Your average billiards player down at the local hall might not be aware that the table he’s playing on represents a complex mathematical problem. Imagine a frictionless (and pocketless) billiard table on which a ball bounces endlessly from wall to wall. Though a seasoned billiards player or mathematician might be able to predict where the ball will hit the side once, possibly twice, soon even tiny differences in angle at each strike will add up and become magnified. Thus the further the time from the original bounce, the more impossible it becomes to predict how the ball will continue to move.
Prof. Vered Rom-Kedar of the Institute’s Computer Science and Applied Mathematics Department is one of a group of mathematicians who study the theoretical properties of billiards. More than just an interesting thought problem, billiard mathematics can describe the physics of everyday systems. It had its beginnings in the theories of such scientists as Ludwig Bolzmann, who in the late 19th century suggested that the molecules of gas in a closed container are similar to hard spheres in motion. Like the balls on a table, their constant bouncing off each other and the container walls results in pnpredictable, erratic trajectories. This insight led him to formulate his basic law of gases, which roughly states that gas molecules, on average, will scatter evenly throughout a space.
In a highly chaotic system such as Bolzmann’s gas, one cannot foresee where any one “ball” will go, but it’s possible to predict the average outcome for a large number of balls. On the other hand, on a perfectly round table with one ball bouncing off its circular boundary, one can predict the ball’s endlessly repeating course with precision. But what can be said about systems that have elements of both?
Rom-Kedar and Prof. Dmitry Turaev of Ben-Gurion University of the Negev investigate so-called “mixed systems,” in which “islands of stability” (areas of predictable, repeating orbits) can coexist with chaotic movement. They begin with the shape of the table, which strongly influences movement. The two work with a fancifully shaped group of tables that have inward-curving sides, making them look something like fat cartoon stars. Called Sinai or dispersing tables, these are variations on a well-studied model composed of two moving disks in a rectangle and they were proved to be inevitably chaotic. Rom-Kedar and Turaev have been demonstrating how, with the slightest of modifications, these highly chaotic systems can give way to mixed ones.
They suggest mixed systems might arise, for instance, if the balls resemble electrons or ions more than billiard balls. Unlike hard balls, these particles do not bang into one another because they carry repellent charges – the particles are deflected before they can make contact. Another way to think of the problem, says Rom-Kedar, is as a system with one ball and elastic walls. As opposed to a sharp impact, which not only changes a ball’s direction but jolts its energy potential, the potential energy of a deflected electron, or of a ball rebounding from an elastic wall, will change in an even, unbroken curve, termed “smooth” or “soft” potential.
Rom-Kedar and Turaev proved that smooth potential allows islands of stability to be created on dispersing tables; they found, furthermore, how the relative hardness or softness of the wall affects the island’s size. Islands of stability come into being, in this case, if a part of the ball’s path forms a tangent to one of the concave boundaries. In another collaborative paper, the two proved that hitting corners – as long as those corners have set, finite angles – can also send the ball on a non-chaotic, stable, repeating path.
The published billiards research relates to two-dimensional tables. Now, with graduate student Anna Rappaport, the mathematicians are working on a theory for smooth, billiard-type systems of higher dimensions. “If we accomplish that,” says Rom-Kedar, “we’ll be closing in on a mathematical understanding of Bolzmann’s hypothesis.”
Prof. Nir Davidson of the Physics of Complex Systems Department has been working for the last few years on trapping atoms in what are arguably the world’s smallest “billiard tables.” These are so-called dark optical traps – a small number of atoms held in a dark space surrounded by a thin wall of laser beam light. He and his team were working on refining the walls of their traps, changing the shapes and trying to thin down the width of the beams.
But the atoms bouncing off the experimental walls were behaving unpredictably. The laser walls were “spongy” – the atoms sank into them a little way before bouncing back, and the walls’ slopes, never exactly 90°, seemed to affect the way the atoms moved.
At this point Prof. Uzy Smilansky of the same department suggested that Rom-Kedar and Turaev’s work on soft-sided tables and islands of stability might hold answers to some of the physicists’ questions. Indeed, the billiards formulas were able to predict how changes in the thickness and slope of the laser enclosure would affect the motions of the entrapped atoms, giving Davidson’s group a solid mathematical basis for its observations.
Later, Davidson’s group noticed that atoms hitting the corners of the traps tended to come back to their starting points, and mentioned this to Rom-Kedar. This observation led her and Turaev to their mathematical study of the conditions needed to create islands of stability for corners.