A Universe Out of Chaos

How did the universe come to be? We don’t know yet, of course, but we know enough about cosmology, gravitation, and quantum mechanics to put together models that standing a fighting chance of capturing some of the truth.

Stephen Hawking‘s favorite idea is that the universe came out of “nothing” — it arose (although that’s not really the right word) as a quantum fluctuation with literally no pre-existing state. No space, no time, no anything. But there’s another idea that’s at least as plausible: that the universe arose out of something, but that “something” was simply “chaos,” whatever that means in the context of quantum gravity. Space, time, and energy, yes; but no order, no particular arrangement.

It’s an old idea, going back at least to Lucretius, and contemplated by David Hume as well as by Ludwig Boltzmann. None of those guys, of course, knew very much of our modern understanding of cosmology, gravitation, and quantum mechanics. So what would the modern version look like?

That’s the question that Anthony Aguirre, Matt Johnson and I tackled in a paper that just appeared on arxiv. (Both of my collaborators have also been guest-bloggers here at CV.)

Out of equilibrium: understanding cosmological evolution to lower-entropy states
Anthony Aguirre, Sean M. Carroll, Matthew C. Johnson

Despite the importance of the Second Law of Thermodynamics, it is not absolute. Statistical mechanics implies that, given sufficient time, systems near equilibrium will spontaneously fluctuate into lower-entropy states, locally reversing the thermodynamic arrow of time. We study the time development of such fluctuations, especially the very large fluctuations relevant to cosmology. Under fairly general assumptions, the most likely history of a fluctuation out of equilibrium is simply the CPT conjugate of the most likely way a system relaxes back to equilibrium. We use this idea to elucidate the spacetime structure of various fluctuations in (stable and metastable) de Sitter space and thermal anti-de Sitter space.

It was Boltzmann who long ago realized that the Second Law, which says that the entropy of a closed system never decreases, isn’t quite an absolute “law.” It’s just a statement of overwhelming probability: there are so many more ways to be high-entropy (chaotic, disorderly) than to be low-entropy (arranged, orderly) that almost anything a system might do will move it toward higher entropy. But not absolutely anything; we can imagine very, very unlikely events in which entropy actually goes down.

In fact we can do better than just imagine: this has been observed in the lab. The likelihood that entropy will increase rather than decrease goes up as you consider larger and larger systems. So if you want to do an experiment that is likely to observe such a thing, you want to work with just a handful of particles, which is what experimenters succeeded in doing in 2002. But Boltzmann teaches us than any system, no matter how large, will eventually fluctuate into a lower-entropy state if we wait long enough. So what if we wait forever?

It’s possible that we can’t wait forever, of course; maybe the universe spends only a finite time in a lively condition like we see around us, before settling down to a truly stable equilibrium that never fluctuates. But as far as we currently know, it’s equally reasonable to imagine that it does last forever, and that it is always fluctuating. This is a long story, but a universe dominated by a positive cosmological constant (dark energy that never fades away) behaves a lot like a box of gas at a fixed temperature. Our universe seems to be headed in that direction; if it stays there, we will have fluctuations for all eternity.

Which means that empty space will eventually fluctuate into — well, anything at all, really. Including an entire universe.

This basic story has been known for some time. What Anthony and Matt and I have tried to add is a relatively detailed story of how such a fluctuation actually proceeds — what happens along the way from complete chaos (empty space with vacuum energy) to something organized like a universe. Our answer is simple: the most likely way to go from high-entropy chaos to low-entropy order is exactly like the usual way that systems evolve from low entropy to high-, just played backward in time.

Here is an excerpt from the paper:

The key argument we wish to explore in this paper can be illustrated by a simple example. Consider an ice cube in a glass of water. For thought-experiment purposes, imagine that the glass of water is absolutely isolated from the rest of the universe, lasts for an infinitely long time, and we ignore gravity. Conventional thermodynamics predicts that the ice cube will melt, and in a matter of several minutes we will have a somewhat colder glass of water. But if we wait long enough … statistical mechanics predicts that the ice cube will eventually re-form. If we were to see such a miraculous occurrence, the central claim of this paper is that the time evolution of the process of re-formation of the ice cube will, with high probability, be roughly equivalent to the time-reversal of the process by which it originally melted. (For a related popular-level discussion see From Eternity to Here, ch. 10.) The ice cube will not suddenly reappear, but will gradually emerge over a matter of minutes via unmelting. We would observe, therefore, a series of consecutive statistically unlikely events, rather than one instantaneous very unlikely event. The argument for this conclusion is based on conventional statistical mechanics, with the novel ingredient that we impose a future boundary condition — an unmelted ice cube — instead of a more conventional past boundary condition.

Let’s imagine that you want to wait long enough to see something like the Big Bang fluctuate randomly out of empty space. How will it actually transpire? It will not be a sudden WHAM! in which nothingness turns into the Big Bang. Rather, it will be just like the observed history of our universe — just played backward. A collection of long-wavelength photons will gradually come together; radiation will focus on certain locations in space to create white holes; those white holes will spit out gas and dust that will form into stars and planets; radiation will focus on the stars, which will break down heavy elements into lighter ones; eventually all the matter will disperse as it contracts and smooths out to create a giant Big Crunch. Along the way people will un-die, grow younger, and be un-born; omelets will convert into eggs; artists will painstakingly remove paint from their canvases onto brushes.

Now you might think: that’s really unlikely. And so it is! But that’s because fluctuating into the Big Bang is tremendously unlikely. What we argue in the paper is simply that, once you insist that you are going to examine histories of the universe that start with high-entropy empty space and end with a low-entropy Bang, the most likely way to get there is via an incredible sequence of individually unlikely events. Of course, for every one time this actually happens, there will be countless times that it almost happens, but not quite. The point is that we have infinitely long to wait — eventually the thing we’re waiting for will come to pass.

And so what?, you may very rightly ask. Well for one thing, modern cosmologists often imagine enormously long-lived universes, and events like this will be part of them, so they should be understood. More concretely, we are of course all interested in understanding why our actual universe really does have a low-entropy boundary condition at one end of time (the end we conventionally refer to as “the beginning”). There’s nothing in the laws of physics that distinguishes between the crazy story of the fluctuation into the Big Crunch and the perfectly ordinary story of evolving away from the Big Bang; one is the time-reverse of the other, and the fundamental laws of physics don’t pick out a direction of time. So we might wonder whether processes like these help explain the universe in which we actually live.

So far — not really. If anything, our work drives home (yet again!) how really unusual it is to get a universe that passes through such a low-entropy state. So that puzzle is still there. But if we’re ever going to solve it, it will behoove us to understand how entropy works as well as we can. Hopefully this paper is a step in that direction.