Straighten two sides of a circle and order collapses. Bunimovich’s stadium, Sinai’s scatterer, and a mushroom that is half of each
Cut a circle in half and push the halves apart, joining them with straight sides. The rule of the game is unchanged, but the order of the last lesson vanishes. Leonid Bunimovich proved in the 1970s that this stadium is completely chaotic.
Launch two balls whose starting points differ by a billionth of the table’s width. Watch how long they stay together in the ellipse, and in the stadium.
Both balls leave the same wall in the same direction; the blue one starts 10^-9 to the right. On the logarithmic chart, a straight climbing line means the gap multiplies by the same factor in each unit of time — sensitive dependence on initial conditions.
In the ellipse the two balls drift apart slowly, at a rate proportional to time: they ride slightly different caustics, so one gradually gets ahead of the other. In the stadium the gap is multiplied by roughly the same factor every unit of time, so a gap of 10⁻⁹ becomes a gap the size of the table in a few dozen bounces. On the logarithmic chart that is a straight climbing line, whose slope is the Lyapunov exponent.
This is why the demo’s floating-point numbers do not matter for the picture but do matter for any single path: after a few dozen bounces the computed stadium trajectory has nothing to do with the true one from the same starting point. What survives is the statistics, which is where the next demo goes.
The surprise is where the chaos comes from. Yakov Sinai showed in 1970 that walls curving inward (like the central disc of his table) scatter nearby paths apart, and expected outward-curving walls to focus them. Bunimovich found that focusing walls cause chaos too, provided the beam has room to pass through its focus and spread out again before the next bounce. The straight sides of the stadium give it that room.
Release thousands of balls from one small spot, almost in step. In the ellipse they keep their shape for ever. In the stadium they spread until the whole table is evenly grey.
Every ball starts within 0.04 of the same point, heading within 2.5° of the same direction. Occupancy counts how many cells of a 24 × 24 grid hold at least one ball, out of the cells a single long orbit visits. Balls released in the mushroom’s stem spread chaotically too, but in the phase portrait below they never enter the orderly orbits that circle the cap.
A billiard is ergodic if almost every single orbit spends time in each region in proportion to its size, and mixing if any blob of starting states eventually spreads evenly over everything. Sinai’s table and Bunimovich’s stadium are both, along with stronger properties.
This was the point of Sinai’s work. Boltzmann had assumed in the nineteenth century that a gas explores all its possible states evenly, the ergodic hypothesis behind statistical mechanics. Sinai’s billiard is a model of two hard discs on a torus, and proving that even this tiny gas is ergodic was a breakthrough. For many hard balls the problem took decades more; Nándor Simányi proved ergodicity for typical masses and radii in the 2000s.
Bunimovich’s mushroom (2001) splits the difference. Balls released in the stem spread chaotically, but they can never get into the orderly orbits that circle the cap without ever dipping into the stem. One table, two kinds of behaviour, each occupying a region of positive size in phase space.
The same portrait as before, for chaotic tables. A single orbit no longer draws a curve; it sprays across the whole cylinder. In the mushroom, regular curves and chaotic spray share the picture.
Horizontal: where the ball hits, as a fraction of the way round the wall. Vertical: the cosine of the angle it leaves at, from gliding one way (top) through straight across (middle) to gliding the other way (bottom). Each orbit is one colour.
Billiards are also the standard test case for quantum chaos: what happens to a quantum particle in a box whose classical motion is chaotic. Shnirelman’s theorem says that for an ergodic table, almost all high-energy standing waves spread evenly over the table, mirroring the even spray in the portrait.
Eric Heller noticed in 1984 that some stadium waves are visibly concentrated along unstable periodic orbits, which he called scars. Whether every high-energy wave eventually spreads out evenly — quantum unique ergodicity — fails for almost every stadium shape, as Andrew Hassell proved in 2010, while for other chaotic systems it is still a major open question.
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