From Eternity to Here: The Quest for the Ultimate Theory of Time - Sean Carroll (2010)

Part III. ENTROPY AND TIME’S ARROW

Chapter 10. RECURRENT NIGHTMARES

Nature is a big series of unimaginable catastrophes.

—Slavoj Žižek

In Book Four of The Gay Science, written in 1882, Friedrich Nietzsche proposes a thought experiment. He asks us to imagine a scenario in which everything that happens in the universe, including our lives down to the slightest detail, will eventually repeat itself, in a cycle that plays out for all eternity.

What if some day or night a demon were to steal into your loneliest loneliness and say to you: “This life as you now live it and have lived it you will have to live once again and innumerable times again; and there will be nothing new in it, but every pain and every joy and every thought and sigh and everything unspeakably small or great in your life must return to you, all in the same succession and sequence—even this spider and this moonlight between the trees, and even this moment and I myself. The eternal hourglass of existence is turned over again and again, and you with it, speck of dust!”167

Nietzsche’s interest in an eternally repeating universe was primarily an ethical one. He wanted to ask: How would you feel about knowing that your life would be repeated an infinite number of times? Would you feel dismayed—gnashing your teeth is mentioned—at the horrible prospect, or would you rejoice? Nietzsche felt that a successful life was one that you would be proud to have repeated in an endless cycle.168

The idea of a cyclic universe, or “eternal return,” was by no means original with Nietzsche. It appears now and again in ancient religions—in Greek myths, Hinduism, Buddhism, and some indigenous American cultures. The Wheel of Life spins, and history repeats itself.

But soon after Nietzsche imagined his demon, the idea of eternal recurrence popped up in physics. In 1890 Henri Poincaré proved an intriguing mathematical theorem, showing that certain physical systems would necessarily return to any particular configuration infinitely often, if you just waited long enough. This result was seized upon by a young mathematician named Ernst Zermelo, who claimed that it was incompatible with Boltzmann’s purported derivation of the Second Law of Thermodynamics from underlying reversible rules of atomic motion.

In the 1870s, Boltzmann had grappled with Loschmidt’s “reversibility paradox.” By comparison, the 1880s were a relatively peaceful time for the development of statistical mechanics—Maxwell had died in 1879, and Boltzmann concentrated on technical applications of the formalism he had developed, as well as on climbing the academic ladder. But in the 1890s controversy flared again, this time in the form of Zermelo’s “recurrence paradox.” To this day, the ramifications of these arguments have not been completely absorbed by physicists; many of the issues that Boltzmann and his contemporaries argued about are still being hashed out right now. In the context of modern cosmology, the problems suggested by the recurrence paradox are still very much with us.

POINCARÉ’S CHAOS

Oscar II, king of Sweden and Norway, was born on January 21, 1829. In 1887, the Swedish mathematician Gösta Mittag-Leffler proposed that the king should mark his upcoming sixtieth birthday in a somewhat unconventional way: by sponsoring a mathematical competition. Four different questions were proposed, and a prize would be given to whoever submitted the most original and creative solution to any of them.

One of these questions was the “three-body problem”—how three massive objects would move under the influence of their mutual gravitational pull. (For two bodies it’s easy, and Newton had solved it: Planets move in ellipses.) This problem was tackled by Henri Poincaré, who in his early thirties was already recognized as one of the world’s leading mathematicians. He did not solve it, but submitted an essay that seemed to demonstrate a crucial feature: that the orbits of the planets would be stable. Even without knowing the exact solutions, we could be confident that the planets would at least behave predictably. Poincaré’s method was so ingenious that he was awarded the prize, and his paper was prepared for publication in Mittag-Leffler’s new journal, Acta Mathematica.169

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Figure 52: Henri Poincaré, pioneer of topology, relativity, and chaos theory, and later president of the Bureau of Longitude.

But there was a slight problem: Poincaré had made a mistake. Edvard Phragmén, one of the journal editors, had some questions about the paper, and in the course of answering them Poincaré realized that he had left out an important case in constructing his proof. Such tiny mistakes occur frequently in complicated mathematical writing, and Poincaré set about correcting his presentation. But as he tugged at the loose thread, the entire argument became completely unraveled. What Poincaré ended up proving was the opposite of his original claim—three-body orbits were not stable at all. Not only are orbits not periodic; they don’t even approach any sort of regular behavior. Now that we have computers to run simulations, this kind of behavior is less surprising, but at the time it came as an utter shock. In his attempt to prove the stability of planetary orbits, Poincaré ended up doing something quite different—he invented chaos theory.

But the story doesn’t quite end there. Mittag-Leffler, convinced that Poincaré would be able to fix things up in his prize essay, had gone ahead and printed it. By the time he heard from Poincaré that no such fixing-up would be forthcoming, the journal had already been mailed to leading mathematicians throughout Europe. Mittag-Leffler swung into action, telegraphing Berlin and Paris in an attempt to have all copies of the journal destroyed. He basically succeeded, but not without creating a minor scandal in elite mathematical circles across the continent.

In the course of revising his argument, Poincaré established a deceptively simple and powerful result, now known as the Poincaré recurrence theorem. Imagine you have some system in which all of the pieces are confined to some finite region of space, like planets orbiting the Sun. The recurrence theorem says that if we start with the system in a particular configuration and simply let it evolve according to Newton’s laws, we are guaranteed that the system will return to its original configuration—again and again, infinitely often into the future.

That seems pretty straightforward, and perhaps unsurprising. If we have assumed from the start that all the components of our system (planets orbiting the Sun, or molecules bouncing around inside a box) are confined to a finite region, but we allow time to stretch on forever, it makes sense that the system is going to keep returning to the same state over and over. Where else can it go?

Things are a bit more subtle than that. The most basic subtlety is that there can be an infinite number of possible states, even if the objects themselves don’t actually run away to infinity.170 A circular orbit is confined to a finite region, but there are an infinite number of points along it; likewise, there are an infinite number of points inside a finite-sized box of gas. In that case, a system will typically not return to precisely the original state. What Poincaré realized is that this is a case where “almost” is good enough. If you decide ahead of time how close two states need to be so that you can’t tell the difference between them, Poincaré proved that the system would return at least that close to the original state an infinite number of times.

Consider the three inner planets of the Solar System: Mercury, Venus, and Earth. Venus orbits the Sun once every 0.61520 years (about 225 days), while Mercury orbits once every 0.24085 years (about 88 days). As shown in Figure 53, imagine that we started in an arrangement where all three planets were arranged in a straight line. After 88 days have passed, Mercury will have returned to its starting point, but Venus and Earth will be at some other points in their orbits. But if we wait long enough, they will all line up again, or very close to it. After 40 years, for example, these three planets will be in almost the same arrangement as when they started.

Poincaré showed that all confined mechanical systems are like that, even ones with large numbers of moving parts. But notice that the amount of time we have to wait before the system returns close to its starting point keeps getting larger as we add more components. If we waited for all nine of the planets to line up,171 we would have to wait much longer than 40 years; that’s partly because the outer planets orbit more slowly, but in large part it simply takes longer for more objects to conspire in the right way to re-create any particular starting configuration.

This is worth emphasizing: As we consider more and more particles, the time it takes for a system to return close to its starting point—known, reasonably enough, as the recurrence time—quickly becomes unimaginably huge.172Consider the divided box of gas we played with in Chapter Eight, where individual particles had a small chance of hopping from one side of the box to the other every second. Clearly if there are only two or three particles, it won’t take long for the system to return to where it started. But once we consider a box with 60 total particles, we find that the recurrence time has become as large as the current age of the observable universe.

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Figure 53: The inner Solar System in a configuration with Mercury, Venus, and Earth all aligned (bottom), and 88 days later (top). Mercury has returned to its original position, but Venus and Earth are somewhere else along their orbits.

Real objects usually have a lot more than 60 particles in them. For a typical macroscopic-sized object, the recurrence time would be at least

101,000,000,000,000,000,000,000,000 seconds.

That’s a long time. For the total number of particles in the observable universe, the recurrence time would be even much longer—but who cares? The recurrence time for any interestingly large object is much longer than any time relevant to our experience. The observable universe is about 1018 seconds old. An experimental physicist who put in a grant proposal suggesting that they would pour a teaspoon of milk into a cup of coffee and then wait one recurrence time for the milk to unmix itself would have a very hard time getting funding.

But if we waited long enough, it would happen. Nietzsche’s Demon isn’t wrong; it’s just thinking long-term.

ZERMELO VERSUS BOLTZMANN

Poincaré’s original paper in which he proved the recurrence theorem was mainly concerned with the crisp, predictable world of Newtonian mechanics. But he was familiar with statistical mechanics, and within a short while realized that the idea of eternal recurrence might, at first blush, be incompatible with attempts to derive the Second Law of Thermodynamics. After all, the Second Law says that entropy only ever goes one way: It increases. But the recurrence theorem seems to imply that if a low-entropy state evolves to a higher-entropy state, all we have to do is wait long enough and the system will return to its low-entropy beginnings. That means it must decrease somewhere along the way.

In 1893, Poincaré wrote a short paper that examined this apparent contradiction more closely. He pointed out that the recurrence theorem implied that the entropy of the universe would eventually start decreasing:

I do not know if it has been remarked that the English kinetic theories can extract themselves from this contradiction. The world, according to them, tends at first toward a state where it remains for a long time without apparent change; and this is consistent with experience; but it does not remain that way forever, if the theorem cited above is not violated; it merely stays that way for an enormously long time, a time which is longer the more numerous are the molecules. This state will not be the final death of the universe, but a sort of slumber, from which it will awake after millions of millions of centuries. According to this theory, to see heat pass from a cold body to a warm one, it will not be necessary to have the acute vision, the intelligence, and the dexterity of Maxwell’s demon; it will suffice to have a little patience.173

By “the English kinetic theories,” Poincaré was presumably thinking of the work of Maxwell and Thomson and others—no mention of Boltzmann (or for that matter Gibbs). Whether it was for that reason or just because he didn’t come across the paper, Boltzmann made no direct reply to Poincaré.

But the idea would not be so easily ignored. In 1896, Zermelo made a similar argument to Poincaré’s (referencing Poincaré’s long 1890 paper that stated the recurrence theorem, but not his shorter 1893 paper), which is now known as “Zermelo’s recurrence objection.”174 Despite Boltzmann’s prominence, atomic theory and statistical mechanics were not nearly as widely accepted in the late-nineteenth-century German-speaking world as they were in the English-speaking world. Like many German scientists, Zermelo thought that the Second Law was an absolute rule of nature; the entropy of a closed system would always increase or stay constant, not merely most of the time. But the recurrence theorem clearly implied that if entropy initially went up, it would someday come down as the system returned to its starting configuration. The lesson drawn by Zermelo was that the edifice of statistical mechanics was simply wrong; the behavior of heat and entropy could not be reduced to the motions of molecules obeying Newton’s laws.

Zermelo would later go on to great fame within mathematics as one of the founders of set theory, but at the time he was a student studying under Max Planck, and Boltzmann didn’t take the young interloper’s objections very seriously. He did bother to respond, although not with great patience.

Zermelo’s paper shows that my writings have been misunderstood; nevertheless it pleases me for it seems to be the first indication that these writings have been paid any attention in Germany. Poincaré’s theorem, which Zermelo explains at the beginning of his paper, is clearly correct, but his application of it to the theory of heat is not.175

Oh, snap. Zermelo wrote another paper in response to Boltzmann, who replied again in turn.176 But the two were talking past each other, and never seemed to reach a satisfactory conclusion.

Boltzmann, by this point, was completely comfortable with the idea that the Second Law was only statistical in nature, rather than absolute. The main thrust of his response to Zermelo was to distinguish between theory and practice. In theory, the whole universe could start in a low entropy state, evolve toward thermal equilibrium, and eventually evolve back to low entropy again; that’s an implication of Poincaré’s theorem, and Boltzmann didn’t deny it. But the actual time you would have to wait is enormous, much longer than what we currently think of as “the age of the universe,” and certainly much longer than any timescales that were contemplated by scientists in the nineteenth century. Boltzmann argued that we should accept the implications of the recurrence theorem as an interesting mathematical curiosity, but not one that was in any way relevant to the real world.

TROUBLES OF AN ETERNAL UNIVERSE

In Chapter Eight we discussed Loschmidt’s reversibility objection to Boltzmann’s H-Theorem: It is impossible to use reversible laws of physics to derive an irreversible result. In other words, there are just as many high-entropy states whose entropy will decrease as there are low-entropy states whose entropy will increase, because the former trajectories are simply the time-reverses of the latter. (And neither is anywhere near as numerous as high-entropy states that remain high-entropy.) The proper response to this objection, at least within our observable universe, is to accept the need for a Past Hypothesis—an additional postulate, over and above the dynamical laws of nature, to the effect that the early universe had an extremely low entropy.

In fact, by the time of his clash with Zermelo, Boltzmann himself had cottoned on to this realization. He called his version of the Past Hypothesis “assumption A,” and had this to say about it:

The second law will be explained mechanically by means of assumption A (which is of course unprovable) that the universe, considered as a mechanical system—or at least a very large part of it which surrounds us—started from a very improbable state, and is still in an improbable state.177

This short excerpt makes Boltzmann sound more definitive than he really is; in the context of this paper, he offers several different ways to explain why we see entropy increasing around us, and this is just one of them. But notice how careful he is—not only admitting up front that the assumption is unprovable, but even limiting consideration to “a very large part of [the universe] which surrounds us,” not the whole thing.

Unfortunately, this strategy isn’t quite sufficient. Zermelo’s recurrence objection is closely related to the reversibility objection, but there is an important difference. The reversibility objection merely notes that there are an equal number of entropy-decreasing evolutions as entropy-increasing ones; the recurrence objection points out that the entropy-decreasing processes will eventually happen some time in the future. It’s not just that a system could decrease in entropy—if we wait long enough, it is eventually guaranteed to do so. That’s a stronger statement and requires a better comeback.

We can’t rely on the Past Hypothesis to save us from the problems raised by recurrence. Let’s say we grant that, at some point in the relatively recent past—perhaps billions of years ago, but much more recently than one recurrence time—the universe found itself in a state of extremely low entropy. Afterward, as Boltzmann taught us, the entropy would increase, and the time it would take to do so is much shorter than one recurrence time. But if the universe truly lasts forever, that shouldn’t matter. Eventually the entropy is going to go down again, even if we’re not around to witness it. The question then becomes: Why do we find ourselves living in the particular part of the history of the universe in the relatively recent aftermath of the low-entropy state? Why don’t we live in some more “natural” time in the history of the universe?

Something about that last question, especially the appearance of the word natural, opens a can of worms. The basic problem is that, according to Newtonian physics, the universe doesn’t have a “beginning” or an “end.” From our twenty-first-century post-Einsteinian perspective, the idea that the universe began at the Big Bang is a familiar one. But Boltzmann and Zermelo and contemporaries didn’t know about general relativity or the expansion of the universe. As far as they were concerned, space and time were absolute, and the universe persisted forever. The option of sweeping these embarrassing questions under the rug of the Big Bang wasn’t available to them.

That’s a problem. If the universe truly lasts forever, having neither a beginning nor an end, what is the Past Hypothesis supposed to mean? There was some moment, earlier than the present, when the entropy was small. But what about before that? Was it always small—for an infinitely long time—until some transition occurred that allowed the entropy to grow? Or was the entropy also higher before that moment, and if so, why is there a special low-entropy moment in the middle of the history of the universe? We seem to be stuck: If the universe lasts forever, and the assumptions underlying the recurrence theorem are valid, entropy can’t increase forever; it must go up and then eventually come back down, in an endless cycle.

There are at least three ways out of this dilemma, and Boltzmann alluded to all three of them.178 (He was convinced he was right but kept changing his mind about the reason why.)

First, the universe might really have a beginning, and that beginning would involve a low-entropy boundary condition. This is implicitly what Boltzmann must have been imagining in the context of “assumption A” discussed above, although he doesn’t quite spell it out. But at the time, it would have been truly dramatic to claim that time had a beginning, as it requires a departure from the basic rules of physics as Newton had established them. These days we have such a departure, in the form of general relativity and the Big Bang, but those ideas weren’t on the table in the 1890s. As far as I know, no one at the time took the problem of the universe’s low entropy at early times seriously enough to suggest explicitly that time must have had a beginning, and that something like the Big Bang must have occurred.

Second, the assumptions behind the Poincaré recurrence theorem might simply not hold in the real world. In particular, Poincaré had to assume that the space of states was somehow bounded, and particles couldn’t wander off to infinity. That sounds like a technical assumption, but deep truths can be hidden under the guise of technical assumptions. Boltzmann also floats this as a possible loophole:

If one first sets the number of molecules equal to infinity and allows the time of the motion to become very large, then in the overwhelming majority of cases one obtains a curve [for entropy as a function of time] which asymptotically approaches the abscissa axis. The Poincaré theorem is not applicable in this case, as can easily be seen.179

But he doesn’t really take this option seriously. As well he shouldn’t, as it avoids the strict implication of the recurrence theorem but not the underlying spirit. If the average density of particles through space is some nonzero number, you will still see all sorts of unlikely fluctuations, including into low-entropy states; it’s just that the fluctuations will typically consist of different sets of particles each time, so that “recurrence” is not strictly occurring. That scenario has all of the problems of a truly recurring system.

The third way out of the recurrence objection is not a way out at all—it’s a complete capitulation. Admit that the universe is eternal, and that recurrences happen, so that the universe witnesses moments when entropy is increasing and moments when it is decreasing. And then just say: That’s the universe in which we live.

Let’s put these three possibilities in the context of modern thinking. Many contemporary cosmologists subscribe, often implicitly, to something like the first option—conflating the puzzle of our low-entropy initial conditions with the puzzle of the Big Bang. It’s a viable possibility but seems somewhat unsatisfying, as it requires that we specify the state of the universe at early times over and above the laws of physics. The second option, that there are an infinite number of things in the universe and the recurrence theorem simply doesn’t apply, helps us wriggle out of the technical requirements of the theorem but doesn’t give us much guidance concerning why our universe looks the particular way that it does. We could consider a slight variation on this approach, in which there were only a finite number of particles in the universe, but they had an infinite amount of space in which to evolve. Then recurrences would truly be absent; the entropy would grow without limit in the far past and far future. This is somewhat reminiscent of the multiverse scenario I will be advocating later in the book. But as far as I know, neither Boltzmann nor any of his contemporaries advocated such a picture.

The third option—that recurrences really do happen, and that’s the universe we live in—can’t be right, as we will see. But we can learn some important lessons from the way in which it fails to work.

FLUCTUATING AROUND EQUILIBRIUM

Recall the divided box of gas we considered in Chapter Eight. There is a partition between two halves of the box that occasionally lets gas molecules pass through and switch sides. We modeled the evolution of the unknown microstate of each particle by imagining that every molecule has a small, fixed chance of moving from one side of the box to the other. We can use Boltzmann’s entropy formula to show how the entropy evolves with time; it has a strong tendency to increase, at least if we start the system by hand in a low-entropy state, with most of the molecules on one side. The natural tendency is for things to even out and approach an equilibrium state with approximately equal numbers of molecules on each side. Then the entropy reaches its maximum value, labeled as “1” on the vertical axis of the graph.

What if we don’t start the system in a low-entropy state? What happens if it starts in equilibrium? If the Second Law were absolutely true, and entropy could never decrease, once the system reached equilibrium it would have to strictly stay there. But in Boltzmann’s probabilistic world, that’s not precisely right. With high probability, a system that is in equilibrium will stay in equilibrium or very close to it. But there will inevitably be random fluctuations away from the state, if we wait long enough. And if we wait very long, we could see some rather large fluctuations.

In Figure 54, we see the evolution of the entropy in a divided box of gas with 2,000 particles, but now at a later time, after it has reached equilibrium. Note that this is an extreme close-up on the change in entropy; whereas the plots in Chapter Eight showed the entropy evolving from about 0.75 up to 1, this plot shows the entropy ranging from between 0.997 and 1.

What we see are small fluctuations from the equilibrium value where the entropy is maximal and the molecules are equally divided. This makes perfect sense, the way we’ve set up the situation; most of the time, there will be equal numbers of particles on the right side of the box and the left side, but occasionally there will be a slight excess on one side or the other, corresponding to a slightly lower entropy. It’s exactly the same idea as flipping a coin—on average, a sequence of many coin flips will average to half heads and half tails, but if we wait long enough, we will see sequences of the same result many times in a row.

The fluctuations seen here are very small, but on the other hand we didn’t wait very long. If we stretched out the plot to much longer times—and here we’re talking much longer times—the entropy would eventually dip down to its original value, representing a state with 80 percent of the particles on one side and only 20 percent on the other. Keep in mind that this graph shows what happens with 2,000 particles; in the real world, with many more particles in any macroscopic object, fluctuations in entropy are correspondingly smaller and more rare. But they will be there; that’s an inevitable consequence of the probabilistic nature of entropy.

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Figure 54: The evolution of the entropy of a divided box of gas, starting from equilibrium. The state spends most of its time near maximum entropy, but there are occasional fluctuations to lower-entropy states. Note from the vertical axis that we have zoomed up close; typical fluctuations are very small. The point x marks a return to equilibrium from a relatively large fluctuation.

So here is Boltzmann’s final, dramatic suggestion: Maybe the universe is like that. Maybe time does last forever, and the underlying laws of physics are Newtonian and reversible, and maybe the assumptions underlying the recurrence theorem are valid.180 And maybe, therefore, the plot of entropy versus time shown in Figure 54 is how the entropy of the real universe actually evolves.

THE ANTHROPIC APPEAL

But—you say—that can’t be right. On that graph, entropy goes up half the time and goes down half the time. That’s not at all like the real world, where entropy only ever goes up, as far as we can see.

Ah, replies Boltzmann, you have to take a wider view. What we’ve shown in the plot are tiny fluctuations in entropy over a relatively short period of time. When we’re talking about the universe, we are obviously imagining a huge fluctuation in entropy that is very rare and takes an extremely long time to play itself out. The overall graph of the entropy of the universe looks something like Figure 54, but the entropy of our local, observable part of universe corresponds to only a very tiny piece of that graph—near the point marked x, where a fluctuation has occurred and is in the process of bouncing back to equilibrium. If the entire history of the known universe were to fit there, we would indeed see the Second Law at work over our lifetimes, while over ultra-long times the entropy is simply fluctuating near its maximum value.

But—you say again, not quite ready to give up—why should we live at that particular part of the curve, in the aftermath of a giant entropy fluctuation? We’ve already admitted that such fluctuations are exceedingly rare. Shouldn’t we find ourselves at a more typical period in the history of the universe, where things basically look like they are in equilibrium?

Boltzmann, naturally, has foreseen your objection. And at this point he makes a startlingly modern move—he invokes the anthropic principle. The anthropic principle is basically the idea that any sensible account of the universe around us must take into consideration the fact that we exist. It comes in various forms, from the uselessly weak—“the fact that life exists tell us that the laws of physics must be compatible with the existence of life”—to the ridiculously strong—“the laws of physics had to take the form they do because the existence of life is somehow a necessary feature.” Arguments over the status of the anthropic principle—Is it useful? Is it science?—grow quite heated and are rarely very enlightening.

Fortunately, we (and Boltzmann) need only a judicious medium-strength version of the anthropic principle. Namely, imagine that the real universe is much bigger (in space, or in time, or both) than the part we directly observe. And imagine further that different parts of this bigger universe exist in very different conditions. Perhaps the density of matter is different, or even something as dramatic as different local laws of physics. We can label each distinct region a “universe,” and the whole collection is the “multiverse.” The different universes within the multiverse may or may not be physically connected; for our present purposes it doesn’t matter. Finally, imagine that some of these different regions are hospitable to the existence of life, and some are not. (That part is inevitably a bit fuzzy, given how little we know about “life” in a wider context.) Then—and this part is pretty much unimpeachable—we will always find ourselves existing in one of the parts of the universe where life is allowed to exist, and not in the other parts. That sounds completely empty, but it’s not. It represents a selection effect that distorts our view of the universe as a whole—we don’t see the entire thing; we see only one of the parts, and that part might not be representative.

Boltzmann appeals to exactly this logic. He asks us to imagine a universe consisting of some collection of particles moving through an absolute Newtonian spacetime that exists for all eternity. What would happen?

There must then be in the universe, which is in thermal equilibrium as a whole and therefore dead, here and there relatively small regions of the size of our galaxy (which we call worlds), which during the relatively short time of eons deviate significantly from thermal equilibrium. Among these worlds the state probability [entropy] increases as often as it decreases. For the universe as a whole the two directions of time are indistinguishable, just as in space there is no up or down. However, just as at a certain place on the earth’s surface we can call “down” the direction toward the centre of the earth, so a living being that finds itself in such a world at a certain period of time can define the time direction as going from less probable to more probable states (the former will be the “past” and the latter the “future”) and by virtue of this definition he will find that this small region, isolated from the rest of the universe, is “initially” always in an improbable state.181

This is a remarkable paragraph, which would be right at home in a modern cosmology discussion, with just a few alterations in vocabulary. Boltzmann imagines that the universe (or the multiverse, if you prefer) is basically an infinitely big box of gas. Most of the time the gas is distributed uniformly through space, at constant temperature—thermal equilibrium. The thing is, we can’t live in thermal equilibrium—it’s “dead,” as he bluntly puts it. From time to time there will be random fluctuations, and eventually one of these will create something like the universe we see around us. (He refers to “our galaxy,” which at the time was synonymous with “the observable universe.”) It’s in those environments, the random fluctuations away from equilibrium, where we can possibly live, so it’s not much surprise that we find ourselves there.

Even in the course of a fluctuation, of course, the entropy is only increasing half the time—in the other half it’s decreasing, moving from equilibrium down to the minimum value it will temporarily reach. But this sense of “increasing” or “decreasing” describes the evolution of entropy with respect to some arbitrarily chosen time coordinate, which—as we discussed in the last chapter—is completely unobservable. As Boltzmann correctly points out, what matters is that the current universe is in the middle of a transition from a low-entropy state to one of thermal equilibrium. In the midst of such a transition, any living beings who pop up will always label the direction of lower entropy “the past,” and the direction of higher entropy “the future.”

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Figure 55: Boltzmann’s “multiverse.” Space is mostly a collection of particles in equilibrium, but there are occasional local fluctuations to low-entropy states. (Not at all to scale.) We live in the aftermath of one exceptionally large fluctuation.

This is a provocative picture of the universe. On large scales, matter is almost always in a dilute collection of gas at some temperature. But every so often, over the course of billions of years, a series of random accidents conspire to create pockets of anomalously low entropy, which then relax back to equilibrium. You, and I, and all the bustling activity we see around us, are epiphenomena riding the wave of entropy as it bounces back from a random excursion into a wildly improbable state.182

So what does a typical downward fluctuation in entropy look like? The answer, of course, is that it looks exactly like the time-reverse of a typical evolution from a low-entropy state back to a high-entropy one. The whole universe wouldn’t suddenly zoom from a thin gas of particles into a dense Big-Bang-like state in a matter of minutes; it would, most of the time, experience a series of unlikely accidents spread over billions of years, all of which would decrease the entropy just a little bit. Stars and galaxies would un-form, omelets would turn into eggs, objects in equilibrium would spontaneously develop substantial temperature gradients. All of these would be completely independent events, each individually unlikely, and the combination of all of them is fantastically unlikely. But if you truly have eternity to wait, even the most unlikely things will eventually happen.

SWERVING THROUGH ANTIQUITY

Boltzmann wasn’t actually the first to think along these lines, if we allow ourselves a little poetic license. Just as Boltzmann was concerned with understanding the world in terms of atoms, so were his predecessors in ancient Greece and Rome. Democritus (c. 400 B.C.E.) was the most famous atomist, but his teacher Leucippus was probably the first to propose the idea. They were materialists, who hoped to explain the world in terms of objects obeying rules, rather than being driven by an underlying “purpose.” In particular, they were interested in rising to the challenge raised by Parmenides, who argued that change was an illusion. The theory of unchanging atoms moving through a void was meant to account for the possibility of motion without imagining that something arises from nothing.

One challenge that the atomists of antiquity faced was to explain the messy complexity of the world around them. The basic tendency of atoms, they believed, was to fall downward in straight lines; that doesn’t make for a very interesting universe. It was left to the Greek thinker Epicurus (c. 300 B.C.E.) to propose a solution to this puzzle, in the form of an idea called “the swerve” (clinamen).183 Essentially Epicurus suggested that, in addition to the basic tendency of atoms to move along straight lines, there is a random component to their motion that occasionally kicks them from side to side. It’s vaguely reminiscent of modern quantum mechanics, although we shouldn’t get carried away. (Epicurus didn’t know anything about blackbody radiation, atomic spectra, the photoelectric effect, or any of the other experimental results motivating quantum mechanics.) Part of Epicurus’s reason for introducing the swerve was to leave room for free will—basically, to escape the implications of Laplace’s Demon, long before that mischievous beast had ever reared his ugly head. But another motivation was to explain how individual atoms could come together to form macroscopic objects, rather than just falling straight down to Earth.

The Roman poet-philosopher Lucretius (c. 50 B.C.E.) was an avid atomist and follower of Epicurus; he was a primary inspiration for Virgil’s poetry. His poem “On the Nature of Things (De Rerum Natura)” is a remarkable work, concerned with elucidating Epicurean philosophy and applying it to everything from cosmology to everyday life. He was especially interested in dispelling superstitious beliefs; imagine Carl Sagan writing in Latin hexameter. A famous section of “On the Nature of Things” counsels against the fear of death, which he sees as simply a transitional event in the endless play of atoms.

Lucretius applied atomism, and in particular the idea of the swerve, to the question of the origin of the universe. Here is what he imagines happening:

For surely the atoms did not hold council, assigning 
Order to each, flexing their keen minds with 
Questions of place and motion and who goes where. 
But shuffled and jumbled in many ways, in the course 
Of endless time they are buffeted, driven along, 
Chancing upon all motions, combinations. 
At last they fall into such an arrangement 
As would create this universe.184

The opening lines here should be read in a semi-sarcastic tone of voice. Lucretius is mocking the idea that the atoms somehow planned the cosmos; rather, they just jumbled around chaotically. But through those random motions, if we wait long enough we will witness the creation of our universe.

The resemblance to Boltzmann’s scenario is striking. We should always be careful, of course, not to credit ancient philosophers with a modern scientific understanding; they came from a very different perspective, and worked under a different set of presuppositions than we do today. But the parallelism between the creation scenarios suggested by Lucretius and Boltzmann is more than just a coincidence. In both cases, the task was to explain the emergence of the apparent complexity we see around us without appealing to an overall design, but simply by considering the essentially random motions of atoms. It is no surprise that a similar conclusion is reached: the idea that our observable universe is a random fluctuation in an eternal cosmos. It’s perfectly fair to call this the “Boltzmann-Lucretius scenario” for the origin of the universe.

Can the real world possibly be like that? Can we live in an eternal universe that spends most of its time in equilibrium, with occasional departures that look like what we see around us? Here we need to rely on the mathematical formalism developed by Boltzmann and his colleagues, to which Lucretius didn’t have recourse.

UN - BREAKING AN EGG

The problem with the Boltzmann-Lucretius scenario is not that you can’t make a universe that way—in the context of Newtonian spacetime, with everlasting atoms bumping against one another and occasionally giving rise to random downward fluctuations of entropy, it’s absolutely going to happen that you create a region of the size and shape of our universe if you wait long enough.

The problem is that the numbers don’t work. Sure, you can fluctuate into something that looks like our universe—but you can fluctuate into a lot of other things as well. And the other things win, by a wide margin.

Rather than weighing down our brains with the idea of a huge collection of particles fluctuating into something like the universe we see around us (or even just our galaxy), let’s keep things simple by considering one of our favorite examples of entropy in action: a single egg. An unbroken egg is quite orderly and has a very low entropy; if we break the egg the entropy increases, and if we whisk the ingredients together the entropy increases even more. The maximum-entropy state will be a soup of individual molecules; details of the configuration will depend on the temperature, the presence of a gravitational field, and so on, but none of that will matter for our present purposes. The point is that it won’t look anything like an unbroken egg.

Imagine we take such an egg and seal it in an absolutely impenetrable box, one that will last literally forever, undisturbed by the rest of the universe. For convenience, we put the egg-in-a-box out in space, far away from any gravity or external forces, and imagine that it floats undisturbed for all eternity. What happens inside that box?

Even if we initially put an unbroken egg inside the box, eventually it would break, just through the random motions of its molecules. It will spend some time as a motionless, broken egg, differentiated into yolk and white and shell. But if we wait long enough, further random motions will gradually cause the yolk and white and even the shell to disintegrate and mix, until we reach a truly high-entropy state of undifferentiated egg molecules. That’s equilibrium, and it will last an extraordinarily long time.

But if we continue to wait, the same kind of random motions that caused the egg to break in the first place will stir those molecules into lower-entropy configurations. All of the molecules may end up on one side of the box, for example. And after a very long time indeed, random motions will re-create something that looks like a broken egg (shell, yolk, and white), or even an unbroken egg! That seems preposterous, but it’s the straightforward implication of Poincaré’s recurrence theorem, or of taking seriously the ramifications of random fluctuations over extraordinarily long timescales.

Most of the time, the process of forming an egg through random fluctuations of the constituent molecules will look just like the time-reverse of the process by which an unbroken egg decays into high-entropy goop. That is, we will first fluctuate into the form of a broken egg, and then the broken pieces will by chance arrange themselves into the form of an unbroken egg. That’s just a consequence of time-reversal symmetry; the most common evolutions from high entropy to low entropy look exactly like the most common evolutions from low entropy to high entropy, just played in reverse.

Here is the rub. Let’s imagine that we have such an egg sealed in an impenetrable box, and we peek inside after it’s been left to its own devices for an absurdly long time—much greater than the recurrence time. It’s overwhelmingly likely that what we will see is something very much like equilibrium: a homogeneous mixture of egg molecules. But suppose we get extremely lucky, and we find what looks like a broken egg—a medium-entropy state, with some shards of eggshell and a yolk running into the egg whites. A configuration, in other words, that looks exactly what we would expect if there had recently been a pristine egg, which for some reason had been broken.

060

Figure 56: An egg trapped for all eternity in an impenetrable box. Most of the time the box will contain egg molecules in high-entropy equilibrium. Occasionally it will fluctuate into the medium-entropy configuration of a broken egg, as in the top row. Much more rarely, it will fluctuate all the way to the low-entropy form of an unbroken egg, and then back again, as in the bottom row.

Could we actually conclude, from this broken egg, that there had recently been an unbroken egg in the box? Not at all. Remember our discussion at the end of Chapter Eight. Given a medium-entropy configuration, and no other knowledge or assumptions such as a Past Hypothesis (which would clearly be inappropriate in the context of this ancient sealed box), it is overwhelmingly likely to have evolved from a higher-entropy past, just as it is overwhelmingly likely to evolve toward a higher-entropy future. Said conversely, given a broken egg, it is no more likely to have evolved from an unbroken egg than it is likely to evolve to an unbroken egg. Which is to say, not bloody likely.

BOLTZMANN BRAINS

The egg-in-a-box example illustrates the fundamental problem with the Boltzmann-Lucretius scenario: We can’t possibly appeal to a Past Hypothesis that asserts the existence of a low-entropy past state, because the universe (or the egg) simply cycles through every possible configuration it can have, with a predictable frequency. There is no such thing as an “initial condition” in a universe that lasts forever.

The idea that the universe spends most of its time in thermal equilibrium, but we can appeal to the anthropic principle to explain why our local environment isn’t in equilibrium, makes a strong prediction—and that prediction is dramatically falsified by the data. The prediction is simply that we should be as close to equilibrium as possible, given the requirement that we (under some suitable definition of “we”) be allowed to exist. Fluctuations happen, but large fluctuations (such as creating an unbroken egg) are much more rare than smaller fluctuations (such as creating a broken egg). We can see this explicitly back in Figure 54, where the curve exhibits many small fluctuations and only a few larger ones. And the universe we see around us would have to be a large fluctuation indeed.185

We can be more specific about what the universe would look like if it were an eternal system fluctuating around equilibrium. Boltzmann invoked the anthropic principle (although he didn’t call it that) to explain why we wouldn’t find ourselves in one of the very common equilibrium phases: In equilibrium, life cannot exist. Clearly, what we want to do is find the most common conditions within such a universe that are hospitable to life. Or, if we want to be a bit more careful, perhaps we should look for conditions that are not only hospitable to life, but hospitable to the particular kind of intelligent and self-aware life that we like to think we are.

Maybe this is a way out? Maybe, we might reason, in order for an advanced scientific civilization such as ours to arise, we require a “support system” in the form of an entire universe filled with stars and galaxies, originating in some sort of super-low-entropy early condition. Maybe that could explain why we find such a profligate universe around us.

No. Here is how the game should be played: You tell me the particular thing you insist must exist in the universe, for anthropic reasons. A solar system, a planet, a particular ecosystem, a type of complex life, the room you are sitting in now, whatever you like. And then we ask, “Given that requirement, what is the most likely state of the rest of the universe in the Boltzmann-Lucretius scenario, in addition to the particular thing we are asking for?”

And the answer is always the same: The most likely state of the rest of the universe is to be in equilibrium. If we ask, “What is the most likely way for an infinite box of gas in equilibrium to fluctuate into a state containing a pumpkin pie?,” the answer is “By fluctuating into a state that consists of a pumpkin pie floating by itself in an otherwise homogeneous box of gas.” Adding anything else to the picture, either in space or in time—an oven, a baker, a previously existing pumpkin patch—only makes the scenario less likely, because the entropy would have to dip lower to make that happen. By far the easiest way to get a pumpkin pie in this context is for it to gradually fluctuate all by itself out of the surrounding chaos.186

Sir Arthur Eddington, in a lecture from 1931, considered a perfectly reasonable anthropic criterion:

A universe containing mathematical physicists [under these assumptions] will at any assigned date be in the state of maximum disorganization which is not inconsistent with the existence of such creatures.187

Eddington presumes that what you really need to make a good universe is a mathematical physicist. Sadly, if the universe is an eternally fluctuating collection of molecules, the most frequently occurring mathematical physicists will be all by themselves, surrounded by randomness.

We can take this logic to its ultimate conclusion. If what we want is a single planet, we certainly don’t need a hundred billion galaxies with a hundred billion stars each. And if what we want is a single person, we certainly don’t need an entire planet. But if in fact what we want is a single intelligence, able to think about the world, we don’t even need an entire person—we just need his or her brain.

So the reductio ad absurdum of this scenario is that the overwhelming majority of intelligences in this multiverse will be lonely, disembodied brains, who fluctuate gradually out of the surrounding chaos and then gradually dissolve back into it. Such sad creatures have been dubbed “Boltzmann brains” by Andreas Albrecht and Lorenzo Sorbo.188 You and I are not Boltzmann brains—we are what one might call “ordinary observers,” who did not fluctuate all by ourselves from the surrounding equilibrium, but evolved gradually from an earlier state of very low entropy. So the hypothesis that our universe is a random fluctuation around an equilibrium state in an eternal spacetime seems to be falsified.

You may have been willing to go along with this line of reasoning when only an egg was involved, but draw up short when we start comparing the number of disembodied brains to the number of ordinary observers. But the logic is exactly the same, if (and it’s a big “if ”) we are considering an eternal universe full of randomly fluctuating particles. In such a universe, we know what kinds of fluctuations there are, and how often they happened; the more the entropy changes, the less likely that fluctuation will be. No matter how many ordinary observers exist in our universe today, they would be dwarfed by the total number of Boltzmann brains to come. Any given observer is a collection of particles in some particular state, and that state will occur infinitely often, and the number of times it will be surrounded by high-entropy chaos is enormously higher than the number of times it will arise as part of an “ordinary” universe.

Just to be careful, though—are you really sure you are not a Boltzmann brain? You might respond that you feel the rest of your body, you see other objects around you, and for that matter you have memories of a lower-entropy past: All things that would appear inconsistent with the idea that you are a disembodied brain recently fluctuated out of the surrounding molecules. The problem is, these purported statements about the outside world are actually just statements about your brain. Your feelings, visions, and memories are all contained within the state of your brain. We could certainly imagine that a brain with exactly these sensations fluctuated out of the surrounding chaos. And, as we have argued, it’s much more likely for that brain to fluctuate by itself than to be part of a giant universe. In the Boltzmann-Lucretius scenario, we don’t have recourse to the Past Hypothesis, so it is overwhelmingly likely that all of our memories are false.

Nevertheless, we are perfectly justified in dismissing this possibility, as long as we think carefully about what precisely we are claiming. It’s not right to say, “I know I am not a Boltzmann brain, so clearly the universe is not a random fluctuation.” The right thing to say is, “If I were a Boltzmann brain, there would be a strong prediction: Everything else about the universe should be in equilibrium. But it’s not. Therefore the universe is not a random fluctuation.” If we insist on being strongly skeptical, we might wonder whether not only our present mental states, but also all of the additional sensory data we are apparently accumulating, represent a random fluctuation rather than an accurate reconstruction of our surroundings. Strictly speaking, that certainly is possible, but it’s cognitively unstable in the same sense that we discussed in the last chapter. There is no sensible way to live and think and behave if that is the case, so there is no warrant for believing it. Better to accept the universe around us as it appears (for the most part) to be.

This point was put with characteristic clarity by Richard Feynman, in his famous Lectures on Physics:

[F]rom the hypothesis that the world is a fluctuation, all of the predictions are that if we look at a part of the world we have never seen before, we will find it mixed up, and not like the piece we just looked at. If our order were due to a fluctuation, we would not expect order anywhere but where we have just noticed it . . .

We therefore conclude that the universe is not a fluctuation, and that the order is a memory of conditions when things started. This is not to say that we understand the logic of it. For some reason, the universe at one time had a very low entropy for its energy content, and since then the entropy has increased. So that is the way toward the future. That is the origin of all irreversibility, that is what makes the processes of growth and decay, that makes us remember the past and not the future, remember the things which are closer to that moment in history of the universe when the order was higher than now, and why we are not able to remember things where the disorder is higher than now, which we call the future.189

WHO ARE WE IN THE MULTIVERSE?

There is one final loophole that must be closed before we completely shut the door on the Boltzmann-Lucretius scenario. Let’s accept the implications of conventional statistical mechanics, that small fluctuations in entropy happen much more frequently than large fluctuations, and that the overwhelming majority of intelligent observers in a universe fluctuating eternally around equilibrium will find themselves alone in an otherwise high-entropy environment, not evolving naturally from a prior configuration of enormously lower entropy.

One might ask: So what? Why should I be bothered that most observers (under any possible definition of “observers”) find themselves alone as freak fluctuations in a high-entropy background? All I care about is who I am, not what most observers are like. As long as there is one instance of the universe I see around me somewhere in the eternal lifetime of the larger world (which there will be), isn’t that all I need to proclaim that this picture is consistent with the data?

In other words, the Boltzmann-brain argument makes an implicit assumption: that we are somehow “typical observers” in the universe, and that therefore we should make predictions by asking what most observers would see.190That sounds innocuous, even humble. But upon closer inspection, it leads to conclusions that seem stronger than we can really justify.

Imagine we have two theories of the universe that are identical in every way, except that one predicts that an Earth-like planet orbiting the star Tau Ceti is home to a race of 10 trillion intelligent lizard beings, while the other theory predicts there are no intelligent beings of any kind in the Tau Ceti system. Most of us would say that we don’t currently have enough information to decide between these two theories. But if we are truly typical observers in the universe, the first theory strongly predicts that we are more likely to be lizards on the planet orbiting Tau Ceti, not humans here on Earth, just because there are so many more lizards than humans. But that prediction is not right, so we have apparently ruled out the existence of that many observers without collecting any data at all about what’s actually going on in the Tau Ceti system.

Assuming we are typical might seem like a self-effacing move on our part, but it actually amounts to an extremely strong claim about what happens throughout the rest of the universe. Not only “we are typical observers,” but “typical observers are like us.” Put that way, it seems like a stronger assumption than we have any right to make. (In the literature this is known as the “Presumptuous Philosopher Problem.”) So perhaps we shouldn’t be comparing the numbers of different kinds of observers in the universe at all; we should only ask whether a given theory predicts that observers like us appear somewhere, and if they do we should think of the theory as being consistent with the data. If that were the right way to think about it, we wouldn’t have any reason to reject the Boltzmann-Lucretius scenario. Even though most observers would be alone in the universe, some would find themselves in regions like ours, so the theory would be judged to be in agreement with our experience.191

The difficulty with this minimalist approach is that it offers us too little handle on what is likely to happen in the universe, instead of too much. Statistical mechanics relies on the Principle of Indifference—the assumption that all microstates consistent with our current macrostate are equally likely, at least when it comes to predicting the future. That’s essentially an assumption of typicality: Our microstate is likely to be a typical member of our macrostate. If we’re not allowed to assume that, all sorts of statistical reasoning suddenly become inaccessible. We can’t say that an ice cube is likely to melt in a glass of warm water, because in an eternal universe there will occasionally be times when the opposite happens. We seem to have taken our concerns about typicality too far.

Instead, we should aim for a judicious middle ground. It’s too much to ask that we are typical among all observers in the universe, because that’s making a strong claim about parts of the universe we’ve never observed. But we can at least say that we are typical among observers exactly like us—that is, observers with the basic physiology and the same set of memories that we have, the same coarse-grained experience of the universe.192 That assumption doesn’t allow us to draw any unwarranted conclusions about the possible existence of other kinds of intelligent beings elsewhere in the universe. But it is more than enough to rule out the Boltzmann-Lucretius scenario. If the universe fluctuates around thermal equilibrium for all eternity, not only will most observers appear all by themselves from the surrounding chaos, but the same is true for the subset of observers with precisely the features that you or I have—complete with our purported memories of the past. Those memories will generally be false, and fluctuating into them is very unlikely, but it’s still much more unlikely than fluctuating the entire universe. Even this minimal necessary condition for carrying out statistical reasoning—we should take ourselves to be chosen randomly from the set of observers exactly like us—is more than enough to put the Boltzmann-Lucretius scenario to rest.

The universe we observe is not a fluctuation—at least, to be more careful, a statistical fluctuation in an eternal universe that spends most of its time in equilibrium. So that’s what the universe is not; what it is, we still have to work out.

ENDINGS

On the evening of September 5, 1906, Ludwig Boltzmann took a piece of cord, tied it to a curtain rod in the hotel room where he was vacationing in Italy with his family, and hanged himself. His body was discovered by his daughter Emma when she returned to their room that evening. He was sixty-two years old.

The reasons for Boltzmann’s suicide remain unclear. Some have suggested that he was despondent over the failure of his ideas concerning atomic theory to gain wider acceptance. But, while many German-speaking scientists of the time remained skeptical about atoms, kinetic theory had become standard throughout much of the world, and Boltzmann’s status as a major scientist was unquestioned in Austria and Germany. Boltzmann had been suffering from health problems and was prone to fits of depression; it was not the first time he had attempted suicide.

But his depression was intermittent; only months before his death, he had written an engaging and high-spirited account of his previous year’s trip to America to lecture at the University of California at Berkeley, and circulated it among his friends. He referred to California as “Eldorado,” but found American water undrinkable, and would drink only beer and wine. This was problematic, as the Temperance movement was strong in America at the time, and Berkeley in particular was completely dry; a recurring theme in Boltzmann’s account is his attempts to smuggle wine into various forbidden places.193 We will probably never know what mixture of failing health, depression, and scientific controversy contributed to his ultimate act.

On the question of the existence of atoms and their utility in understanding the properties of macroscopic objects, any lingering doubts that Boltzmann was right were rapidly dissipating when he died. One of Albert Einstein’s papers in his “miraculous year” of 1905 was an explanation of Brownian motion (the seemingly random motion of small particles suspended in air) in terms of collisions with individual atoms; most remaining skepticism on the part of physicists was soon swept away.

Questions about the nature of entropy and the Second Law remain with us, of course. When it comes to explaining the low entropy of our early universe, we won’t ever be able to say, “Boltzmann was right,” because he suggested a number of different possibilities without ever settling on one in particular. But the terms of the debate were set by him, and we’re still arguing over the questions that puzzled him more than a century ago.