The Sciences

A Glimpse Into Boltzmann's Actual Brain

Cosmic VarianceBy Sean CarrollMay 4, 2007 4:03 AM


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You've heard the "Boltzmann's Brain" argument (here and here, for example). It's a simple idea, which is put forward as an argument against the notion that our universe is just a thermal fluctuation. If the universe is an ordinary thermodynamic system in equilibrium, there will be occasional fluctuations into low-entropy states. One of these might look like the Big Bang, and you might be tempted to conclude that such a process explains the arrow of time in our universe. But it doesn't work, because you don't need anything like such a huge fluctuation. There will be many smaller fluctuations that do just as well; the minimal one you might imagine would be a single brain-sized collection of particles that just has time to look around and go Aaaaaagggghhhhhhh before dissolving back into equilibrium. (These days a related argument is being thrown around in the context of eternal inflation -- not exactly the same, because we're not assuming the ensemble is in equilibrium, but similar in spirit.) Boltzmann wasn't the one to come up with the "brain" argument; I'm not sure who did, but I first heard it articulated clearly in a paper by Albrecht and Sorbo. It's the maybe-our-universe-is-a-fluctuation idea that goes back to Boltzmann. Except it's not actually his, as we can see by looking at Boltzmann's original paper! (pdf) The reference is Nature51, 413 (1895), as tracked down by Alex Vilenkin. Don Page copied it from a crumbling leather-bound volume in his local library, and the copy was scanned in by Andy Albrecht. The discussion is just a few paragraphs at the very end of a short paper.

I will conclude this paper with an idea of my old assistant, Dr. Schuetz. We assume that the whole universe is, and rests for ever, in thermal equilibrium. The probability that one (only one) part of the universe is in a certain state, is the smaller the further this state is from thermal equilibrium; but this probability is greater, the greater is the universe itself. If we assume the universe great enough, we can make the probability of one relatively small part being in any given state (however far from the state of thermal equilibrium), as great as we please. We can also make the probability great that, though the whole universe is in thermal equilibrium, our world is in its present state. It may be said that the world is so far from thermal equilibrium that we cannot imagine the improbability of such a state. But can we imagine, on the other side, how small a part of the whole universe this world is? Assuming the universe great enough, the probability that such a small part of it as our world should be in its present state, is no longer small. If this assumption were correct, our world would return more and more to thermal equilibrium; but because the whole universe is so great, it might be probable that at some future time some other world might deviate as far from thermal equilibrium as our world does at present. Then the afore-mentioned H-curve would form a representation of what takes place in the universe. The summits of the curve would represent the worlds where visible motion and life exist.

So even Boltzmann doesn't want credit for the idea, which he attributes to his old assistant. Andy Albrecht points out that, in order to preserve the all-important alliteration, perhaps we should be calling them "Schuetz's Schmartz."

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