The Boltzmann Brain paradox is an argument against the idea that the universe around us, with its incredibly low-entropy early conditions and consequential arrow of time, is simply a statistical fluctuation within some eternal system that spends most of its time in thermal equilibrium. You can get a universe like ours that way, but you’re overwhelmingly more likely to get just a single galaxy, or a single planet, or even just a single brain — so the statistical-fluctuation idea seems to be ruled out by experiment. (With potentially profound consequences.)
The first invocation of an argument along these lines, as far as I know, came from Sir Arthur Eddington in 1931. But it’s a fairly straightforward argument, once you grant the assumptions (although there remain critics). So I’m sure that any number of people have thought along similar lines, without making a big deal about it.
One of those people, I just noticed, was Richard Feynman. At the end of his chapter on entropy in the Feynman Lectures on Physics, he ponders how to get an arrow of time in a universe governed by time-symmetric underlying laws.
So far as we know, all the fundamental laws of physics, such as Newton’s equations, are reversible. Then were does irreversibility come from? It comes from order going to disorder, but we do not understand this until we know the origin of the order. Why is it that the situations we find ourselves in every day are always out of equilibrium?
Feynman, following the same logic as Boltzmann, contemplates the possibility that we’re all just a statistical fluctuation.
One possible explanation is the following. Look again at our box of mixed white and black molecules. Now it is possible, if we wait long enough, by sheer, grossly improbable, but possible, accident, that the distribution of molecules gets to be mostly white on one side and mostly black on the other. After that, as time goes on and accidents continue, they get more mixed up again.
Thus one possible explanation of the high degree of order in the present-day world is that it is just a question of luck. Perhaps our universe happened to have had a fluctuation of some kind in the past, in which things got somewhat separated, and now they are running back together again. This kind of theory is not unsymmetrical, because we can ask what the separated gas looks like either a little in the future or a little in the past. In either case, we see a grey smear at the interface, because the molecules are mixing again. No matter which way we run time, the gas mixes. So this theory would say the irreversibility is just one of the accidents of life.
But, of course, it doesn’t really suffice as an explanation for the real universe in which we live, for the same reasons that Eddington gave — the Boltzmann Brain argument.
We would like to argue that this is not the case. Suppose we do not look at the whole box at once, but only at a piece of the box. Then, at a certain moment, suppose we discover a certain amount of order. In this little piece, white and black are separate. What should we deduce about the condition in places where we have not yet looked? If we really believe that the order arose from complete disorder by a fluctuation, we must surely take the most likely fluctuation which could produce it, and the most likely condition is not that the rest of it has also become disentangled! Therefore, from the hypothesis that the world is a fluctuation, all of the predictions are that if we look at a part of the world we have never seen before, we will find it mixed up, and not like the piece we just looked at. If our order were due to a fluctuation, we would not expect order anywhere but where we have just noticed it.
After pointing out that we do, in fact, see order (low entropy) in new places all the time, he goes on to emphasize the cosmological origin of the Second Law and the arrow of time:
We therefore conclude that the universe is not a fluctuation, and that the order is a memory of conditions when things started. This is not to say that we understand the logic of it. For some reason, the universe at one time had a very low entropy for its energy content, and since then the entropy has increased. So that is the way toward the future. That is the origin of all irreversibility, that is what makes the processes of growth and decay, that makes us remember the past and not the future, remember the things which are closer to that moment in history of the universe when the order was higher than now, and why we are not able to remember things where the disorder is higher than now, which we call the future.
And he closes by noting that our understanding of the early universe will have to improve before we can answer these questions.
This one-wayness is interrelated with the fact that the ratchet [a model irreversible system discussed earlier in the chapter] is part of the universe. It is part of the universe not only in the sense that it obeys the physical laws of the universe, but its one-way behavior is tied to the one-way behavior of the entire universe. It cannot be completely understood until the mystery of the beginnings of the history of the universe are reduced still further from speculation to scientific understanding.
We’re still working on that.
Stat mech works best when there’s a thermodynamic limit. And that certainly does care about the details of the state space and the hamiltonian.
I’m not sure what you mean. The thermodynamic limit is when there are a large number of states. That’s not really a problem for the whole universe.
There are many things we don’t know about gravitational entropy, but we know more than enough to say “the entropy of the early universe was small.”
I think that Aaron is referring to the existence of (the possibility of) thermodynamic equilibrium. That, to say the least, is problematic in gravitational systems.
I suppose it is problematic, although I would argue that de Sitter is the correct equilibrium state in the presence of a positive cosmological constant. Regardless, non-equilibrium statistical mechanics certainly exists, and nothing stops me from calculating the entropy using Boltzmann’s formula. (And certainly entropy exists, and tends to increase in a closed system.)
Classically, that is certainly false.
Quantum mechanically … who know? Nobody understands quantum gravity in de Sitter space.
I’m hardly an expert on the subject, but I don’t see how (what little I know of ) the existing machinery of non-equilibrium stat mech helps you here.
Sean: OK — and thanks for your patience.
I guess my problem was that I’d tacitly assumed that we’ve observed so much disequilibrium already that we’re reasonably entitled to frame the hypothesis “We live 14 billion years after a genuine low-entropy Big Bang, in a universe that might or might not have 10^100 Boltzmann brains in its far future”. Once you allow that hypothesis as reasonable, the further evidence we gather just keeps on supporting it.
But I think I can see now why you don’t believe I’m entitled to start from that point — or at least, you’re saying I ought to give this hypothesis a stupendously low prior probability.
Greg Egan: Yes, as Sean said, what I am driving at is that we have to proceed in this way if we want to do stat mech at all.
Aaron Bergman: You might find it helpful, as I did, to look at this:
http://arxiv.org/abs/physics/0402040
The question of the definition of gravitational entropy, while interesting, is not very relevant really.
To me, the really interesting point is this. In my reading of Feynman, he claims that there is something *absolutely fundamental* that we don’t understand about the early universe. Question: by not understanding this, and, worse, by not even recognising that there is a problem here, are we in grave danger of talking complete nonsense when we discuss the early universe? Can we really get away with ignoring a major new law of physics?
Classically, the cosmic no-hair theorem implies it is pretty darn close to true. You could be unlucky enough to fall into a black hole, but if you manage to avoid that and wait around long enough, your local patch of universe will approach de Sitter.
Quantum mechanically, the black holes will evaporate, so you don’t even have to worry about that. Of course de Sitter might then not be completely stable; in fact I hope it’s not. But it persists for a long time, at least.
There are many things we don’t understand about quantum gravity, but that doesn’t seem like a good reason to completely ignore features of semiclassical gravity that seem pretty robust. I can’t imagine any new insight that would make “Nature doesn’t distinguish between matter and gravitation, as far as entropy is concerned” a false statement, but I could certainly be wrong. But “we don’t understand everything” isn’t enough to prevent me from trying to move forward.
If I could put forth something relating to the basic conceptualization of, “is flow of time relative”: I think it isn’t. Consider a world where events are “moving backwards” and something intervenes in the flow of those events. The results are not symmetrical with what we expect for our universe. Consider yourself an “outsider” that is not part of the reversing time flow. A bullet that (to the backwards universe) was fired “out of” a gun is now approaching that gun (in your reckoning) to go back into it. You push the bullet, or the gun, out of the way. Now what? It is imaginable in a world of normal time flow, what really happens if I fire a gun and then push the bullet out of the way later as commonly understood in time sequence. But if we allow the intervention to be conceived as happening in a world where time is running backwards the result is absurd: the bullet now misses the gun and does, what? It’s re-reversed behavior would be absurd, it would have to spring out of e.g. a tree that was behind the shooter, etc. With this distinction between whether past or future is affected by an intervention in the chain of events, how can time-reversal be merely relative?
Hi Sean —
Thanks for your prompt reply, which I think does provide a clear answer: the early thermal Universe was disordered in all the matter/radiation degrees of freedom, but highly ordered in its gravitational degrees of freedom. I agree that why the latter is true is the interesting question (the word “constraint” to describe a smooth metric in my comment was perhaps unfortunate; all I really meant was that given a smooth metric the matter/radiation had reached equilibrium, not that a smooth metric was a constant condition). This does bring up the next natural question, though: what would thermal equilibrium with gravitational degrees of freedom included look like?
Presumably the answer to this question is equivalent to identifying (a prelude to counting) the microstates of gravity, which you say is a hard problem and so one shouldn’t expect a simple answer. But let me give it a partial shot, with some intro-level GR, and it may then be instructive for you to point out where I’ve gone wrong.
Perturbations from a flat Minkowski metric (or from a Robertson-Walker metric on scales much smaller than the Hubble radius) can, according to Penrose, be divided into two types: volume-changing and volume-preserving. The former are in general tied to the matter/energy distribution and so are not really independent degrees of freedom; the exception among volume-changing perturbations is black holes, which can be entirely vacuum (outside the singularity). The volume-preserving perturbations are basically gravity waves, which can exist in vacuum or in non-empty space. So the independent gravitational degrees of freedom that Feynman warns us not to forget can, crudely, be categorized as black holes and gravity waves.
Looking at gravity waves first, is there any fundamental error in decomposing gravity waves into gravitons? ie a configuration of spin-2 particles? If that’s OK then I should be able to count this class of gravity’s microstates since I know how to count particle states, at least in thermal equilibrium. At any given temperature the energy and entropy densities in the early Universe are directly proportional to the number of particle types with mass less than the temperature, counting each combination of spin, color, flavor, etc. as an independent type. At temperatures above the QCD transition this number is on the order of 100 for the particles in the standard model. If we simply naively add a parallel, ideal gas of spin-2 gravitons then it raises the effective number of types by 5; and so if those gravitons are absent then their “missing gravitational entropy” in that early epoch is something like a 5% correction to what’s in matter/energy — not much to get excited about if you ask me, and certainly not the difference between the Universe being “very ordered” versus “very disordered”.
This, then, leaves black holes as the main, significant expression of independent gravitational degrees of freedom in thermal equilibrium. And so, if there are no great errors in the foregoing then I would conclude that your general question “Why is the early Universe so highly ordered in the gravitational sector?” really boils down to the more specific question “Why wasn’t the early Universe, and why isn’t our present-day Universe, dominated by black holes?” Do you think that’s a fair conclusion? if not, then what have I missed?
Paul– I can only suggest that you have a look at hep-th/0410270 !
(I think a lot of what you say is right, but reality is a bit more complicated, because the existence of gravity changes what counts as “equilibrium” just in the matter sector, due to the Jeans instability.)
Since I’m not planning on leaving our galaxy (much less our local cluster, which is also, I believe, a gravitationally bound system), my local patch of the universe will never approach de Sitter.
Neil B: Hey! Somebody actually read my post! At least…I assume it’s my post that you’re referring to…
Optimistically assuming that you were:
So the particles all reverse trajectory, and then something causes the gun to move…which wasn’t in the forward version of events, but now is in the reverse version of events.
When I first started to write this reply, I started working back through all the consequences. For example, prior to being struck by the bullet, the tree was going about it’s business absorbing oxygen and using that to break sugars into carbon dioxide and water, plus some photons which it shoots off towards the sun which is absorbing them so as to break helium atoms into hydrogen atoms.
BUT, then it occured to me…this world is in a very fragile, very special state of decreasing entropy. As long as nothing disturbed the process, it would proceed along fine with entropy decreasing back towards a bing-bang event (assuming deterministic physics).
BUT, the force that moved the gun has now disturbed the extremely delicate state of affairs.
By moving the gun you’ve opened the door for entropy to begin increasing again. And it will start to increase immediately. Their whole world will now eventually disintegrate into chaos.
It was a very finely tuned situation to start with, and you untuned it with your gun push. Way to go Neil. You killed them all.
As to how this would be percieved by the people who are living in reverse…who knows. As the chaos spreading out from the bullet-tree collision starts to affect their brains…probably brief mass confusion followed rapidly by oblivion.
But my original post was really more about how we percieve time and reality than anything else. It may have been a tiny bit off topic. Oops.
THOUGH, there’s no reason why that scenario couldn’t play out during the “entropy decreasing” part of a Boltzmann-style statistical fluctuation of entropy, which would remove my “trajectory reversal” gimmick and make the post more on-topic.
“As to how this would be percieved by the people who are living in reverse…who knows. As the chaos spreading out from the bullet-tree collision starts to affect their brains…probably brief mass confusion followed rapidly by oblivion.”
Actually it will be perceived by them as:
Oblivion, followed by brief disorientation, followed by them assuming a full coherent set of memories of a non-existent past, and then them proceding on with their lives with no recollection that anything strange had ever happened, until they hit the trajectory reversal point (in the original post) or the beginning of the low entropy statistical fluctuation (in the revised version), at which point they return to oblivion.
From oblivion, to oblivion. The inevitable lot of all mortals.
Jacques, our galaxy is certainly not a stable system. Even at the level of Newtonian galactic dynamics a la Binney & Tremaine, the galaxy will continually eject some stars, as others become more tightly bound and eventually fall into the black hole. From there, see previous flow chart.
Suggesting an isolated collapsed object like a planet or a white dwarf is a better bet. But ultimately those only resist gravitational collapse through the miracle of quantum mechanics. And once you have quantum mechanics, there is some amplitude for tunneling to a black hole.
de Sitter is your best bet, believe me.
I think that Aaron is referring to the existence of (the possibility of) thermodynamic equilibrium.
At least when I took stat mech from Elliot Lieb way back when, the existence of the thermodynamic limit was the statement that, in the large various things limit, the usual thermodynamic quantities exist and obey the expected properties. The existence of this limit for interacting systems is very nontrivial and fails for gravitational systems. This is different, I believe, from the stability of matter which is, of course, another interesting nontrivial result (although I seem to recall that the tempering of the Coulomb force is an essential part of each proof.)
As for non-equilibrium stat mech, last I checked at least, it is a mess including such fun things as local entropies that, when integrated, do not give the correct global entropy.
Anyways, this is my yearly objection to talking about the entropy for the universe — everyone can go back to talking about it now, and I’ll be quiet.
Fubaris, I first imagined that thought experiment (intervention in a time-reversed world) literally decades ago while reading “One, two, three … Infinity” by George Gamow. It is a perplexing and challenging idea for anyone who wants to consider time flow a purely relative matter. An intervention changes “the continued past” of the time reversed world and “the continued future” of a normally progressing world. Sure, even the intervention itself wouldn’t be modeled the same way in both worlds, but that isn’t the point. The point is, the intervention can be done in the TRW and the effects work backward to ruin the rational structure of the supposed past of that world. How can a TRW be vulnerable, even in principle, to such an action if not really different in principle from a normal world? This question is indeed relevant to the idea of “thermodynamic reversibility” since the latter tends to a statistical challenge to TRWs. IOW, they are said to have a tiny change of maintaining the reverse flow, but if anything “went wrong” it all falls apart. However, we consider our universe “robust” and interventions would only affect the “true future.” That makes common sense, but violates the supposed inherent physical equivalency of time-reversed processes.
I don’t think the effects would be what you imagine. An intervention in a TRW does not apply to the events its dwellers consider “after” the intervention, but to what they presumptively had a right to consider their “past.” So it is hard to imagine how they experience it. Things would actually move normally after the intervention and there’d be no trace of it, it would be like the world suddenly changing in the past here, and we can’t tell. But before: the bullet missing its barrel and seeming to spring out of a tree, and then working backward we might ruin all rational history, it would be like a surreal dream that the inhabitants woke from with no way to know – so I can’t prove it didn’t happen to us! We just have that certain faith in the coherence of what we see, kind of like presuming I’m not a weird construct with simulated experiences, like a Boltzmann Brain?
Aaron, there are a lot of thermodynamic quantities that are hard to talk about far from equilibrium — temperature being an obvious example. But, I would argue (and I’m happy to hear other points of view, as the matters are by no means settled), entropy is not one of them. Given a coarse graining on the space of states, the entropy is k log W, just like Boltzmann says it is. And it tends to go up, for fairly obvious reasons, so long as it starts small. None of that story relies intimately on the properties of equilibrium.
In particular circumstances, you may be much more ambitious, and talk about the specific ways that entropy and other thermodynamic variables evolve in specific systems, and in that case the issues you raise become crucial. But pointing out the tiny entropy of the early universe doesn’t require that much care.
So the key thing is that Time didn’t reverse. The trajectories of the particles did, which would make it appear to an outside observer that time was moving backwards.
But in actually, it’s just that the particles are retracing their previous paths, and thus entropy is decreasing.
But for things to work out, everything has to happen exactly the right way. Any deviation by any particle will cause a whole chain reaction of increasing entropy, because that particle won’t be in the correct place to interact with other particles which themselves then won’t be in the right place to interact with yet other particles, and so on.
So pretty soon, the whole system goes off the rails and you have chaos…and increasing entropy.
It wouldn’t just be a matter that the bullet isn’t in the right place, but everyone acts as though it were. That bullet not being in the right place will have a long chain of disruptive effects that will eventually cause the whole march towards lower entropy to unravel.
But the main point of my original post was just to point out that (barring your outside interference) people in the reverse world would see increasing entropy even in a world with decreasing entropy, IF you take as a given that what we perceive is entirely a matter of our brain states.
What is your definition of a macrostate outside of thermodynamic equilibrium?
An equivalence class of microstates under some coarse-graining.
Again, in practice, it might be convenient to define your coarse-graining by reference to macroscopic observables that are not defined out of equilibrium. But that’s a matter of convenience, not a fundamental part of the definition Coarse grain any way you like.
Fubaris you get the issue of what goes wrong in the “time” reversed world. I suppose you are also right about experiencing such a world, since the processes are relative to the beings there and they should feel the same as we do, experiencing their world as moving normally in time. But that equivalence is just one part of the fundamental conceptual problem: if you look at the reversibility of microscopic events, you supposedly get the idea that there is no “true forward direction of time” – it is relative and no more literally real than “true velocity”.
But then what accounts for the thermodynamic process going the way it does? You can say “chance” etc, but why does the chance favor the “correct” flow of time and make our world robust under alterations, but leave the world in reversal at constant risk for drastic screwup if any little thing doesn’t fit together right? If there’s no real absolute time, how can the worlds even be different from each other? That is a deep philosophical issue and I wish the pros around here would deign to remark on it …
One issue inadequately brought up in thermodynamic discussions, and made by Roger Penrose is that wave functions aren’t time symmetrical. They spread out from the “emission point” and then the “collapse” is not like that. But some would say the WFs aren’t ultimately real (what is, then?) and claim that decoherence solves the problem of collapse. But “decollusion” doesn’t, because it involves a circular argument (inserting probabilities at the outset – the thing you’re trying to explain) as well as applying ensemble context to the single event. There is still no valid or non-surreal way to produce the one result while excluding the other (like at distantly separated detectors) without a conventional spread-out WF that goes “poof” when detection obtains at one of the possible hitting points.
Maybe I’m stupid but i don’t understand why entropy increases or decreases when the total content of the universe does not change. Why are Ice-cubes more ordered than water. The total energy and information contained in the ice doesn’t vanish it’s still there. So for the universe as a whole doesn’t lose any information and I don’t see how one state of matter and of the universe as a whole is more or less structured. Is it only structured by our standards or is there some physical meaning and process to the system that the system would register that it was ordered or disordered. From a particles perspective does it matter if it is water or ice? What physical state of a system makes it ordered or not. I don’t know if
I’m explaining it well. But the total energy of the universe is zero, matter being positive and gravity being negative, why is there this conception of entropy, from matter/energy point of view? What does it mean to be ordered from the universes perspective.
>> But, I would argue (and I’m happy to hear other points of view, as the matters are by no means settled), entropy is not one of them.
And I would agree with you.
Temperature (mean kinetic energy) may not be well defined for a system out of equilibrium, but entropy certainly is.
If entropy would only be defined for a system in equilibrium, we would not need the 2nd law, because we would always have dS/dt = 0.
But there is an issue with gravity, which you glossed over. We simply do not know how to calculate the entropy of spacetime. Penrose made the proposal to equate it with the Weyl curvature, but one can show that there are problems with that.
Thus his proposal that low entropy in the early stages of the universe means vanishing Weyl curvature is on shaky grounds…
wolfgang– I don’t want to gloss over that at all; calculating the entropy of a system in which gravity is important is something we don’t know how to do, and that’s a problem. However, we have fairly reliable estimates in certain circumstances of interest: homogeneous plasma in an expanding universe, or a black hole, or de Sitter space. Armed just with that, we can tell a pretty reliable story of how entropy evolves over the observable history of the universe, even if the details remain to be filled in.
David– Water vs. ice cubes isn’t the best example; if you had a truly isolated ice cube, it would just stay an ice cube, since there would be no energy available to heat it up. A better example is a warm glass of water with an ice cube, vs. a cold glass of water into which the ice cube has melted. The total energy is the same in both cases, and we can imagine such a system isolated from the rest of the universe. But the configuration in which there is an ice cube floating in the warm glass of water clearly has lower entropy, because it’s out of equilibrium. We are making some macroscopic statement (at the location of the ice cube, the average kinetic energy of a water molecule is much less than at the location of the water) which dramatically restricts the number of microstates that could satisfy that condition. There are many more ways to arrange the molecules in a glass of water at constant temperature than to separate them into water + ice cube.
The early universe is the same way; there are many ways to arrange the degrees of freedom in the universe that don’t look anything like “packed into an incredibly small region of space with no significant local gravitational fields.”