A world you can add up

The chain that remembered

In the early 1950s the Los Alamos laboratory in New Mexico had one of the first electronic computers, MANIAC I, built to do the arithmetic of nuclear weapons. Between jobs, three scientists gave it a different kind of problem. Enrico Fermi, the Italian-born physicist who had led the first nuclear chain reaction in Chicago, John Pasta, a physicist who worked on the new machine, and the Polish-born mathematician Stanislaw Ulam wanted to watch how heat spreads through a solid. A solid is a lattice of atoms held together by something like springs, so they modeled a short piece of one: a row of masses between two fixed walls, joined by springs. The report describes a chain of up to 64 masses (the run that everyone plots has 32). Mary Tsingou, a young mathematician, wrote the program and ran it on the machine. The report thanks her for the coding and the computations, though her name is not among its authors, and since a 2008 article by Thierry Dauxois the problem is often called FPUT, for all four names.

Their springs were nearly perfect. A perfect spring obeys Hooke’s law, the rule that a stretched spring pulls back in proportion to the stretch. Fermi, Pasta, and Ulam added a small bend to that rule, so that a stretched spring pulled back slightly more than proportion would give. Then they started the chain in the smoothest possible motion, a single arch swinging up and down, with all of the energy in that one pattern. Statistical physics said what should follow. The bend lets the patterns of motion trade energy, so the energy should leak from the first pattern into the others, and keep leaking, until every pattern held its fair share. That is the computer version of heat spreading through a metal bar, and it was the result everyone expected.

At first it looked that way: the second, third, and fourth patterns were excited in turn. Then the calculation kept running. (The story that they left it running by accident is oral history, told by Nicholas Metropolis many years later; it is not in the report.) When they looked again, the energy had come back. In the standard run, after about 157 periods of the slowest pattern, nearly all of it was in the slowest pattern again: later accounts say all but about 3 percent, and the report itself says the system returned to within about 1 percent of its start. Instead of forgetting where it began, the chain remembered.

That outcome is the “FPU paradox.” Nonlinearity alone is not enough to share energy equally. The explanation came in two pieces over the next decade. In 1965 Norman Zabusky and Martin Kruskal looked not at the patterns of waves but at the chain itself, stretched out in space. Starting from the equations of the chain, they showed that the smooth arch breaks into sharp lumps that travel, pass through one another without changing shape, and meet again at the start, which makes the recurrence. They coined the word soliton for such a lump. In the same years a mathematical theorem of Kolmogorov, Arnold, and Moser, together with work by Izrailev and Chirikov in 1966, explained why weak nonlinearity leaves most motion orderly and why stronger nonlinearity finally does thermalize the chain. Both stories are still being refined; no one calls the problem finished.

There is a quiet lesson in the delay. Dauxois and his colleagues point out that the paradox might not have lasted ten years if someone had watched the chain in space earlier. Physicists were so used to splitting problems into waves, and adding the nonlinearity afterward as a coupling between them, that nobody had looked at the lattice as it moved. The habit of linear thinking had hidden the answer. Ulam later said that Fermi thought the result minor. The calculation was not minor: it became one of the landmark computer experiments, and it showed that a computer could be used as a laboratory, to run an idea and see what it does, when pencil and paper cannot say. (Other calculations of 1953 on the same machine also have a claim to be early; this one is the best-known for nonlinear science.)

What “linear” means

Mathematically, a relationship is linear if it has two properties. If you double the cause, you double the effect. And if two causes act together, the effect is just the sum of the two effects acting alone. The second property is called superposition. A graph of a linear relationship is a straight line, which is why the word comes from line, but the property that matters is the adding.

Superposition is what lets a hard problem be cut into easy ones. Robert Hooke announced his law of springs in 1676 as an anagram, the letters ceiiinosssttuv, and revealed the solution, ut tensio, sic vis (“as the extension, so the force”), in 1678. It works up to a limit, the elastic limit, beyond which the spring is permanently bent. Two centuries later the French mathematician Joseph Fourier turned the idea into a method. A piece of music, a wave on water, or a temperature profile along a bar can be written as a sum of simple sine waves. Solve the problem for each sine wave, which is easy, then add the answers back. His treatise on heat appeared in 1822, and the same method now underlies the Laplace transform, the wavelet analysis of Ingrid Daubechies (1988), linear algebra, the electrical engineering of circuits, and Aleksandr Lyapunov’s 1892 way of testing stability near an equilibrium by replacing the problem with its linear approximation.

Even the strangest part of physics is linear. The Schrödinger equation of Course 6 is linear, which is why the wave function of an electron can be a superposition of states. (What is not linear is the awkward step in which a measurement yields one definite outcome; whether and how that happens is the measurement problem, and some proposals for solving it add nonlinear terms.) The linear toolbox is large and it works. It is the reason that a century of physics and engineering could solve so many problems exactly.

Models are nets

Why does science love the linear? Partly because science works by simplifying. In this reading, a scientist looks at the world through models, much as everyone looks through a pair of glasses, and a model is a deliberately simple picture that can be worked out and used to predict. Waves are pictured as sine waves, motion as a regular cycle, a spread of heights as the bell curve. And gases are pictured as an ideal gas: a crowd of points that never attract each other and bounce off the walls without loss. Every one of those assumptions is false for real gas, and the ideal gas law is nonetheless one of the most useful models in science. Computed from standard tables, the pressure of air at one atmosphere and room temperature differs from the ideal-gas prediction by about 0.03 percent. Nitrogen squeezed to 200 atmospheres is off by about 5 percent, and carbon dioxide at 50 atmospheres by more than a third, close to turning into a liquid. A model is trusted inside a range, not everywhere.

The statistician George Box put the point in a line that is now famous: all models are wrong, but some are useful. (The first half appears in his 1976 paper on science and statistics, and the full line in his 1979 and 1987 writing.) Much of the series so far makes the same point. Kepler’s circles, Newton’s clockwork, and the quantum atom were each a model, and each was good for some purposes and failed at others.

The source course draws a philosophical conclusion from this: what scientists find are scientific laws, which are human inventions, and not the laws of nature itself. In this reading, a law is a tool for catching nature, like a fishing net, and nets have holes. The philosopher Karl Popper, in Logik der Forschung (1934; English edition 1959), argued that a claim is scientific only if it could be shown wrong by some observation, so scientific laws are fallible by design. Linear models are the favorite nets, simple, effective, and easy to learn, and a student can fill a notebook with them. The trouble, the source course says, is that nature’s catch is not all linear. The “Another view” section below says what some philosophers would answer.

Right mathematics, wrong premise

Fourier’s heat equation is linear, and it was made for exactly this kind of problem: how heat flows through a solid. In 1862 William Thomson, who became Lord Kelvin, used it to ask how old the Earth is. He pictured the Earth as a solid ball that began molten and uniformly hot, with a fixed surface temperature, losing heat only by conduction, and he measured how steeply temperature rises as one goes down into the rock. From that gradient the mathematics gave an age. For his best guess at the melting temperature of rock, he found about 98 million years, and his stated limits were 20 million and 400 million. In 1897 he narrowed it to between 20 and 40 million years, probably much nearer 20.

The geologists and naturalists would not have it. Darwin’s first edition of On the Origin of Species (1859) had estimated about 300 million years for the erosion of one valley in southern England, the Weald; he dropped the figure from the third edition (1861), before Kelvin’s paper, but in later years the short ages troubled him greatly. In 1869 Thomas Huxley attacked the assumptions behind Kelvin’s numbers. And Kelvin’s mathematics was never the problem. The weak point was the premise: that the Earth is a rigid solid with no other heat source. In 1895 John Perry showed that if rock deep inside flows, even very slowly, heat is carried upward far faster than Kelvin allowed, and the surface gradient says little about the age. (The usual story credits radioactivity, discovered in 1896, and radioactivity did show that nature held energy sources no one had imagined, which mattered most for the Sun. For the Earth’s conduction model, though, it was Perry’s convection that broke Kelvin’s argument.) In 1956 Clair Patterson measured lead isotopes in meteorites and gave 4.55 billion years, give or take 70 million; the accepted value today is 4.54 billion years, give or take 50 million. Kelvin’s last answer was more than a hundred times too young.

Set the two stories side by side. The ideal gas has false premises and works well. Kelvin’s Earth had false premises and failed by a factor of a hundred. What separates them is not how simple the model is, and not how good the mathematics is. It is whether the premises were tested over the range of use, and whether anyone checked what the model left out. The source course adds that the geologists’ feeling that the Earth was very old was right while their numbers were no better than Kelvin’s, and it concludes that a good scientific feeling is worth training. That is fair as far as it goes, but what actually corrected Kelvin was a new physical idea (convection, and later isotopes), not feeling.

When adding fails

The mathematician Stanislaw Ulam is credited with a remark about the name of this field. Calling it “nonlinear science,” he is said to have observed, is like calling most of zoology “the study of non-elephant animals.” (The remark appears in a 1985 paper by James Crutchfield, Doyne Farmer, and colleagues, at second hand; no earlier printed source has been found.) The joke makes a real point. Linear relationships are the special case, the elephant, and nonlinear ones are everything else. Here are six, each with a number to check.

• Volume discounts. The price of fish is not proportional to the number of fish when the stall gives a discount for buying a crate. A bill that does not double when the order doubles is already nonlinear.
• The elastic limit. Hooke’s law holds only up to a point. Ordinary structural steel yields at about one eighth of one percent stretch, and beyond that the response bends.
• Stopping a car. Braking distance grows with the square of the speed, because the energy to remove does. Twice as fast takes four times the distance.
• A pedestrian’s chance. A 2011 study for the AAA Foundation of US crashes found that the risk of death for a pedestrian struck by a car is about 10 percent at 23 miles per hour, 25 percent at 32, 50 percent at 42, and 90 percent at 58. A speed 80 percent higher raises the risk fivefold.
• Wind power. The power in moving air grows with the cube of its speed, so a wind twice as strong carries eight times the power (up to the speed a turbine is built for).
• Earthquakes. Each whole step on the moment-magnitude scale is about 32 times more energy. A magnitude 8 releases about a thousand times the energy of a magnitude 6.

People are poor at feeling nonlinearity, and the best-documented case is compound growth. In 2009 Victor Stango and Jonathan Zinman reported, in the Journal of Finance, that many households “linearize” exponential growth: asked about the cost of a loan, they behave as if interest adds up in a straight line instead of building on itself, and the more biased a household is, the more it borrows and the less it saves. (A 2022 critique found flaws in how some of the bias has been measured, though not in the direction of the effect.) This is one of the cases Course 10 on thinking fast and slow would call a systematic error of intuition, since mental arithmetic runs on straight lines.

There is a second way that adding fails, and the source course illustrates it with a picture. The Dutch artist M. C. Escher’s lithograph Waterfall (October 1961) shows a stream that runs down a channel, drives a mill wheel, and then flows along a channel that somehow brings it back to the top. Look at any short stretch of the picture and it is fine; the trick shows only when you follow the whole loop. Escher built it from the “impossible triangle,” which the mathematician Roger Penrose and his father, the geneticist Lionel Penrose, described in 1958 (the Swedish artist Oscar Reutersvärd had drawn one in 1934, which the Penroses did not know). The source course uses the picture as a parable: a life or a policy can be made of steps that each look sensible, and the whole can still be absurd, because the rules change as you go and you cannot extend a locally correct step forever. It is a parable, not a proof, but it is a useful warning against assuming that a trend continues in a straight line.