Illustrated Physics
physics, drawn
Concept

Determinism — where it appears

The property of a law that fixes the whole future of a system from its present state, with nothing left to chance. It does not imply predictability: a double pendulum is entirely deterministic and its trajectory separates from a neighbour's at 2.05 per second, so a forecast lasts only as long as the logarithm of the precision allows.

Named by 4 essays across 2 fields — each of them below, with the objects they name alongside it.

Four predictions no list of answers can keep. The four measurements a three-particle GHZ state predicts with certainty, and what the best possible list of pre-agreed answers does with them. Each row is a choice of which quantity to measure on each of the three particles; the quantum value is not an average but a certainty, so a single run of any row has a determined outcome. Every list of answers assigns a value to X and to Y at each particle, which is sixty-four lists in all, and the enumeration finds that 32 of them get three of the four right and 32 get one. None gets four, and none can: multiplying the four left-hand sides together gives every X and every Y twice, so the product is +1 for any list whatever, while the product of the four quantum values is −1. The column on the right is one such list, which agrees with the first three rows and is then forced into the opposite of the fourth. That is the whole argument, and it needs no inequality, no average and no repetition: measure the first three rows on three copies, and the fourth is predicted with certainty and comes out the other way.

The disagreement that one run settles

Bell's argument is a statistical one — a correlation of 2.828 where a pre-agreed list of answers is stuck at 2, dug out of hundreds of coincidences. Add a third particle and the argument stops being about how often. Three measurements predict a fourth with certainty, every list of answers that gets the three right gets the fourth exactly backwards, and one run of the experiment is enough.

quantum · Entanglement Indistinguishable for 10.2 seconds, then not. The path of the lower bob for two double pendulums released 1e-8° apart, over 11 seconds, with the arms drawn at the final instant. The two traces lie on top of each other for the first 10.2 seconds — the point at which they are two pixels apart on this canvas — and after that they have nothing to do with one another. Neither is more correct: both are exact solutions of the same equations, differing only in a release angle that no apparatus could set apart. The separation is growing at 2.05 per second the whole time, including during the stretch where the picture shows one curve.

The error that doubles on a schedule

Two double pendulums released a hundred-millionth of a degree apart follow one curve for ten seconds and then have nothing to do with each other. The separation grows exponentially the whole time, including while the picture shows a single trace — which turns unpredictability into a rate, and makes the length of a forecast the logarithm of the precision rather than anything proportional to it.

mechanics · Chaos The set an orbit that never repeats settles onto. 24,000 successive positions of one orbit of the map x' = 1 − 1.4x² + y, y' = 0.3x, after five hundred steps of transient have been discarded. Nearby points separate at e^0.4188 per step, so the orbit is unpredictable in the way a doubling error measures; and every one of the 24,000 points lies inside a box 2.558 by 0.767, a diagonal of 2.670, so it is going nowhere. Those two statements are not compatible with a smooth stretching: something has to bring the separated points back, and the bringing back is the visible fold at the left-hand end. The curve is not a curve. Every strand of it is a bundle of strands at any magnification, which is what an area contraction of 0.3 per step leaves behind when the stretching along the other direction is e^0.419. The 24,000 points paint 7,352 distinct marks at the resolution this is drawn at, which is itself a measurement of how little of the plane the set occupies.

The fold that has to be there

Two trajectories that separate exponentially, in a region they can never leave, are being asked to do two incompatible things. The resolution is that the motion is folded back on itself over and over, and the object that survives infinitely many foldings is neither a curve nor a patch of surface — it has a dimension between the two, and the number can be measured two entirely different ways.

mechanics · Chaos Every toss already decided. The outcome of a coin toss as a map of how it was launched: the upward speed in metres per second across, the spin in radians per second up, with the region where the coin shows the face it started with shaded and the region where it shows the other face left clear. A coin spinning about a diameter and caught at the height it left is in the air for 2v/g and turns 2vω/g radians, so the boundaries are the hyperbolae vω = nπg/2, one for every half-turn. Near the origin the bands are wide: a slow toss with little spin is easy to control, and a conjuror's "honest" toss lives there. Nothing in the map is random. Every point is a definite outcome, and the bands are smooth curves — there is no fractal boundary, no region where a tiny change has an unpredictable effect. What makes a toss a fair bet is only how narrow the bands become further out.

The toss that is random without chaos

A coin toss is the textbook random event and the textbook deterministic one at the same time. Its flight is a few lines of Newtonian mechanics with no chaos anywhere: two tosses that start almost alike stay almost alike, drifting apart only in proportion to time. Its outcome is fixed the moment it leaves the thumb, by a map of launch speed and spin carved into bands of heads and tails. It is a fair bet because the bands, at a real toss, are only a few per cent wide and no hand repeats itself that well — and it is not quite fair, because a coin wobbles as it flips, and the wobble keeps its starting face up a little more than half the time: 50.8 per cent, predicted in 2007 and counted in 2023 over 350,757 tosses.

mechanics · Chaos

Named alongside it

The objects these essays reach for when they reach for this one.

ChaosLyapunov exponentPredictabilityProbabilityAngular momentumAttractorBell inequalityBox countingCorrelationDissipationEntanglementExponential sensitivity

All concepts