From Eternity to Here (39 page)

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.