A Glimpse Into Boltzmann's Actual Brain

By Sean Carroll | May 4, 2007 12:03 am

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 Nature 51, 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.”

  • http://www.sunnykalara.com/blog Sunny

    Didn’t realize that Boltzmann was so explicit in his articulation of the universe as a “random fluctuation.” I thought people talked about universe as a fluctuation in the context of Boltzmann Equilibrium and thats where the name Boltzmann Brain originated.

    If this was 1895 and I was putting forward an idea like this, I would want to retain “plausible deniability” by attributing the idea to my trusted assistant!

    Andy’s “Schuetz’s Schmartz” part is funny; haven’t seen him since we shared an office at Los Alamos eons ago, probably at the same time when Boltzmann was working on these ideas!!!

  • http://hronir.blogspot.com hronir

    Since probability is a ratio, even if you can make the universe as bis as you want, the probability of a particular thermal fluctuation does not change! What can change is the number of actual realizations of such a fluctuation…
    Am I missing something?

  • http://hronir.blogspot.com hronir

    errata corrige: “as big as you want”

  • CaptainExcess

    As I understand it, what we care about is the probability of at least one interesting event occurring. Say that the universe is one coin that’s flipped. If it’s heads we’re at high entropy and if it’s tails something interesting (say, a Schuetz Schmart) happens. Then the probability that something interesting happens is 1 in 2. Suppose then that we make the universe bigger, to comprise 3 coins. The probability that the coin comes up tails at least once in 3 coin flips is 7/8. Because we care only about the occurrence of the event and not where it occurs this probability gets higher as we increase the number of locations in which the event could occur. The probability that something interesting occurs gets close to one as the size of the whole universe (the number of coins) gets large.

  • http://theeternaluniverse.blogspot.com Joseph Smidt

    Someone should write a new-age version of Macbeth where the “man not born of woman” is a Boltzmann Brain.

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  • Jason Dick

    The thing you may possibly be missing, hornir, is that the visible part, the part that we can actually see and be sure is far from equilibrium, may only be a tiny part of the whole. If the universe beyond our horizon is large enough, it doesn’t matter how unusual our universe is: the probability for some region like ours to exist among all possible regions will necessarily approach unity. It’s rather like CaptainExcess’ coin flipping analogy, but perhaps a more apt analogy would be rolling a die. Imagine that die rolls 2-6 are “typical”, but a die roll of 1 is unusual. If we are looking for die rolls of 1, and roll a die 100 times, then the probability that we will get this “unusual” die roll at least once is about 0.999999988.

  • andy.s

    A Boltzmann’s brain that maintains an illusion of a universe for 80 years and then dies is many thousands of orders of magnitude more likely than an entire universe.

    So any consciousness picked at random is infinitely more likely to be a Hallucinating Boltzmann’s Brain than a real universe.

    Damn. I think I just became a solipsist.

  • http://lablemminglounge.blogspot.com/ Lab Lemming

    “I will conclude this paper with the idea of my old assistant”…

    Isn’t that just the scientific version of the teenage classic, “Hi, I’m asking this [potentially embarrasing] question for my friend…”

  • http://backreaction.blogspot.com/ B

    Hi Sean, thanks for the link to the Monty Python video :-) I am still giggling about it (even though I probably saw the movie ten times or so). Haven’t yet succeeded in reading the rest (my attention span is approaching that of a BB I suspect).

    Have a nice weekend,


  • Jason Dick

    Well, andy.s, this doesn’t appear to be the case if the universe began with cosmic inflation. Since we have detected cosmic inflation in the early stages of our universe, the Boltzmann’s Brain argument for the non-existence of the universe doesn’t appear to hold water.

  • Jason Dick

    Whoops, forgot to mention: the Albrecht and Sorbo paper mentioned above goes into how inflation is more likely than you being a Boltmann’s Brain. Also, a little clarification: the actual idea doesn’t require the illusion of a universe for 80 years. You just need an instant and the memories of a coherent past.

  • http://tyrannogenius.blogspot.com Neil B.

    “The” universe, created by a thermal fluctation in what? There has to be some conceptual accountability, whatever physicists might want to imagine. One can’t just say, you are referring to the virtual fluctations of the same sort happening now in our existing space-time, since that is in effect supposed to have been created by something going on in… ? What sort of “realm” was that, and why could it have just been there, static I presume? Otherwise it would need to have a chance or rule of being created by something else. (BTW, for those who think that everthing is either lawful or probabilistic, just consider a number sequence with just one of the series changed to some other number….) In any case, once you have a chance of a universe, then there has to be many, as I argued previously:

    If “our universe” (implying perhaps the possibility of there being others….) somehow erupts from some sort of superspace or background: I won’t pass judgment on weird orobouros style self-looping schemes, which look fishy to me. However, if you imagine that in order for an “event” to happen, even one like the formation of a/the “universe,” then here has to be a probability of occurance within some range, even if you don’t want to call it just like time. (Otherwise, the pre-condition would just have to stay that way, wouldn’t it, unless something like “time” could operate to allow events….) Unless that background/mother reality etc. itself is confined or only existed for awhile, in which case it was created at some time-like moment, then any probability of any universe has to be played out over and over again – there can’t be just one. That doesn’t explain the principles behind the background, or prove that some universes would have different laws (i.e., ways for things to act) than others etc., or answer the modal realists’ attacks on the very idea of what “existing” means in distinguishing some possible realities from others, or the problems thereof, etc.

  • http://whenindoubtdo.blogspot.com/ Eugene

    Interesting. I always thought that this paper was a bit off the wall, and I went “who is Schuetz??” :


    The Thermodynamical Arrow of Time: Reinterpreting the Boltzmann-Schuetz Argument

    Cirkovic, Milan M.

    He did cite Boltzy’s 1895 nature article, so I figured that’s he accurately attributed it to Schuetz.

  • Paul Valletta

    The Universe cannot come from an initially High Entropic State, a H.E.S, will “NOT” have available future energy?..thus the arrow of time would cease to point from a past state to future state. A reverted Arrow of Time would tend to move from a Cold state(present time) to Hotter state (past time), the thermometer would be running “up”?, thermodynamical arrow takes more energy to point in the reverted/reverse direction.

    The Universe takes a measured time to arrive at a specific temperature of the background energy detected today, if one was to reverse this process, there would be a definate difference in timescales to arrive at the initial moment of the Big Bang, the reason for this would be due to the fact it takes MORE energy to go from cold bodies >> hot bodies, and the thermodynamic arrow of time, if reverted, would fall outside the specific timescales, its quicker for heat to transfer to colder bodies, than it is for heat to be converted from cloder bodies to hotter bodies.

    Thus again to reverse the thermodynamic arrow in time, would mean there has to be a pool of available hidden “extra” Energy lurking somewhere in the Universe.

    The conversion rate to exchange available energies from a Hot body (past tense) to a Colder body (present tense) would differ vastly from that of a Cold body to a hotter body, due to the fact the Universe was Hotter initially, and it will become Colder, eventually?

    In the future, the thermodynamic arrow of spacetime will have a lot of available “extra” cold dark energy, this will assist the Universe to transist out from heat death scenario..in a simple process, there will be light!

  • http://hronir.blogspot.com hronir

    @CaptainExcess: I don’t think that what we care about is the probability of at least one interesting event occurring. Otherwise we could simply care about the possibility of such an event occurring, and this take us back to the “original” Boltzmann’s argument. In fact, following your argument, for any rare-event we could make the universe as big as we want so that its probability can approach one as well as we want. But this simply means that, given a big enough universe, anything could happen.On the contrary, what the Boltzmann’s Brain argument points out is that we have to care about the Bayesian probability, i. e. we have to care about the most probable past that can take us to the present we see. Even if we have a bigger universe, such that the chance of get our present scene is significant, still our present scene would be accurately reproduced maaaaaany many many more times in a fake manner a’ la single glancing Boltzmann soul.

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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] cosmicvariance.com .


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