Order that flows

Two arrows of time

Course 5 ended with a puzzle, and this course begins with it. The second law of thermodynamics says that in an isolated system, entropy, a measure of disorder, never decreases. Smoke from a stick of incense spreads through the room and never gathers itself back into the stick. Perfume from an open bottle drifts until it fills the air. A living body ages, and in the end it stops. Everything we know about heat says that the world runs down.

Yet look at the history of the world and a second arrow seems to point the other way. Simple chemistry organized itself into cells, cells into animals, animals into societies. An oak grows from an acorn and a city grows from a crossroads. Over billions of years the earth has become more intricate, not less. One arrow points toward disorder and the other, at least in some places, toward order. Do they contradict each other?

They would if the things that build order were isolated. They are not. A living thing, a city, and the earth itself are open systems: matter and energy flow in, and waste and heat flow out. Think of a refrigerator. Its inside gets colder and more orderly, and no law is broken, because a motor pumps heat out of the back into the kitchen. Count the whole ledger, the refrigerator plus the kitchen, and entropy has gone up. A cell does the same. It takes in energy-rich food, ultimately energy from sunlight, and sends out low-grade heat and waste. The order inside is paid for by a larger disorder outside.

The ledger has a second column that is easy to forget: the outflow. A system that takes energy in but cannot get rid of the waste does not become more ordered; it overheats, or chokes on its own garbage. In this course’s words, openness means a flow in and a flow out.

One caution before going on. The “second arrow” is not a law that makes everything grow more orderly. Most of the universe is simply cooling and spreading out, and even where order grows, life does not always become more complex; Lesson 4 returns to that. What the idea says is narrower: where energy flows through a system, order can appear and last without breaking the second law.

A dish of liquid heated from below

You can watch order appear from nothing more than a flow of heat. In 1900 the French physicist Henri Bénard, working in a laboratory in Paris, spread a layer of melted wax about a millimeter deep, open to the air, over a metal floor that he heated evenly from below, so that the bottom was warmer than the top. When the difference was small, nothing happened. The liquid sat still, and heat crept upward from molecule to molecule by collisions, the way a handshake passes along a line of people. But when Bénard raised the difference past a critical value, the surface suddenly broke into a honeycomb of small cells, mostly hexagons, each about twice as wide as the layer was deep. In each cell the wax slowly turned over, rising in the middle and sinking at the edges.

Why does the honeycomb appear? Collisions carry heat well when the temperature difference is small. Past the critical difference, they cannot carry it away fast enough, and the liquid changes its method: instead of passing heat from molecule to molecule, it moves in bulk, with warm liquid rising and cooler liquid sinking. Seen from above, those rolling motions fall into a regular pattern of hexagons.

Notice what did not happen. Nobody told the molecules to make hexagons, and no molecule knows what a hexagon is. A smooth layer of liquid looks the same everywhere. The honeycomb does not: it has cells here and not there. Which way each cell turns is decided by chance, by tiny random disturbances that the heating amplifies. Physicists call this symmetry breaking. The pattern is a property of the whole liquid, not of any part of it, which is why it is the textbook example of emergence.

The details are more subtle than the first story. Bénard’s own layers were very thin, with an open top surface, and in 1956 and 1958 M. J. Block and J. R. A. Pearson showed that in such layers differences in surface tension do much of the pushing: the surface is pulled from the warm spots toward the cool ones, and the liquid below follows. In deeper layers, as Lord Rayleigh calculated in 1916, it is buoyancy: warm liquid is lighter and rises, and the usual pattern there is stripes, though hexagons can appear too. The two cases look alike and behave alike. Both have a threshold, and past it a pattern appears out of a smooth layer. The same kind of pattern shows on a grander scale at the surface of the sun, whose bright “granules,” each roughly a thousand kilometers across, are the tops of convection cells.

Structures that are flows

The Belgian chemist Ilya Prigogine gave this kind of pattern a name. In work from the late 1960s he called it a dissipative structure: an organized pattern that is kept up by a flow of energy through it, and that exists only for as long as the flow lasts. Such a structure pays for its order by exporting entropy, so the total entropy of structure plus surroundings still rises. In Bénard’s layer the cells actually speed up the flow of heat, which means the layer produces entropy faster than it would have without them. Prigogine received the 1977 Nobel Prize in Chemistry “for his contributions to non-equilibrium thermodynamics, particularly the theory of dissipative structures.”

Patterns in the world sort into two kinds. A static pattern, such as a salt crystal, sits in equilibrium and needs nothing to stay as it is. Its parts do not move, and it will last as long as it is left alone. A flowing pattern is quite different. Crouch beside a stream and look at the standing wave that forms behind a rock. The wave stays in the same place, steady enough to photograph, while every drop of water in it is new within a second. The shape is permanent and the stuff is not. Kayakers and surfers ride such waves for hours, for example at the whitewater park in Glenwood Springs, Colorado. A candle flame is the same: a shape that persists while wax, air, and heat pour through it. So is a hurricane, and so is a living cell.

The river and the body

Confucius, standing by a river, is recorded as saying 逝者如斯夫，不舍昼夜, “It passes on like this, never stopping day or night” (Analects 9.17). In Greece, Heraclitus is credited with the saying that you cannot step into the same river twice. Scholars think that is Plato’s paraphrase; the closest surviving words are “On those stepping into rivers staying the same other and other waters flow,” which can be read as saying that a river stays the same river only because its water keeps changing. Both men were looking at a flowing structure.

Your body is built the same way. Food and air flow in, heat and waste flow out, and the materials of the body are continually replaced. You may have heard that “every cell in your body is replaced every seven years,” or even every four. That is not true as stated, and the real picture is more interesting. A 2021 estimate by Ron Sender and Ron Milo is that about 330 billion cells, about 80 grams’ worth, die and are replaced every day, close to a percent of all the cells in the body, and nearly all of them are blood cells and cells of the gut lining. The lining of your gut turns over in a few days. Red blood cells last about four months, and the surface of the skin renews in a matter of weeks. The whole skeleton is rebuilt at roughly 5 to 10 percent a year. At the other extreme, studies that dated cells using the carbon-14 left in the air by 1950s nuclear tests found that most neurons in the outer layer of the brain, the cortex, were made around the time of birth and are not replaced, and that the core proteins of the lens of the eye are as old as you are. Heart muscle renews at about one percent a year in young adults and less later, so most of it is with you for life. Even the atoms are on the move: tracer experiments since the 1930s show that atoms in the body are constantly exchanged with food, air, and water, though a famous 1950s estimate that 98 percent are replaced in a year is only a rough figure, and some structures, like tooth enamel, keep theirs for life.

So you are not the same stuff you were, and you are not entirely new stuff either. Philosophers ask the old question of the ship of Theseus, whose planks were replaced one by one: is it still the same ship? Seen as a dissipative structure, the answer is that what persists is the pattern: the organization of the body and the slow-changing parts, like neurons and memories, that carry it. The pattern is not a thing. It is more like the standing wave, a shape sustained by a flow.