In his new book, Cosmological Koans, Anthony Aguirre uses Zen poetry to ponder key questions about the world and physics. In this extract, he explains how the second law of thermodynamics holds that entropy – or disorder – is always increasing, and so order must be endowed at the beginning of the Universe. In that case, where did it come from?
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INSTRUCTIONS FROM THE COOK
(ZUIŌ-JI TEMPLE, JAPAN, 1625)
On a crisp autumn afternoon of falling leaves, you overhear the master Zenjo talking with the monastery cook, also one of his most senior students.
Zenjo: The Honored One states that all ordered things are impermanent, and tend to fall into disorder.
Cook: Indeed. Look at this moldy rice!
Zenjo: The world is old. Why has the kitchen not long ago dissolved to dust?
Cook: Perhaps order arises from disorder.
Zenjo: Does the kitchen clean itself?
Cook: It does when I am in the kitchen.
Zenjo: But where does the dirt go?
Cook: There is plenty of space behind the shed to empty the dirty water.
Zenjo: So there is. But where does your breakfast come from?
Cook: The sunlight and the rain.
Zenjo: Why does the Sun shine?
Cook: Because it was born.
Zenjo: Who gave birth to the Sun?
Cook: The universe.
Zenjo: And who cleans the universe?
Your desk gets messier; the kitchen nearly always needs cleaning. Cars break; bridges fall. Your body grows older, even as you read this. It sickens; it will die. We are all subject to decay and dissolution. Yet the chaos is not complete: the kitchen gets cleaned, cars built and repaired, bridges reconstructed, babies born.
Why does the kitchen never clean itself? And if it does not, how does it get cleaned, if the cleaner is also subject to decay? If all is subject to dissolution, why did the world not dissolve into complete disorder long ago? Does the law of dissolution brook exceptions? If so, how? If not, what gives order to the world? These basic experiences, we shall see, touch some of the foundations of physics, as well as the most vast scales of space and time.
Let’s begin in the kitchen. What does it mean for it to be tidy? It means that the pots are put away, not strewn on the counter; the rice is on its shelf; the floor is swept; the counters are clear. A messy kitchen is one in which many of the kitchen’s components are out of their proper place. Looked at another way, there are many, many ways the utensils, spices, pans, and staples could be arranged in the kitchen, but only a very few of these ways make the kitchen clean. Clean is a special property, too easily lost.
To be more precise, we can imagine enumerating all the possible configurations of the kitchen— that is, all the possible ways that all of the plates, utensils, jars, and so on could be arranged. Now suppose that for each one of these, we assign a judgment that the configuration is “very clean” or “somewhat clean” or “somewhat dirty” or “very dirty.” It is quite clear that these four groupings will contain progressively more and more of our kitchen configurations.
Now let’s imagine the kitchen in one of its “somewhat clean” states, and let it be acted on by an agent that rearranges things with no regard whatsoever to the assignations we have made. (You might imagine, in increasing order of random rearranging power, an earthquake, a hurricane, or a four-year-old child.) You’ll see that with extremely high probability, under the action of the agent, the kitchen will inexorably move into the “somewhat dirty,” then “very dirty” states. It is possible that by pure dumb luck it will become cleaner, but this is stupendously unlikely.
This tendency for a system to spontaneously become “disordered” is the so-called second law of thermodynamics in action. Physicists formalize the second law in terms of entropy: “Entropy in a closed system is nondecreasing.” But what is entropy? There is a confusing array of definitions that physicists use, but they can be boiled down to two basic notions. The first (we’ll return to the second later) might be called disorder entropy, or just disorder, and was devised by Ludwig Boltzmann in the nineteenth century, much in line with our discussion of the kitchen.
Let’s call each particular configuration of the kitchen a microstate, so all possible configurations of the kitchen correspond to a set of all possible microstates. Now we’ll call the four levels of cleanliness macrostates, where “macro” is a reminder that these are “big” states. We can sort all of the microstates into macrostates, each of which is a collection of microstates. Put another way, each macrostate is essentially a label given to a certain set of microstates, such that each microstate has exactly one label (see the figure below). We can also count the number of microstates with a given label, and physicists would assign a “Boltzmann entropy” (which we will call disorder) to the macrostate, determined by this count. The second law states that as time progresses, the system evolves to macrostates of equal or greater disorder; the progressively less tidy kitchen follows this law in a way that we can now quantify.
Thus far, we have spoken just about the various objects in the kitchen. But each of these is composed of many pieces, and if we, say, break a plate, then we suddenly have a new state that is not one of our former set of possibilities. To repair this limitation, we can consider a much larger set of possible (micro)states, describing the pieces of the objects in the kitchen. Carrying this progression to its natural conclusion, we could create a very complete set by defining our microstates in terms of the kitchen’s tiniest constituents: the atoms and molecules that make it up. In doing this, we might imagine that the number of states is infinite, as there is a continuum of positions that, say, an atom at the edge of a rice paddle might occupy. But quantum mechanics has taught us that, in general, physical systems have a finite amount of information to give— hence, effectively, a finite set of states.
Finally, once we have these states, we can also consider the laws of physics that evolve one state into another. In classical physics, the microstates specify the position and velocity of each atom; the physical laws that tell us how the atoms move then correspond to a rule for how one microstate evolves into another one. In quantum physics the states would just be quantum states, with the Schrödinger equation telling us how they evolve.
With this new combination of states and dynamics, exactly the same rules and reasoning apply. The numbers, however, are unimaginably larger: there are probably on the order of N = 1030 atoms in a kitchen, and on the order of 10N configurations that they could have. We can define a much more detailed set of macrostates (distinguishable states that we’re interested in) that includes not just rearrangements of items but also different states of each item— new rice, cooked rice, old rice, refried rice, rotten rice, and so on. Even in this much more detailed description, each macrostate has many, many, many, many microstates associated with it; moreover, macrostates with higher disorder contain very many more microstates than those with lower disorder. Thus, while evolving to lower-disorder macrostates (like rice unrotting) is not strictly impossible, the probability is so incredibly tiny that you should never, never, ever expect it to actually happen: you would be far, far more likely to win the lottery for the 10th time in a row, while simultaneously being hit by an asteroid and struck by lightning, than to see a macroscopic violation of the second law. Disorder will increase— hence the “law” in the second law.
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Now we come to Zenjo’s question. Suppose we make the kitchen a closed system by completely shielding it from any outside influence whatsoever: no thing or influence at all either enters or leaves. As the kitchen sits there, it still evolves, under the laws of physics, from state to state. But—and here is the crux of the matter—once again those laws know nothing of the particular set of macrostates we care about: they don’t know or care whether the kitchen is clean or dirty, or whether the pots are shiny, rusted, or decayed into dust. Just like the four-year-old, they just do what they do. Thus the kitchen evolves to macrostates of greater and greater disorder.
This being so, why is any actual kitchen relatively clean rather than extremely run-down, decayed, even turned to dust? A kitchen left messy at night will not, alas, clean itself as we sleep. Yet, as the cook points out, once a person is in the kitchen, the person can clean it. Does the fact that a kitchen can be cleaned and a clean kitchen has lower disorder than a dirty one violate the second law, which holds that disorder never decreases? On one level it does, and on a different, perhaps more fundamental level, it does not. At the level of configurations of all the items in the kitchen, and our four macrostates of cleanliness, the law is violated. The dynamics of the system (the actions of the kitchen cleaner) purposely evolve the system so that it tends toward “clean” configurations; this is quite unlike the four-year-old’s or earthquake’s dynamics.
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Yet it is also clear that this description of the system is not complete, and cannot work forever. Suppose you are locked in the “closed” kitchen for a long time. You could keep it clean for a while, but eventually, the garbage would start to pile up, the rice would go moldy, and you would get hungry. That is, no matter how we move stuff around, the kitchen eventually attains a state that was not part of our original set of arrangements. This always happens if we wait long enough, unless we use an ultra-fine-grained set of states— that is, the detailed description in terms of 1030 atoms and molecules. But in this more detailed description, states such as fresh and moldy are macrostates, and we can no longer easily evolve from highly disordered macrostates to more ordered ones. Even the most fastidious cleaner cannot turn moldy rice into fresh rice. For this reason, a clean closed kitchen will last only so long; we eventually find that we are in trouble if we do not connect our closed system to some larger system that can provide fresh food and accept waste to hide behind the shed.
Likewise, the cleaner cannot act in isolation forever. People are able to see, think, and move things around because we metabolize food. And a full stomach represents low disorder; thus, even while cleaning externally, a cleaner internally generates disorder while digesting and metabolizing breakfast. Therefore, even if one decreases the disorder of the kitchen a bit, the increase in order always comes at the expense of creating at least as much disorder through one’s metabolism. Eventually, this reservoir of order runs out, and the kitchen cannot remain ordered without some connection to a larger system that can provide new and low-disorder food, and a place to put the waste.
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Let’s take stock. The kitchen won’t clean itself. A person can keep it clean as long as the person is provided with a somewhat larger environment, perhaps including a garden and shed, from which highly ordered material such as food can be drawn, and to which disordered “waste” material can be removed. Yet, a moment’s thought will reveal that we have just moved the problem to a larger scale. Why is this larger system so ordered? Again, we can appeal to a yet larger system: sunlight and rain allow the food to grow, and the atmosphere and weather eventually remove the waste. In fact, all of these actions are ultimately tied to the Sun, which provides the Earth with a huge supply of order in the form of the relatively ordered radiation of sunlight.
But why stop there? Where is the order coming from that allows the Sun to exist, rather than there being just a big chaotic mess? The Sun has two sources of order. First, the Sun contains lots of material of low chemical disorder, because its primary constituents, hydrogen and helium, can fuse into much heavier elements while releasing energy and increasing disorder. If the Sun were a large ball of iron (which cannot fuse or fission while releasing energy), it would not be able to do this. The second source of order is gravitational. Gravity, being attractive, tends to want to compress or collapse objects. The Sun feels this strong inward force, but it is balanced by the pressure forces that are due to the heat of the gas composing the Sun. If not for this pressure, the Sun would collapse in about 20 minutes. But as it did so, each solar atom would speed up as it fell toward the Sun’s center. Such an increase in speed makes each atom more energetic, so in this way we can see that as objects collapse, they tend to release the sort of energy associated with the movement of atoms, which is heat, which is highly disordered.
This ability to do work and create heat suggests that uncollapsed objects are ordered, and that, in terms of gravity, a uniform medium, while structureless, is highly ordered. The presence of this order in the current Sun is a relic of the fact that the gas that formed it was quite uniform, rather than maximally clumpy. Putting the chemical and gravitational aspects together, then, we can conclude that the Sun’s ability to shine depends on a large reservoir of uniform, chemically simple gas. In fact, the universe on the largest scales is just like this.
In short, the kitchen can stay clean because the universe is big, simple, and uniform. Amazing. But we can’t help but ask: “Why is the universe so orderly?”
Should we appeal to a larger system?
Cosmological Koans: A Journey to the Heart of Physics by Anthony Aguirre is available now (£20, Allen Lane)