Boltzmann's Universe

By Sean Carroll | January 14, 2008 11:32 pm

Boltzmann’s Brain CV readers, ahead of the curve as usual, are well aware of the notion of Boltzmann’s Brains — see e.g. here, here, and even the original paper here. Now Dennis Overbye has brought the idea to the hoi polloi by way of the New York Times. It’s a good article, but I wanted to emphasize something Dennis says quite explicitly, but (from experience) I know that people tend to jump right past in their enthusiasm:

Nobody in the field believes that this is the way things really work, however.

The point about Boltzmann’s Brains is not that they are a fascinating prediction of an exciting new picture of the multiverse. On the contrary, the point is that they constitute a reductio ad absurdum that is meant to show the silliness of a certain kind of cosmology — one in which the low-entropy universe we see is a statistical fluctuation around an equilibrium state of maximal entropy. According to this argument, in such a universe you would see every kind of statistical fluctuation, and small fluctuations in entropy would be enormously more frequent than large fluctuations. Our universe is a very large fluctuation (see previous post!) but a single brain would only require a relatively small fluctuation. In the set of all such fluctuations, some brains would be embedded in universes like ours, but an enormously larger number would be all by themselves. This theory, therefore, predicts that a typical conscious observer is overwhelmingly likely to be such a brain. But we (or at least I, not sure about you) are not individual Boltzmann brains. So the prediction has been falsified, and that kind of theory is not true. (For arguments along these lines, see papers by Dyson, Kleban, and Susskind, or Albrecht and Sorbo.)

I tend to find this kind of argument fairly persuasive. But the bit about “a typical observer” does raise red flags. In fact, folks like Hartle and Srednicki have explicitly argued that the assumption of our own “typicality” is completely unwarranted. Imagine, they say, two theories of life in the universe, which are basically indistinguishable, except that in one theory there is no life on Jupiter and in the other theory the Jovian atmosphere is inhabited by six trillion intelligent floating Saganite organisms.

In the second theory, a “typical” intelligent observer in the Solar System is a Jovian, not a human. But I’m a human. Have we therefore ruled out this theory? Pretty clearly not. Hartle and Srednicki conclude that it’s incorrect to imagine that we are necessarily typical; we are who we observe ourselves to be, and any theory of the universe that is compatible with observers like ourselves is just as good as any other such theory.

This is an interesting perspective, and the argument is ongoing. But it’s important to recognize that there is a much stronger argument against the idea that Boltzmann’s Brains were originally invented to counter — that our universe is just a statistical fluctuation around an equilibrium background. We might call this the “Boltzmann’s Universe” argument.

Here’s how it goes. Forget that we are “typical” or any such thing. Take for granted that we are exactly who we are — in other words, that the macrostate of the universe is exactly what it appears to be, with all the stars and galaxies etc. By the “macrostate of the universe,” we mean everything we can observe about it, but not the precise position and momentum of every atom and photon. Now, you might be tempted to think that you reliably know something about the past history of our local universe — your first kiss, the French Revolution, the formation of the cosmic microwave background, etc. But you don’t really know those things — you reconstruct them from your records and memories right here and now, using some basic rules of thumb and your belief in certain laws of physics.

The point is that, within this hypothetical thermal equilibrium universe from which we are purportedly a fluctuation, there are many fluctuations that reach exactly this macrostate — one with a hundred billion galaxies, a Solar System just like ours, and a person just like you with exactly the memories you have. And in the hugely overwhelming majority of them, all of your memories and reconstructions of the past are false. In almost every fluctuation that creates universes like the ones we see, both the past and the future have a higher entropy than the present — downward fluctuations in entropy are unlikely, and the larger the fluctuation the more unlikely it is, so the vast majority of fluctuations to any particular low-entropy configuration never go lower than that.

Therefore, this hypothesis — that our universe, complete with all of our records and memories, is a thermal fluctuation around a thermal equilibrium state — makes a very strong prediction: that our past is nothing like what we reconstruct it to be, but rather that all of our memories and records are simply statistical flukes created by an unlikely conspiracy of random motions. In this view, the photograph you see before you used to be yellow and wrinkled, and before that was just a dispersed collection of dust, before miraculously forming itself out of the chaos.

Note that this scenario makes no assumptions about our typicality — it assumes, to the contrary, that we are exactly who we (presently) perceive ourselves to be, no more and no less. But in this scenario, we have absolutely no right to trust any of our memories or reconstructions of the past; they are all just a mirage. And the assumptions that we make to derive that conclusion are exactly the assumptions we really do make to do conventional statistical mechanics! Boltzmann taught us long ago that it’s possible for heat to flow from cold objects to hot ones, or for cream to spontaneously segregate itself away from a surrounding cup of coffee — it’s just very unlikely. But when we say “unlikely” we have in mind some measure on the space of possibilities. And it’s exactly that assumed measure that would lead us to conclude, in this crazy fluctuation-world, that all of our notions of the past are chimeric.

Now, just like Boltzmann’s Brain, nobody believes this is true. In fact, you can’t believe it’s true, by any right. All of the logic you used to tell that story, and all of your ideas about the laws of physics, depend on your ability to reliably reconstruct the past. This scenario, in other words, is cognitively unstable; useful as a rebuke to the original hypothesis, but not something that can stand on its own.

So what are we to conclude? That our observed universe is not a statistical fluctuation around a thermal equilibrium state. That’s very important to know, but doesn’t pin down the truth. If the universe is eternal, and has a maximum value for its entropy, then we it would (almost always) be in thermal equilibrium. Therefore, either it’s not eternal, or there is no state of maximum entropy. I personally believe the latter, but there’s plenty of work to be done before we have any of this pinned down.


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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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