The obstacle that makes a table chaotic

Every table so far on this ladder has been orderly in some way that could be named. A square unfolds to a torus and its trajectories take four directions. A rational polygon unfolds to a surface and takes finitely many. A circle or an ellipse has a conserved quantity that sorts its trajectories into layers.

Put one round post in the middle of the square, and all three kinds of order go at once.

What a curved wall does to a beam #

Take two parallel trajectories, a small distance $d$ apart, and bounce them off a wall.

A flat wall reflects them to two parallel trajectories, still $d$ apart. Nothing happens to the gap at all; the whole growth in an empty polygon comes from the trajectories having slightly different directions, so the gap grows in proportion to the distance travelled and no faster.

A wall curving away from the table — a convex obstacle, seen from inside — reflects them to two trajectories that are no longer parallel. The one that strikes further round the post is turned through a different angle, and the difference is proportional to the gap times the curvature. So the gap is multiplied at each encounter.

Multiplication compounds. Ten encounters with a factor of three each is a factor of nearly sixty thousand, and a gap of a hundred-thousandth becomes a gap of half a table. That is what the figures measure: the same two starting states, the same distance travelled, and a separation four thousand times larger with the post in place.

The factor at a single encounter can be estimated, and the estimate says which quantities matter. Two parallel trajectories a distance $d$ apart strike a circle of radius $r$ at points whose normals differ by about $d/r$, so the outgoing directions differ by about $2d/r$; after travelling a further distance $\ell$ to the next wall, the gap has grown by about $2 \ell d / r$. So the multiplier is roughly $2\ell/r$ — the ratio of the distance between collisions to the obstacle’s radius — and it exceeds one whenever the flight is longer than half the radius, which on any table worth drawing it is. A small obstacle on a large table gives a large multiplier and rare encounters; a large one gives a small multiplier and frequent encounters, and the exponent is the trade between them.

Why the empty table is not chaotic #

The contrast is the point, and it deserves stating carefully, because “the trajectories separate” is true on both tables.

On the empty square, two trajectories with directions differing by $ε$ are $ε s$ apart after travelling a distance $s$. That is unbounded growth, and it is linear: to predict twice as far ahead needs half the starting error. Nothing about the square is unpredictable in the ordinary sense; halving the uncertainty doubles the horizon.

With the post the growth is $e^{λ s}$ for some positive $λ$. Halving the starting error buys a fixed additional distance, not twice the distance, and the additional distance is $(\log 2)/λ$ — a constant. To predict ten times as far ahead needs a starting error smaller by a factor of $e^{9λ s}$, which is why prediction fails absolutely rather than merely getting harder.

That distinction — linear against exponential separation — is the definition of chaos in this collection and everywhere else, and a billiard table is the cleanest place to see it caused by a shape rather than by a formula.

What Sinai proved #

Yakov Sinai proved in 1970 that a billiard in a square with a circular obstacle is ergodic: almost every trajectory spends time in each region in proportion to that region’s area, so the long-run average of any reasonable quantity along one trajectory equals its average over the whole table.

He proved more — the system is mixing, and has positive entropy — and the argument was the first of its kind for a system with a physical interpretation. That interpretation is the reason it mattered.

A gas of hard spheres in a box is a billiard: the configuration of $N$ spheres is a point in a high-dimensional space, the collisions are reflections off the boundaries of that space, and the boundaries corresponding to two spheres touching are convex from inside — exactly the dispersing walls of this essay. Boltzmann’s ergodic hypothesis, which underwrites the whole of statistical mechanics, is the assumption that such a system is ergodic. Sinai’s theorem is the first case where the assumption was proved rather than assumed, and the case is small — two dimensions, one obstacle — while the hypothesis is used for $10^{23}$ spheres.

The three kinds of wall #

Sinai’s mechanism is dispersion, and it is one of three things a wall can do to a beam of nearby trajectories.

Flat walls preserve it. A polygon is made entirely of these, which is why a polygon’s dynamics is at worst intricate and never chaotic in this sense — its Lyapunov exponent is zero.

Dispersing walls, curving away from the table, spread it at every encounter. Any table with one has positive Lyapunov exponent.

Focusing walls, curving into the table, converge it — and might be expected to produce order. Sometimes they do: a circle is entirely focusing and completely integrable. But Bunimovich showed in 1974 that a stadium — two semicircles joined by straight sides — is chaotic, because a beam focused by an arc passes through its focus and then diverges again, and if the straight section is long enough the divergence wins. That is defocusing, and it means chaos does not require dispersing walls; it requires only that convergence be overwhelmed on average.

The stadium is the more surprising of the two results, and it is the one that shows the mechanism is about averages along a trajectory rather than about the sign of the curvature anywhere.

Measuring the rate #

The figures report a separation rather than an exponent, and the difference is worth a paragraph.

The Lyapunov exponent is defined as a limit: the long-run average of the logarithmic growth rate of an infinitesimal separation. Nothing about it can be measured on a finite trajectory with a finite starting gap, because the gap eventually saturates at the size of the table and the average over a short run depends on which collisions happened to occur.

What can be measured is what the figures measure: the gap at sampled distances, the ratio between the tables, and the growth rate over the range where the separation is still small compared with the table. The figures assert three things — that the empty table’s gap stays proportional to the distance travelled, that the obstacle’s gap ends up hundreds of times larger, and that the obstacle’s growth rate over the drawn range is positive and bounded away from zero. All three are honest statements about a finite computation, and none of them is the exponent.

The exponent itself, for a Sinai billiard, is known to be positive by Sinai’s theorem and is computed numerically in the literature. It scales roughly as the logarithm of the ratio between the free path length and the obstacle’s radius — bigger obstacles and longer flights between them both increase it — which is visible in the figures as the drawn rate changing with the post’s size.

How far ahead anybody can predict #

A number worth carrying, because it makes the abstraction concrete.

Michael Berry’s often-quoted estimate concerns a ball on a table with obstacles, perturbed by the gravitational attraction of a single electron at the edge of the observable universe. The perturbation is unimaginably small; the divergence multiplies it by a factor at every collision; and after roughly a dozen collisions it is the size of the ball. After that the trajectory computed without the electron and the trajectory with it have nothing in common.

The point is not the electron. It is that the horizon of prediction depends only logarithmically on the precision of the starting data: an improvement by a factor of a thousand in measurement buys a handful more collisions. Every practical consequence of chaos comes from that logarithm.

The Lorentz gas, and a random walk that is not random #

Repeat the post periodically — a lattice of discs, with a ball moving among them — and the result is the Lorentz gas, introduced in 1905 to model an electron moving through a metal’s ions.

It is the same mechanism as this essay’s table, unrolled onto the whole plane, and its behaviour is the reason the subject is studied. Over long times the ball’s position performs something extremely close to a random walk: its mean squared displacement grows in proportion to time, with a diffusion constant that can be computed. Nothing random happens anywhere in the system — the trajectory is determined by the starting state to arbitrary precision — and the statistics are those of a coin-flipping walk.

That is the cleanest available statement of what deterministic chaos buys. The exponential separation destroys the memory of the starting state so quickly that after a few collisions the direction of travel is, for statistical purposes, freshly chosen; and the sum of many nearly-independent steps is a walk that becomes a curve, with the same limit theorems and the same square-root growth.

The qualification matters: nearly independent. The correlations decay exponentially rather than vanishing, and whether the resulting walk obeys the central limit theorem exactly is a real theorem with a real proof rather than an assumption — Bunimovich and Sinai proved it for the finite-horizon Lorentz gas in 1981. Where the discs are small enough to leave infinite corridors the answer changes, and the displacement grows faster than a random walk’s by a logarithm.

What is preserved, and what is destroyed #

Chaos does not mean nothing survives. A billiard preserves area in the phase space of positions and directions, and by Poincaré’s recurrence theorem almost every trajectory returns arbitrarily close to where it started, however chaotic the table.

So the dispersing billiard is recurrent and unpredictable at the same time, which sounds contradictory and is not. Recurrence says the trajectory comes back near its start eventually; it says nothing about when, and the waiting time is enormous. Prediction asks where the trajectory is at a specified time, and that is what fails.

What is destroyed is every conserved quantity except energy. The circle’s invariant is gone at the first encounter with the post; the polygon’s finite direction set is gone because the post’s normal points in every direction. A system with no conserved quantity beyond the obvious one is a system in which nothing organises the trajectories, and the histogram an orbit leaves is then the only stable thing about it: individual trajectories are unpredictable and their statistics are exactly known.

Why the proof was hard #

Sinai’s theorem took a long paper, and the reason is worth knowing, because the mechanism above sounds like it ought to settle everything in a page.

Exponential separation is a local statement: it says what happens to two trajectories that are already close, over one collision. Ergodicity is a global one: it says that almost every trajectory eventually visits every region, in the right proportion. Getting from one to the other means showing that the local expansion does not conspire with itself — that the directions in which trajectories are stretched fit together across the table into structures that genuinely mix it.

The obstruction is singularities. A trajectory that grazes the post tangentially is a discontinuity of the map: two nearby trajectories on either side of it separate immediately and completely, and the expansion rate there is unbounded rather than merely large. Those grazing trajectories form curves in the phase space, and the whole difficulty of the proof is controlling how the stretching interacts with them — how thin the pieces cut off by successive singularity lines can be, and whether they can accumulate in a way that stops the mixing.

That is the technical heart of the subject and it has not become easy. Extending Sinai’s result to three or more dimensions, or to systems of several spheres, occupied a generation of work, and the general hard-sphere case in a box remained open long after the two-dimensional one was settled. Physicists had assumed ergodicity for a century by then, on the grounds that it was obviously true.

What the pictures cannot show #

The figures draw two trajectories on each table and measure the gap between them at sampled distances. The claim — positive Lyapunov exponent — is an average over almost every pair of nearby trajectories, and two pairs do not establish an average. What the drawing shows is the phenomenon at the parameters drawn, with the numbers reported rather than eyeballed.

Nor is the exponential rate measured properly here. A Lyapunov exponent is a limit as the distance goes to infinity, and the figures follow forty bounces — long enough for the separation to reach the size of the table, which is where the measurement has to stop, since a gap cannot grow past the table’s width. Everything after that saturates, and the figures assert the growth over the range before saturation rather than fitting a rate to it.

Ergodicity is not drawn at all and could not be: it is a statement about almost every trajectory and about time averages over infinite time. What a picture can offer is one long trajectory that looks like it fills the table, and a trajectory that looks like it fills the table is exactly what a non-ergodic system can also produce.

Where the ladder goes next #

Three rungs of this ladder have been about tables where something is known. The last one is about the simplest possible table where nothing is: a triangle. Every acute triangle has a periodic path, every right triangle has one, every triangle with rational angles has one — and whether every triangle has one is open, and has been since the question was asked.