Boltzmann’s Anthropic Brain

A recent post of Jen-Luc’s reminded me of Huw Price and his work on temporal asymmetry. The problem of the arrow of time — why is the past different from the future, or equivalently, why was the entropy in the early universe so much smaller than it could have been? — has attracted physicists’ attention (although not as much as it might have) ever since Boltzmann explained the statistical origin of entropy over a hundred years ago. It’s a deceptively easy problem to state, and correspondingly difficult to address, largely because the difference between the past and the future is so deeply ingrained in our understanding of the world that it’s too easy to beg the question by somehow assuming temporal asymmetry in one’s purported explanation thereof. Price, an Australian philosopher of science, has made a specialty of uncovering the hidden assumptions in the work of numerous cosmologists on the problem. Boltzmann himself managed to avoid such pitfalls, proposing an origin for the arrow of time that did not secretly assume any sort of temporal asymmetry. He did, however, invoke the anthropic principle — probably one of the earliest examples of the use of anthropic reasoning to help explain a purportedly-finely-tuned feature of our observable universe. But Boltzmann’s anthropic explanation for the arrow of time does not, as it turns out, actually work, and it provides an interesting cautionary tale for modern physicists who are tempted to travel down that same road.

The Second Law of Thermodynamics — the entropy of a closed system will not spontaneously decrease — was understood well before Boltzmann. But it was a phenomenological statement about the behavior of gasses, lacking a deeper interpretation in terms of the microscopic behavior of matter. That’s what Boltzmann provided. Pre-Boltzmann, entropy was thought of as a measure of the uselessness of arrangements of energy. If all of the gas in a certain box happens to be located in one half of the box, we can extract useful work from it by letting it leak into the other half — that’s low entropy. If the gas is already spread uniformly throughout the box, anything we could do to it would cost us energy — that’s high entropy. The Second Law tells us that the universe is winding down to a state of maximum uselessness.

Ludwig Boltzmann Boltzmann suggested that the entropy was really counting the number of ways we could arrange the components of a system (atoms or whatever) so that it really didn’t matter. That is, the number of different microscopic states that were macroscopically indistinguishable. (If you’re worried that “indistinguishable” is in the eye of the beholder, you have every right to be, but that’s a separate puzzle.) There are far fewer ways for the molecules of air in a box to arrange themselves exclusively on one side than there are for the molecules to spread out throughout the entire volume; the entropy is therefore much higher in the latter case than the former. With this understanding, Boltzmann was able to “derive” the Second Law in a statistical sense — roughly, there are simply far more ways to be high-entropy than to be low-entropy, so it’s no surprise that low-entropy states will spontaneously evolve into high-entropy ones, but not vice-versa. (Promoting this sensible statement into a rigorous result is a lot harder than it looks, and debates about Boltzmann’s H-theorem continue merrily to this day.)

Boltzmann’s understanding led to both a deep puzzle and an unexpected consequence. The microscopic definition explained why entropy would tend to increase, but didn’t offer any insight into why it was so low in the first place. Suddenly, a thermodynamics problem became a puzzle for cosmology: why did the early universe have such a low entropy? Over and over, physicists have proposed one or another argument for why a low-entropy initial condition is somehow “natural” at early times. Of course, the definition of “early” is “low-entropy”! That is, given a change in entropy from one end of time to the other, we would always define the direction of lower entropy to be the past, and higher entropy to be the future. (Another fascinating but separate issue — the process of “remembering” involves establishing correlations that inevitably increase the entropy, so the direction of time that we remember [and therefore label “the past”] is always the lower-entropy direction.) The real puzzle is why there is such a change — why are conditions at one end of time so dramatically different from those at the other? If we do not assume temporal asymmetry a priori, it is impossible in principle to answer this question by suggesting why a certain initial condition is “natural” — without temporal aymmetry, the same condition would be equally natural at late times. Nevertheless, very smart people make this mistake over and over, leading Price to emphasize what he calls the Double Standard Principle: any purportedly natural initial condition for the universe would be equally natural as a final condition.

The unexpected consequence of Boltzmann’s microscopic definition of entropy is that the Second Law is not iron-clad — it only holds statistically. In a box filled with uniformly-distributed air molecules, random motions will occasionally (although very rarely) bring them all to one side of the box. It is a traditional undergraduate physics problem to calculate how often this is likely to happen in a typical classroom-sized box; reasurringly, the air is likely to be nice and uniform for a period much much much longer than the age of the observable universe.

Faced with the deep puzzle of why the early universe had a low entropy, Boltzmann hit on the bright idea of taking advantage of the statistical nature of the Second Law. Instead of a box of gas, think of the whole universe. Imagine that it is in thermal equilibrium, the state in which the entropy is as large as possible. By construction the entropy can’t possibly increase, but it will tend to fluctuate, every so often diminishing just a bit and then returning to its maximum. We can even calculate how likely the fluctuations are; larger downward fluctuations of the entropy are much (exponentially) less likely than smaller ones. But eventually every kind of fluctuation will happen.

Entropy Fluctuations

You can see where this is going: maybe our universe is in the midst of a fluctuation away from its typical state of equilibrium. The low entropy of the early universe, in other words, might just be a statistical accident, the kind of thing that happens every now and then. On the diagram, we are imagining that we live either at point A or point B, in the midst of the entropy evolving between a small value and its maximum. It’s worth emphasizing that A and B are utterly indistinguishable. People living in A would call the direction to the left on the diagram “the past,” since that’s the region of lower entropy; people living at B, meanwhile, would call the direction to the right “the past.”

During the overwhelming majority of such a universe’s history, there is no entropy gradient at all — everything just sits there in a tranquil equilibrium. So why should we find ourselves living in those extremely rare bits where things are evolving through a fluctuation? The same reason why we find ourselves living in a relatively pleasant planetary atmosphere, rather than the forbiddingly dilute cold of intergalactic space, even though there’s much more of the latter than the former — because that’s where we can live. Here Boltzmann makes an unambiguously anthropic move. There exists, he posits, a much bigger universe than we can see; a multiverse, if you will, although it extends through time rather than in pockets scattered through space. Much of that universe is inhospitable to life, in a very basic way that doesn’t depend on the neutron-proton mass difference or other minutiae of particle physics. Nothing worthy of being called “life” can possibly exist in thermal equilibrium, where conditions are thoroughly static and boring. Life requires motion and evolution, riding the wave of increasing entropy. But, Boltzmann reasons, because of occasional fluctuations there will always be some points in time where the entropy is temporarily evolving (there is an entropy gradient), allowing for the existence of life — we can live there, and that’s what matters.

Here is where, like it or not, we have to think carefully about what anthropic reasoning can and cannot buy us. On the one hand, Boltzmann’s fluctuations of entropy around equilibrium allow for the existence of dynamical regions, where the entropy is (just by chance) in the midst of evolving to or from a low-entropy minimum. And we could certainly live in one of those regions — nothing problematic about that. The fact that we can’t directly see the far past (before the big bang) or the far future in such a scenario seems to me to be quite beside the point. There is almost certainly a lot of universe out there that we can’t see; light moves at a finite speed, and the surface of last scattering is opaque, so there is literally a screen around us past which we can’t see. Maybe all of the unobserved universe is just like the observed bit, but maybe not; it would seem the height of hubris to assume that everything we don’t see must be just like what we do. Boltzmann’s goal is perfectly reasonable: to describe a history of the universe on ultra-large scales that is on the one hand perfectly natural and not finely-tuned, and on the other features patches that look just like what we see.

But, having taken a bite of the apple, we have no choice but to swallow. If the only thing that one’s multiverse does is to allow for regions that resemble our observed universe, we haven’t accomplished anything; it would have been just as sensible to simply posit that our universe looks the way it does, and that’s the end of it. We haven’t truly explained any of the features we observed, simply provided a context in which they can exist; but it would have been just as acceptable to say “that’s the way it is” and stop there. If the anthropic move is to be meaningful, we have to go further, and explain why within this ensemble it makes sense to observe the conditions we do. In other words, we have to make some conditional predictions: given that our observable universe exhibits property X (like “substantial entropy gradient”), what other properties Y should we expect to measure, given the characteristics of the ensemble as a whole?

And this is where Boltzmann’s program crashes and burns. (In a way that is ominous for similar attempts to understand the cosmological constant, but that’s for another day.) Let’s posit that the universe is typically in thermal equilibrium, with occasional fluctuations down to low-entropy states, and that we live in the midst of one of those fluctuations because that’s the only place hospitable to life. What follows?

The most basic problem has been colorfully labeled “Boltzmann’s Brain” by Albrecht and Sorbo. Remember that the low-entropy fluctuations we are talking about are incredibly rare, and the lower the entropy goes, the rarer they are. If it almost never happens that the air molecules in a room all randomly zip to one half, it is just as unlikely (although still inevitable, given enough time) that, given that they did end up in half, they will continue on to collect in one quarter of the room. On the diagram above, points like C are overwhelmingly more common than points like A or B. So if we are explaining our low-entropy universe by appealing to the anthropic criterion that it must be possible for intelligent life to exist, quite a strong prediction follows: we should find ourselves in the minimum possible entropy fluctuation consistent with life’s existence.

And that minimum fluctuation would be a “Boltzmann Brain.” Out of the background thermal equilibrium, a fluctuation randomly appears that collects some degrees of freedom into the form of a conscious brain, with just enough sensory apparatus to look around and say “Hey! I exist!”, before dissolving back into the equilibrated ooze.

You might object that such a fluctuation is very rare, and indeed it is. But so would be a fluctuation into our whole universe — in fact, quite a bit more rare. The momentary decrease in entropy required to produce such a brain is fantastically less than that required to make our whole universe. Within the infinite ensemble envisioned by Boltzmann, the overwhelming majority of brains will find themselves disembodied and alone, not happily ensconsed in a warm and welcoming universe filled with other souls. (You know, like ours.)

This is the general thrust of argument with which many anthropic claims run into trouble. Our observed universe has something like a hundred billion galaxies with something like a hundred billion stars each. That’s an extremely expansive and profligate universe, if its features are constrained solely by the demand that we exist. Very roughly speaking, anthropic arguments would be more persuasive if our universe was minimally constructed to allow for our existence; e.g. if the vacuum energy were small enough to allow for a single galaxy to arise out of a really rare density fluctuation. Instead we have a hundred billion such galaxies, not to count all of those outside our Hubble radius — an embarassment of riches, really.

But, returning to Boltzmann, it gets worse, in an interesting and profound way. Let’s put aside the Brain argument for a moment, and insist for some reason that our universe did fluctuate somehow into the kind of state in which we currently find ourselves. That is, here we are, with all of our knowledge of the past, and our observations indicating a certain history of the observable cosmos. But, to be fair, we don’t have detailed knowledge of the microstate corresponding to this universe — the position and momentum of each and every particle within our past light cone. Rather, we know some gross features of the macrostate, in which individual atoms can be safely re-arranged without our noticing anything.

Now we can ask: assuming that we got to this macrostate via some fluctuation out of thermal equilibrium, what kind of trajectory is likely to have gotten us here? Sure, we think that the universe was smaller and smoother in the past, galaxies evolved gradually from tiny density perturbations, etc. But what we actually have access to are the positions and momenta of the photons that are currently reaching our telescopes. And the fact is, given all of the possible past histories of the universe consistent with those photons reaching us, in the vast majority of them the impression that we are observing an even-lower-entropy past is an accident. If all pasts consistent with our current macrostate are equally likely, there are many more in which the past was a chaotic mess, in which a vast conspiracy gave rise to our false impression that the past was orderly. In other words, if we ask “What kind of early universe tends to naturally evolve into what we see?”, the answer is the ordinary smooth and low-entropy Big Bang. But here we are asking “What do most of the states that could possibly evolve into our current universe look like?”, and the answer there is a chaotic high-entropy mess.

Of course, nobody in their right minds believes that we really did pop out of a chaotic mess into a finely-tuned state with false memories about the Big Bang (although young-Earth creationists do believe that things were arranged by God to trick us into thinking that the universe is much older than it really is, which seems about as plausible). We assume instead that our apparent memories are basically reliable, which is a necessary assumption to make sensible statements of any form. Boltzmann’s scenario just doesn’t quite fit together, unfortunately.

Price’s conclusion from all this (pdf) is that we should take seriously the Gold universe, in which there is a low-entropy future collapsing state that mirrors our low-entropy Big Bang in the past. It’s an uncomfortable answer, as nobody knows any reason why there should be low-entropy boundary conditions in both the past and the future, which would involve an absurd amount of fine-tuning of our particular microstate at every instant of time. (Not to mention that the universe shows no sign of wanting to recollapse.) The loophole that Price and many other people (quite understandably) overlook is that the Big Bang need not be the true beginning of the universe. If the Bang was a localized baby universe in a larger background spacetime, as Jennie Chen and I have suggested (paper here), we can comply with the Double Standard Princple by having high-entropy conditions in both the far past and the far future. That doesn’t mean that we have completely avoided the problem that doomed Boltzmann’s idea; it is still necessary to show that baby universes would most often look like what we see around us, rather than (for example) much smaller spaces with only one galaxy each. And this whole “baby universe” idea is, shall we say, a mite speculative. But explaining the difference in entropy between the past and future is at least as fundamental, if not more so, as explaining the horizon and flatness problems with which cosmologists are so enamored. If we’re going to presume to talk sensibly and scientifically about the entire history of the universe, we have to take Boltzmann’s legacy seriously.

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102 Responses to Boltzmann’s Anthropic Brain

  1. Thomas S. says:

    But how many of Boltzmann’s brains worry about the anthropic principle?

  2. Aaron Bergman says:

    I’ve never been convinced that there’s a sensible definition of “the entropy of the universe”, but I’ve never gotten very far with that argument with anyone.

  3. missVolare says:

    Yeah, it sort of seems like shorthand for “variables we don’t know exactly how to manipulate” go here…a cosmic trashcan that appears to empty!

    why can’t we have both low and high entropic conditions at the same time? i dunno…

    all we’re doing is saying that the timespacewave function is only a portion of the resonance of the multiverses, and you are positing static variables in a dynamic equilibrium function. DNC

  4. Allyson says:

    I don’t understand anything in this post. I think whatever fist-sized hunk of brain that is used for understanding this post is either a) missing from my head, or b) holding all of my excitement for Snakes On a Plane, as well as my hatred for the Green Lantern.

  5. John Branch says:

    While reading Sean’s post, I had a thought similar to Aaron Bergman’s. I can imagine what it means for all the gas molecules in a box to be evenly distributed, or for them to be in one half of a box. I’m not sure what it would mean for the universe to be in a state of thermal equilibrium. However, that may be because I’m not trained in these concepts (or because random fluctuations of the universe don’t happen to have crafted for me a brain that’s adequate for the task).

  6. Paul Valletta says:

    Sean, very neat post!

    From the “NOW” moment back to the start-point, Big-Bang, there is a specific (thermal) distance, is this equal to the “now” moment of a theorized big-crunch, or Heat-Death of the Universe? .. just which side of the ‘Bang’ are we at?

  7. Markk says:

    For some reason reading this I am reminded of these duplicate Hubble bubbles that people discuss with parallel worlds because of the the Holographic Principle. How would that effect things? I guess this would be similar to Aaron Bergman’s note – given a universe where there are only finite number of bits inside any surface area (due to Holographic Principle) and an infinite number of regions – or even a finite but big enough set of regions. Then unless most of the regions are constrained won’t there be regions with just about any entropy due to pigeon hole principle arguments? Thus there is no meaningful universal entropy.

    That Holographic Principle seems very Laplacian by the way, very anti-quantum uncertainty feeling and even more anti-continuous space General Relativity feeling.

  8. Aaron Bergman says:

    The thermodynamic limit does not exist for gravitating systems. You can get by with approximate thermodynamic equilibrium on short scales, but on large scales I have no idea what any of this means.

  9. Louise says:

    Another fascinating question! The number of states available for any thermodynamic system is increasing, which should be a clue to the solution.

  10. Cynthia says:

    Keep in mind – when Boltzmann formulated his contributions to the second law, he was without the aid of quantum mechanics… Unlike classical mechanics, quantum mechanics allows for low-entropy fluctuations to emerge from an overall high-entropy universe without violating the second law. Roughly speaking, quantumness safeguards the Universe from a classical heat-death.

    Simply put – if Boltzmann had the luxury of a “quantum brain”, then he probably would have no need for an “anthropic brain”.

  11. island says:

    Asymmetric transitions.

    This occurs with matter gneration in a quasi-static version of Einstein’s static vacuum model, because you have to condense or compress his negative pressure energy down over a finite enough region of space to attain the matter density before you can make a real particle pair from this energy.

    This has a rarefying effect on the vacuum which necessarily increases negative pressure, (e.g., the “hole” leaves a real hole in the vacuum), which drives expansion, while holding the universe flat and stable, since the increase in the matter density and positive gravitational curvature gets proportionally offset by the increase in negative pressure.

    In this case, tension between the vacuum and ordinary matter increases as the universe expands, and this will eventually inevitably compromise the integrity of the forces…


  12. Young-Earth creationists are wrong, of course. Everyone knows the Universe was created Last Thursday. Think of it. It takes alot less effort. The whole 40+ gigalightyear radius Universe didn’t have to be created even once. Just a sphere about a light week in radius. Much smaller. Now then, attention to detail is important. The photons at a light week out have to be just the right frequencies and they have to be pointed in just the right directions, otherwise the effect is ruined. If there are any mistakes, though, they can be corrected Next Thursday.

    I’m proud of God’s accomplishments. His creation of the WMAP probe, the Chandra X-Ray telescope, and HST were perhaps none-too-subtle, but on the whole, the Big Universe illusion really holds together. I’m hoping and praying that next week, He’ll create a little cooler weather.

  13. Count Iblis says:

    Markk, I would agree with that. A similar argument for possible histories of the universe was given here.

  14. beajerry says:

    I suggest we ask Slartibartfast about all this.

  15. Qubit says:

    So if I was the universe and was to look back on my self, I would see my self as perfect and I would be evolving in to my past self, as I travelled into the future. I would be man at the moment, a perfect man in the past and will reach perfection in the future?

  16. Bob E. says:

    Can someone explain why the early universe is believed to have been in a low-entropy state? I have only had a course in undergrad stat mech, but it seems to me that a universe consisting of only a highly energetic uniform gas of elementary particles would be high entropy. Please excuse my ignorance.
    Thanks, Bob

  17. Elliot says:


    I love the final line of the abstract…. “unquestionably real”.

    Nothing like unabashed confidence….


  18. mqk says:

    I am under the (perhaps mistaken) impression that given a sufficiently fine-grained view, entropy is actually a conserved quantity. Certainly this is true for collisionless systems, for which Liouville’s theorem ensures that the phase space volume a system occupies is conserved. Classically this holds even for a gas, if I am able to follow all individual gas particle’s positions and momenta and can describe the particle’s interactions with a Hamiltonian. Does a quantum description somehow save the day and allow entropy to increase?

    If entropy is in fact ultimately conserved, perhaps an explanation of the seeming progression of time should be looked for elsewhere? Or alternatively, is the arrow of time a coarse-grained (i.e. observer-dependent) phenomenon?

  19. Count Iblis says:

    Elliot, indeed :)

    The aticle mentions that Elvis didn’t die prematurely in some sector of the multiverse, so that’s good news for Elvis fans. This was probably the first PRD article that mentioned Elvis :)

    mqk, unitarity time evolution leads to the same conclusion as in the classical case. This is in fact the source of the “information paradox” problem for evaporating black holes. If the “fine grained” entropy of a system could really increase then that would amount to a fundamental source of information loss.

  20. John Baez says:

    Thanks for putting in the energy to write a great post about one of the trickiest and most controversial issues in physics! Shockingly, I agree with almost all of it. Like you, I’m a big fan of Huw Price.

    A few random comments:

    Sean Carroll wrote:

    … nobody knows any reason why there should be low-entropy boundary conditions in both the past and the future, which would involve an absurd amount of fine-tuning of our particular microstate at every instant of time.

    Just for people who haven’t thought about this much, it’s worth restating that even having low-entropy boundary conditions in the past involves an absurd amount of fine-tuning of our particular microstate. Putting low-entropy boundary conditions in both the past and future means we have an “absurd squared” amount of fine-tuning. We gain symmetry at the expense of more fine-tuning.

    There certainly seems to be no evidence for this symmetry. Indeed, if we take what astronomers see today as the last word, everything about the universe suggests a vast asymmetry between the beginning of the universe (Big Bang) and the end (indefinitely accelerating expansion). So, personally, I don’t find the symmetry between past and future philosophically pleasing enough to impose it at the expense of extra fine-tuning.

    On the other hand, nature could have more tricks up her sleeve – she usually does – and I doubt we’re close to the last word on this subject.

    Have you read Huw Price’s ideas on how to search for starlight coming from the future? It sounds like an insane idea until you remember that the equations governing light are symmetric under time reversal, and most starlight formed now should keep zipping through space into the very far future… so if the universe were time-symmetric and there was a time-reversed Big Bang in the far future, stars at that end of time might send light back to the current era.

    It may still be an insane idea, but it’s harder to dismiss than you might at first think. In particular, you have to do some calculations to understand what “light coming from the future” is actually like (technically: advanced rather than retarded solutions of Maxwell’s equations).

    You can read about these ideas in Price’s book.

    Price has some even more speculative ideas about how an “absurd squared” amount of fine tuning could make classical mechanics look quantum… or at least strange. Normally we imagine the experimenter has “free will” when deciding to, say, measure the spin of an electron either along the z axis or the y axis. More precisely, we imagine this decision is uncorrelated with the state of the electron. But with an “absurd squared” amount of fine tuning, is this assumption still justified?

    As you can see, this business gets pretty weird pretty fast….

    Aaron Bergman wrote:

    The thermodynamic limit does not exist for gravitating systems. You can get by with approximate thermodynamic equilibrium on short scales, but on large scales I have no idea what any of this means.

    Yes – there are many layers of subtlety to this arrow of time business, and this is one. People like to use thermodynamic concepts like “equilibrium” when discussing the arrow of time, but in so doing they neglect that even in Newtonian gravity, thermodynamic equilibrium does not exist, because the energy is unbounded below.

    Then, when we bring in general relativity, there’s another layer of subtlety: there’s not even a clear-cut concept of energy anymore! This is called the problem of time. It’s distinct from the problem of the arrow of time… but it gets involved when we try to use thermodynamics while taking general relativity into account.

    Bob E. writes:

    Can someone explain why the early universe is believed to have been in a low-entropy state? I have only had a course in undergrad stat mech, but it seems to me that a universe consisting of only a highly energetic uniform gas of elementary particles would be high entropy. Please excuse my ignorance.

    That’s a great question. A hot uniform gas does indeed seem to have high entropy… but that’s neglecting gravity. Let’s stick to Newtonian gravity at first. It turns out that if you have a big enough box of hot uniform gas, it’s not in thermal equilibrium: gravity will make it clump up, reducing its potential energy and increasing its kinetic energy, making it even hotter, raising its entropy! So, its entropy may look high, but it’s low compared to what it could be.

    In fact, a “clumping” process roughly like this did occur in our universe – that’s basically how galaxies and stars were formed! This clumping process is not over yet, either. In Newtonian gravity it can continue indefinitely – so there is, in fact, no state of thermal equilibrium when we take gravity into account.

    Life gets more complicated when we try to take general relativity into account. We need it to understand the expansion of the universe, and when enough clumping occurs we need it to understand the resulting black holes. But, as I mentioned earlier, nobody really knows how to combine thermodynamics and general relativity.

  21. Gavin Polhemus says:

    I’ll join Aaron Bergman and Bob E. in saying that I don’t quite get what this is all about, although my concerns are a little different.

    Just to be clear, when I think of the “entropy of universe,” I’m thinking about the entropy of the quantum fields, excluding gravity. For the times that we are discussing (inflation to today) gravity can be treated classically, and therefore does not contribute to the entropy. (Correct me if this is wrong.)

    Since entropy is an extensive quantity, it was low in the early universe because the early universe was very small. (It was also hot, but the smallness wins.) The universe expanded from this small, low entropy state very quickly. Since the expansion was not adiabatic, all sorts of great, non-equalibrium stuff got to happen, like life.

    The analogy I have in mind is an insulated chamber with an adjustable volume (via a piston, for example). Start out with a small volume filled with water. The water is in equilibrium, the highest entropy it can achieve given the macroscopic constraints. If I slowly increase the volume, then the water evaporates gradually, everything cools a bit, and the final entropy is the same as the starting entropy. If I rapidly increase the volume of the chamber, the water bubbles and boils and eventually reaches a state of much higher entropy. The evolution of our universe is like the second case, with rapid expansion causing all sorts of fizzing and popping that leads to the non-equilibrium mess of galaxies, planets and physics blogs.

    It seems to me the question is not “Why was the starting entropy so low?” The question is “Why was the starting entropy so high?” How did the universe get into an equilibrium state that could produce the homogeneous universe that we see today? Equilibration takes time, something that you don’t have at the beginning of the universe. I thought that this was one of the problems that inflation solved. Inflation happens after some sort of equilibration, expanding the equilibrated region to cosmic size.

    Other interesting questions are “Why did the universe start out so small?” and “Why did it expand so fast?” but the increasing entropy just seems to come along for the ride.

    Is there something I’m totally missing here? Do I net to treat the gravitational entropy? Is this debate about earlier times?

    Thanks for the thought provoking post. I obviously need more thought before I’m done with it.


  22. Pingback: Ars Mathematica » Blog Archive » Cosmic Variance on Boltzmann

  23. Aaron Bergman says:

    Just to be clear, when I think of the “entropy of universe,” I’m thinking about the entropy of the quantum fields, excluding gravity. For the times that we are discussing (inflation to today) gravity can be treated classically, and therefore does not contribute to the entropy. (Correct me if this is wrong.)

    The entropy of quantum fields unfortunately can mean different things. If you mean that there’s a density matrix and something like

    S = – Tr rho log rho ,

    then that’s a conserved quantity. If you mean some sort of coarse graining, then you’re still stuck with the same problems as in the classical sense.

    Since entropy is an extensive quantity,

    As I remember, this statement depends on the existence of the thermodynamic limit.

    As for homogeneity, I’m not sure I understand what you’re asking, but it turns out that the initial conditions for inflation seem to require uniformity on scales larger than the horizon at the time. In fact, the calmness you need, I think, is a very low entropy situation (if those words mean anything) — Sean wrote about this somewhere, so I’m sure he’ll correct me if I have it wrong.

  24. Gavin Polhemus says:


    I do mean with coarse graining. What are the “same problems as in the classical sense”?


  25. Aaron Bergman says:

    The lack of the thermodynamic limit.

  26. Louise says:

    To answer Gavin’s and other questions: the early Universe expanded quickly because c was much greater. Since hc appears to be constant, Planck value h was very small leading to a low-entropy state. The increasing entropy in quantum mechanics is intimately tied to Relativity.

  27. Jack says:

    I’m not sure I really understand what Aaron is driving at. Sure, it is not known precisely how to define “gravitational entropy”. But surely it is clear that, whether we know how to define it or not, there *is* such a thing. In other words, isn’t it clear that there is a connection between the arrow of time and the extreme smoothness of the geometry of the early Universe, whether or not we can *formulate* this in terms of familiar ideas from thermodynamics?

  28. Elliot says:

    Not to muddy the waters here but gravitational entropy is given an interesting analysis by Roger Penrose. His reasoning “explains” the unusually low entropy in the early universe. He uses what he calls the “Weyl curvature hypothesis” constraining this portion of the tensor.

    All in the “laypersons” Emperors New Mind.

    His argument however does not appear to have achieved widespread acceptance as far as I can determine.

  29. Ijon Tichy says:

    Some of you might find this paper interesting and relevant:

    On the Origins of the Arrow of Time: Why There is Still a Puzzle about the Low Entropy Past by Huw Price (PDF, 63KB)

  30. David Moles says:

    Sean, can you provide any pointers to what you’re talking about with this bit?

    Another fascinating but separate issue — the process of “remembering” involves establishing correlations that inevitably increase the entropy, so the direction of time that we remember [and therefore label “the past”] is always the lower-entropy direction.

    Are we talking about chemical processes in the brain? Or information theory? Or that Star Trek episode where McCoy posits that memory operates via tachyons?

    Okay, probably not that last one. :) But whenever somebody brings up memory or perception in connection with cosmology — paging Julian Barbour — I get confused…

  31. Thomas Dent says:

    One basic disagreement about the ‘entropy of the Universe’ seems to be, what would be the Universe with the highest entropy.

    One argument (essentially from holography) is that in any given volume the maximal entropy state consists of big black holes filling that volume.

    Sean’s counterargument is that black holes evaporate into the surrounding space. Now these two arguments are actually at cross purposes, because there is another fundamental question: how big is the Universe and does it have boundaries?

    If the Universe is finite and closed then the black holes win the day. If it is infinite and tends asymptotically to de Sitter space then any black hole smaller than the horizon size will evaporate leaving just thermal bath.

    I’m not sure how Sean’s proposal for ‘baby inflating Universes’ in a sea of time-symmetric de Sitter space is really different from Boltzmann’s ‘multiverse’.

    Perhaps it is again the question of the total amount of space and stuff in the Universe. If the Universe did have a conserved, finite amount of space and stuff then the Boltzmann picture seems to be unavoidable.

    The special property of inflation is that given the (very unlikely) fluctuation that sets it off, it automatically produces a whole lot more space and stuff with a peculiarly nice and uniform density distribution. I would assume that Sean wants to invoke this special property as solving part or all of the problem.

    Dyson, Kleban, Susskind put forward here:

    the modern version of the ‘Boltzmann’s brain’ picture, and it’s still not clear how to get round it, unless you can show that inflation gives certain types of low-entropy region (like the one we seem to live in) a decisive statistical advantage over other low-entropy regions that arise directly from random fluctuation.

  32. Thomas Dent says:

    …whoops, slight misstatement there: de Sitter space is actually effectively finite because it has a horizon at a fixed distance from the observer.

    Anyway this does not adversely affect the Dyson-Kleban-Susskind arguments.

    For asymptotically de Sitter space with a region inside it undergoing slow-roll inflation, although there is still a de Sitter horizon, the volume inside the inflating region quickly becomes exponentially larger.

  33. Aaron Bergman says:

    ‘m not sure I really understand what Aaron is driving at. Sure, it is not known precisely how to define “gravitational entropy”. But surely it is clear that, whether we know how to define it or not, there *is* such a thing.

    Why? It’s easy to see that thermodynamics works and makes sense on scales where the gravitational interaction is relatively unimportant, but I don’t see why that implies that thermodynamics should make any sense on universal scales.

    (The quantity called entropy in a lot of cosmological calculations is a local thing, and I don’t see any reason why it should integrate to a useful global entropy.)

  34. Elliot says:


    Re: “memory”… I believe the act of storing information requires the expenditure of energy thereby increasing the entropy. That’s from an info-theoretic basis. No free lunch…Maxwell’s demon etc.


  35. Plato says:

    Some might be quick to point out the issues and sources of such illusions about the outcomes of the universe?

    However, don’t be fooled! The charm of the golden number tends to attract kooks and the gullible – hence the term “fool’s gold”. You have to be careful about anything you read about this number. In particular, if you think ancient Greeks ran around in togas philosophizing about the “golden ratio” and calling it “Phi”, you’re wrong. This number was named Phi after Phidias only in 1914, in a book called _The Curves of Life_ by the artist Theodore Cook. And, it was Cook who first started calling 1.618…the golden ratio. Before him, 0.618… was called the golden ratio! Cook dubbed this number “phi”, the lower-case baby brother of Phi.

    There’s alway “fool’s gold” even amidst the nuggets? Really?

    So “no” simplier entropic valuation? No supersymmetry?

    Maybe then having reached such a “supersymmetrical state” the tunneling(QGP assumption here) allowed for this multiversian discription as…a pascalian one?:)

    All “probable outcomes” are satisfactorially, okay? So where did this “point” emerge?

  36. island says:

    John Baez wrote:
    There certainly seems to be no evidence for this symmetry. Indeed, if we take what astronomers see today as the last word, everything about the universe suggests a vast asymmetry between the beginning of the universe (Big Bang) and the end (indefinitely accelerating expansion).

    I don’t think that is what is necessarily what is suggested by astronomers observations. These conclusions require projections that aren’t necessarily the most natural solution.

    For example, the most natural (naive) projection backwards in time doesn’t include inflationary theory, rather, it indicates that the universe had certain volume when the BB occurred.

    That we had a big bang only indicates that big bangs happen, so if a universe with volume can have a big bang, then your conclusion might ought to be that the accelerating universe is once again racing toward that end.

    What we actually know is that entropy always increases, so the most natural observational conclusion is that it always has and will.

    And yes, I know just how naive that appears, but Einstein said that meant that “god” was talking”.

  37. Louise says:

    The Universe is based upon a very simple principle: Expansion of Space is indistinguishable from the forward flow of time. In math terms, R = ct. Scale R of the Universe is its age t multiplied by factor c. That is why the Universe expands, for as t increases the R expands. This is the first arrow of time, the Cosmological Arrow.

    The Universe can’t expand at the same rate c continuously, for gravitation slows it down. Factor c is further related to t by GM = tc^3, where GM combines the Mass and Newton constant of the Universe. When t was tiny, c was enormous and the Universe expanded like a bang. As t increases expansion slows and continues asymptotically to this day.

    Since the product hc is constant, h increases proportional to t^(1/3). Since R increases proportional to t^(2/3), the number of available quantum states in this volume increases. A growing h leads to an increasing entropy. That is the second arrow of time, the Thermodynamic Arrow.

    If one calculates, the total energy of the Universe E = 0! It’s the ultimate free lunch, which is how the Universe expanded from a tiny size to the complexity we observe today. You needn’t add any energy to the system. Increasing h leads to complexity of the Universe.

    Now we can state some initial conditions: At time t = 0, R = 0 and the Universe resembles an initial singularity. We also have h =0, for in zero size there is no uncertainty in the position of anything. The value of c approaches infinity. These initial conditions are extremely unstable, an initial extremum. A nearly infinite c caused the Universe to expand at an enormous rate, slowed by gravity.

    Thoughtful replies are surprising and welcome!

  38. island says:

    Thoughtful replies are surprising and welcome!

    As t increases expansion slows and continues asymptotically to this day.


  39. Ben M says:

    I disagree with the premise: “The lowest entropy state is one in which Boltzmann’s brain materializes out of the vacuum”. Let’s say that the brain consists of 10^27 nucleons. How many different ways are there to configure 10^27 nucleons? Something of order (10^27)!, right, modulo some sort of combinatorics? Only a tiny, tiny subset of those states actually turn out to be (or to self-assemble into) Boltzmann’s brain: other equally likely states are Boltzmann’s recently-deceased brain, five copies of Boltzmann’s medulla tied together with twine, a pot of arsenic-laced coffee, or a remaindered hardcover copy of “The Da Vinci Code”. Of course, a brain is more likely to emerge from a larger ensemble of nucleons—if you had 10^62 nucleons, for example, the chance of finding Boltzmann’s Brain would increase by a factor of 10^35 or so. (Regrettably, the chance of finding “The Da Vinci Code” also scales up. The chance of finding “The Da Vinci Code” micro-printed on Boltzmann’s Brain must not be ignored.)

    But when you’ve got 10^62 nucleons in the universe, there’s a much-more-probable fate for the ensemble. 10^62 is the Jeans mass, the mass which (in our universe, with our Hubble constant and our fluctuation spectrum, etc.) is able to gravitationally self-collapse to form stars. Once you’re gravitationally collapsin, something in the neighborhood of 100% of available states lead to massive stars, and 100% of massive stars lead to supernovae, and 100% of supernovae produce heavy elements. Now, perhaps the odds of getting planets out of an arbitrary one-Jeans-mass collapse are low, but they’re not crazy-exponential-factorial low. So, by thermodynamics standards, we can say that an appreciable fraction of all possible configurations of 10^62 atoms results in planet formation. And—well, once you have planet formation, there’s no way to grab all of those nucleons and stop them from undertaking chemical reactions. At that point, evolution takes over, and it’s (again, in the sense of “it’s not forbidden to the tune of 10^23 factorial”) reasonably likely that intelligent life emerges.

    So: let’s say that “entropy fluctuations” in the early Universe can produce a cool-ish blob of 10^62 atoms. Over the set of all possible states of this blob, very, very (!) few give rise to Boltzmann’s brain in the pop-into-existence scenario, while a largeish fraction give rise to intelligent life, including brains, thermodynamicists, and so on. Therefore, by the anthropic principle, the “average conscious observer” finds him or herself on a planet around a population-I or population-II star, in a universe of at least one Jeans mass, since this is a fairly typical result of random configurations of atoms.

    Of course, this assumes that the laws of physics are fixed, and we’re just wondering where the original matter and entropy came from. Furthermore, if we need to assign probability distribution to the size of the blob, then it’s not obvious that P(get 10^27)xP(make brain) is less than P(get 10^62) … ugh. Even logarithms don’t help me think about these numbers.

  40. Torbjörn Larsson says:

    I was under the undoubtedly very naive impression that endless inflation scenarios explained this if successful inflationary regions are both somewhat initially ordered and expanding. Which Aaron’s commentary seems to suggest: “the initial conditions for inflation seem to require uniformity on scales larger than the horizon at the time”. The system doesn’t develop to the next stage until the right condition occurs.

  41. Louise says:

    Re: huh?
    Expansion and slowing can be predicted from the math. Asymptotically means that expansion will never slow to zero or reverse.

  42. Gavin Polhemus says:


    Let me elaborate a bit on Aaron’s response, and them I’m going to see if I can get past the thermodynamic limit concern.

    It is difficult to define an energy to the gravitational field or a potential energy to objects in a gravitational field unless the space time has certain special properties (asymptotic flatness or a timelike killing vector field), properties that our Universe doesn’t have. You might think that even without those properties there must be an energy of the field, even if we don’t how to write it down. You would be wrong. (Or you might not think that, and you would be right.) There is no energy of the gravitational field and there is no gravitational potential energy in general situations.

    There is, in all situations, a local stress-energy-momentum tensor that is locally conserved (divergenceless). The stress-energy-momentum tensor does not have any terms from the gravitational field, and if you try to add such terns the conservation law is lost. The time-time component of this tensor is the local energy density, so energy density likewise has no contribution from the gravitational field. The “energy of the gravitational field” is a nonsense concept.

    Even worse, this energy density described above cannot be used to generate a total energy of the universe that is conserved. Any integration to find the total energy is going to depend on the choice of coordinates at every point. Even you you pick natural coordinates for our universe, the total energy that you get isn’t conserved.

    My intuition is that the story is about the same for entropy. There is a local entropy density, which may actually be just one component of something more complicated, which obeys a local “non-decreasing” law (the divergence is nonnegative, for example). However, I expect there is no way to make this into a global entropy. Actually this isn’t just my intuition, you can find more in “Gravitation” by Misner, Thorne and Wheeler in the sections on thermodynamics and hydrodynamics, especially problem 22.7 which will guide you through the derivation of an entropy four-vector and the associated local second law of thermodynamics. Good luck.

    Does the above cover your concerns about an “entropy of the universe.” When I’ve seen people claim that the entropy of the early universe was small, I assumed that they meant that the local entropy per co-moving volume was small. (For spectators, the entropy density was large because it was hot, but the co-moving volume scales with the size of the universe, so it was much, much smaller in the early universe, leading to a small entropy per co-moving volume).

    Now that I’ve seen this discussion, it appears that some people do not mean local entropy per co-moving volume; they actually want to talk about the total entropy of the universe. Such a concept requires some serious explaining that I haven’t seen.

    Since local entropy per co-moving volume does not include any gravitational terms, have I satisfied your concerns about the thermodynamic limit?


  43. Aaron Bergman says:

    Again we run into the problem that lots of things go by the name ‘entropy’ and they don’t always agree. There is a quantity, s, often, that shows up in fluid dynamics and the like that acts like a local entropy density. The problem is relating this to something statistical mechanical like the microcanonical or canonical entropy (which don’t necessarily agree, BTW). Even in situations where the thermodynamic limit does exist, my recollection is that the integral of this local quantity does not (except for an ideal gas, maybe) give you the correct global entropy.

    Fluid mechanics works, of course. It’s a long jump from that to stuff like that in the original post, however.

    On the other hand, thermodynamic reasoning has been rather successful in understanding black holes. When I sit down to think about it, I end up concluding that that success is really very mysterious. I feel like there’s something very important that it is telling us, and I have no idea what it is.

  44. Sean says:

    Sorry for being too busy to answer any comments since the original post. Traveling will do that. Briefly:

    — Taking gravitational entropy into account is absolutely necessary, the whole point really. The entropy of our current universe is dominated by gravity. The entropy of all the matter and radiation within a Hubble volume is about 10^88, while the entropy of a single million-solar-mass black hole is 10^90, and there are many such black holes (one per large galaxy).

    — Having said that, it’s certainly true that we don’t have a rigorous definition of the gravitational entropy, or even an understanding of what the degrees of freedom are, and both of those would certainly be nice to have. But even without them, it’s still clear that there is a problem. The early universe doesn’t look anything like the late universe. And more specifically, there aren’t that many ways to take the degrees of freedom in our current Hubble patch and arrange them into a tiny smooth region with GUT-scale energy density, while there are many ways to arrange them in a huge nearly-empty configuration. So it’s the early universe that is very fine-tuned.

    — Inflation doesn’t solve the problem by itself, for the reasons just mentioned. The proto-inflationary state is incredibly finely-tuned; invoking inflation just begs the question.

    — The closest thing to “thermal equilibrium” in the presence of gravity is empty space (which will be de Sitter if you have a positive vacuum energy). It would be exactly equilibrium if it weren’t for the possiblity of baby-universe creation. How do we know? Anything other configuration will evolve into something else, implying that it wasn’t in equilibrium. Alternatively, if you give me your favorite configuration of matter, I can increase its entropy by expanding space and scattering its constituents to the winds.

    — Ben M, your starting point of a Jeans-mass cloud is nowhere near thermal equilibrium, so what arises from fluctuations therein is a very different question. It might seem pretty close to equilibrium, but gravity changes everything.

    And Allyson — Green Lantern is awesome.

  45. island says:

    Louise said:
    Expansion and slowing can be predicted from the math. Asymptotically means that expansion will never slow to zero or reverse.

    No, not that “huh”… I’m fairly sure that we have independently supported observational evidence that exactly the opposite is true. That, “huh”.

  46. Elliot says:


    Please indulge the following gedanken experiment…Your response would be greatly appreciated.

    Suppose at the next instant gravity were to “disappear”. Wouldn’t everything move toward a higher state of entropy (thermal equalibrium) as gravity was no longer holding things together. So doesn’t this pose somewhat of a paradox in that the current state of the universe (including gravity, structure formation etc.) has a much lower entropy than a universe without gravity.

    So my query is that if the description above is correct, and removing gravity would raise the entropy of the universe, then why does gravity add to the entropy of the universe instead of lower it?

    Hopefully this makes sense. (even if incorrect)


  47. Gavin Polhemus says:


    I’ll be sure to include the entropy of black holes. We have a formula for it, so it won’t be any problem.

    Having said that, it’s certainly true that we don’t have a rigorous definition of the gravitational entropy, or even an understanding of what the degrees of freedom are….

    I could say exactly the same thing about the gravitational energy. In that case the reason we don’t have a rigorous definition is because, aside from special cases, the whole concept is nonsense. How do we know that gravitational entropy won’t meet the same fate? It seems to me quite likely that it will, and we will be left to look only at the local entropy density of the remaining, non-gravitational stuff.


  48. Sean says:

    Elliot– according to the Second Law, things are always moving toward greater entropy, in accordance with the laws of physics. If you change the laws of physics (e.g. by removing gravity), everything changes, including what counts as a “high entropy configuration.” Without gravity, the number of degrees of freedom are different, as are the conservation laws.

    It’s not true that “the universe would have a higher entropy if it weren’t for gravity.” It’s just that the process by which entropy increases looks different with and without gravity.

  49. Sean says:

    Gavin, it doesn’t matter. We know that the universe was in a very special state close to the Big Bang, because it rapidly evolved into something else. And nobody thinks that a re-collapsing universe would naturally smooth itself out during a Big Crunch. So why was it like that?