positive radius, it may seem intuitive to assume that no categorical difference exists between the two. A new proof-of-concept paper by Leonid Bunimovich says otherwise.

Bunimovich discovered as the radius of a physical billiard ball increases, the change in the behavior of the entire system is equivalent to modeling mathematical billiards with a smaller table. With increasing radius, the geometry of the system evolves. For instance, some parts of the table may become inaccessible to the ball. This results in a progression in the dynamics of the system between mathematical and physical cases, and it may become more or less chaotic with changing radius.

The author notes all the conclusions drawn in the paper arise from already existing mathematical results, requiring no new complicated proofs. Specifically, one of the subsidiary theorems that follow is there exist billiard tables that can go from strongly chaotic in the mathematical case to non-chaotic once the radius of the ball exceeds a critical value in the physical case.

“Anything is possible,” said Bunimovich. “There are various types of transitions from order to chaos, and chaos to order.”

This study has potential applications in the propagation of particles in channels, such as pipes. In these cases, boundaries are not perfectly smooth, and impurities can affect particle flow. Depending on particle size, this may result in the creation of vortices, halting particle propagation.

Source: “Physical versus mathematical billiards: From regular dynamics to chaos and back,” by L. A. Bunimovich, Chaos (2019). The article can be accessed at https://doi.org/10.1063/1.5122195 .