Have a Thermodynamically Consistent Christmas

By Sean Carroll | December 24, 2008 11:55 pm

The important event this Dec. 25 isn’t celebrating the birthday of Isaac Newton or other historical figures, it’s the release of The Curious Case of Benjamin Button, a David Fincher film starring Brad Pitt and based on the story by F. Scott Fitzgerald. As you all know, it’s a story based on the device of incompatible arrows of time: Benjamin is born old and ages backwards into youth (physically, not mentally), while the rest of the world behaves normally. Some have pretended that scientific interest in the movie centers on issues of aging and longevity, but of course it’s thermodynamics and entropy that take center stage. While entropy increases and the Second Law is respected in the rest of the world, Benjamin Button’s body seems to be magically decreasing in entropy. (Which does not, strictly speaking, violate the Second Law, since his body isn’t a closed system, but it sure is weird.)

Benjamin Button

It’s a great opportunity to address an old chestnut: why do arrows of time have to be compatible? Why can’t we imagine ever discovering another galaxy in which entropy increased toward (what we call) the past instead of the future, as in Greg Egan’s story, “The Hundred Light-Year Diary”? Or why can’t a body age backwards in time?

First we need to decide what the hell we mean. Let’s put aside for the moment sticky questions about collapsing wave functions, and presume that the fundamental laws of physics are perfectly reversible. In that case, given the precise state of the entire universe (or any closed system) at any one moment in time, we can use those laws to determine what the state will be at any future time, or what it was at any past time. That’s just how awesome the laws of physics are. (Of course we don’t know the laws, nor the state of the entire universe, nor could we actually carry out the relevant calculation even if we did, but we’re doing thought experiments here.) We usually take that time to be the “initial” time, but in principle we could choose any time — and in the present context, when we’re worried about arrows of time pointing in different directions, there is no time that is initial for everything. So what we mean is: Why is it difficult/impossible to choose a state of the universe with the property that, as we evolve it forward in time, some parts of it have increasing entropy and some parts have decreasing entropy?

Notice that we can choose conditions that reverse the arrow of time for some individual isolated system. Entropy counts the “typicalness” of the system’s microscopic state, from the point of view of macroscopic observers. And it tends to go up, because there are many more ways to be high-entropy than low entropy. Consider a box of gas, in which the gas molecules are (by some means) all bunched together in the middle of the box, in a low-entropy configuration. If we just let it evolve, the molecules will move around, colliding with each other and with the walls of the box, and ending up (with overwhelmingly probability) in a much higher-entropy configuration.


It’s easy to convince ourselves that there exists some configurations from which the entropy would spontaneously go down. For example, take the state of the above box of gas at any moment after it has become high-entropy, and consider the state in which all of the molecules have exactly the same positions but precisely reversed velocities. From there, the motion of the molecules will precisely re-trace the path that they took from the previous low-entropy state. To an external observer, it will look as if the entropy is spontaneously decreasing. (Of course we know that it took a lot of work to so precisely reverse all of those velocities, and the process of doing so increased the entropy of the wider world, so the Second Law is safe.)


But a merely reversed arrow of time is not the point; we want incompatible arrows of time. That means entropy increasing in some part of the universe while it is decreasing in others.

At first it would seem simple enough. Take two boxes, and prepare one of them in the low entropy state with gas in the middle, and the other in the delicately constructed state with reversed velocities. (That is, the two boxes on the left side of the two figures above.) The entropy will go up in one box, and down in the other, right? That’s true, but it’s kind of trivial. We need to have systems that interact — one system can somehow communicate with the other.

And that ruins everything, of course. Imagine we started with these two boxes, one of which had an entropy that was ready to go up and the other ready to go down. But now we introduced a tiny coupling — say, a few photons moving between the boxes, bouncing off a molecule in one before returning to the other. Certainly the interaction of Benjamin Button’s body with the rest of the world is much stronger than that. (Likewise Egan’s time-reversed galaxy, or Martin Amis’s narrator in Time’s Arrow.)

That extra little interaction will slightly alter the velocities of the molecules with which it interacts. (Momentum is conserved, so it has no choice.) That’s no problem for the box that starts with low entropy, as there is no delicate tuning required to make the entropy go up. But it completely ruins our attempt to set up conditions in the other box so that entropy goes down. Just a tiny change in velocity will quickly propagate through the gas, as one affected molecule hits another molecule, and then they hit two more, and so on. It was necessary for all of the velocities to be very precisely aligned to make the gas miraculously conspire to decrease its entropy, and any interaction we might want to introduce will destroy the required conspiracy. The entropy in the first box will very sensibly go up, while the entropy in the other will just stay high. You can’t have incompatible arrows of time among interacting subsystems of the universe.

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  • http://mirror2image.wordpress.com Serge

    Looks like the box with decreasing entropy is not Lyapunov stable. Wondering if structural stability have something to do with it ?

  • Giotis

    “The important event this Dec. 25 isn’t celebrating the birthday of Isaac Newton or other historical figures”

    I wonder….what could be the purpose of such statement? To provoke maybe?

  • http://www.americafree.tv Marshall Eubanks

    You can’t have incompatible arrows of time among interacting subsystems of the universe.

    Entropy is low in the universe because of the big bang; the universe started with a very low entropy, and it has been increasing since. Now, in a “big crunch” (a gnab gib – a big bang in reverse – what will happen in the far future if the universe is closed with a finite lifetime), the end state will very much resemble a big bang, so entropy decreases in a big crunch. Some people have used this to speculate that once the big crunch starts time will appear to reverse.

    I personally think this is silly. Except for very close to the time of the bang / crunch, the openness of the universe, and whether it is expanding or contracting, is a global phenomenon, not a local one – there is no measurement at a single point that will tell you the sign of the expansion of the universe. In the exact same way, an event horizon, the boundary around a local, not big, crunch, is not a local observable. You need not know (immediately) if you have passed through an event horizon, even though, if you do, a dramatic lowering of entropy is in your future.

    So, consider two regions of space time – one is on its way to collapse, and everything in it will eventually form a black hole, the other will continue to share in the overall expansion of the universe. Let there be two observers, call them Alice and Bob, with Alice in the open region, and Bob in the collapsing region. In the beginning, they can communicate freely. At some time, the collapse of Bob’s region begins, and the overall entropy of Bob’s area of spacetime starts to decrease. This time being not a local observable, Bob may not even know it. At some later time, Bob will pass through the event horizon, and Alice will stop receiving his messages. (Bob will for a while continue to get messages from Alice, and, if the black hole is big enough, will survive passing through the event horizon and will not be able to tell
    from any local observation that he has passed the event horizon.)

    In the early days, entropy is decreasing (over all) around Bob, and yet he can still communicate with Alice. To me, this indicates that there is more to the arrow of time than just the direction in which entropy increases, and also means that it is not clear that “[y]ou can’t have incompatible arrows of time among interacting subsystems of the universe.”

  • http://arcsecond.wordpress.com meichenl

    It makes sense to me that if you finely-tune a box to decrease in entropy, and then introduce some coupling, you destroy the fine tuning.

    But who says we have to do it that way? The coupling itself is totally deterministic and totally predictable. So why don’t we fine-tune that box so that it would have increasing (or constant) entropy in the absence of coupling, but we arrange things just so that the coupling itself pushes the second box over into a state where entropy spontaneously decreases?

    I seems like you could do this like so:
    First, set the two boxes up so one is high entropy and the other is low entropy. Let them evolve in time, coupling and all. Then after some time they’ll both be high entropy. Now reverse the velocities of all the particles, including the ones that tranfer energy between the systems. Then the two boxes should evolve back to their original states, meaning on has decreasing entropy and the other is pretty much normal (stays high entropy the whole time).

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    meichenl– that’s a good point, and I thought about including it in the original post, but Santa was coming. Your specific proposal would induce one “reversed” arrow of time and one box without any arrow of time at all (since the entropy just stays high).

    But you could just impose boundary conditions by fixing positions and not velocities at two times: early and late. So you could insist that one box had low entropy at time 1, and the second box had low entropy at time 2 (and the other had high entropy at that time). This case is a little more subtle, but it comes close to working. One problem with it is that there’s no reason for the individual entropies to continually, smoothly evolve in opposite senses, as in Benjamin Button et al. It’s more likely that both entropies will remain high for most of the evolution, and one will suddenly go down at the end. More importantly, reverse evolution of entropy in one box will depend intimately on its interactions with the other box. There is no sense in which the backwards box has its own reversed arrow of time all by itself; we would interpret its evolution as being nudged into a lower-entropy state by the rest of the world. But it’s the best you can do, I think.

    Marshall– there’s no reason to think that entropy would decrease if there were to be a Big Crunch. It could very easily increase along the way.

    Giotis– trust me, there is no fun in provoking the kind of person who would be provoked by that.

    Serge– it’s completely unstable. That’s one reason why things like this don’t happen in the real world.

  • Eric

    “Let’s put aside for the moment sticky questions about collapsing wave functions, and presume that the fundamental laws of physics are perfectly reversible.”
    Bah! Why only for the moment? What would possibly make you even consider a non-linear, non-unitary, non-differentiable, non-local, non-CPT-symmetric, non-relativistic, acausal, faster-than-light phenomenon?

    And if I remember correctly, theoretically, entropy doesn’t increase or decrease, phase space is conserved. It just spreads out into a more “complex” shape so drawing a simple boundary will give you a larger space.

  • http://physicsmuse.wordpress.com Sandy

    Hi Sean, thanks for the post.
    There could be an arrow of time set into motion at the big bang (along with space) however, using the Second Law of Thermodynamics to explain the arrow of time is taking a law for very specific situations and applying it to the fabric of the universe….I think that is over reaching. In some parts of the universe entropy is increasing, in other parts decreasing. Adding energy to any system can increase or decrease the entropy of that system. Even if the entropy of the universe can be shown to be decreasing over all, and even if these processes are shown to be irreversible in time, what mathematical or experimental evidence does therefore show these processes to be (or to represent) the arrow of time itself??

    Heat will never flow from a cold object to a hot one. So the lack of motion wont flow into areas of higher motion. Well, the lack of motion isn’t really anything, so how could it flow?? These are attributes of matter, not time.

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  • http://www.gregegan.net/ Greg Egan

    Sean, I certainly wouldn’t defend the premise of The Hundred Light Year Diary as being likely in our own universe, but the assumption (admittedly not made explicit in the story) is that the time-reversed galaxy is one of the, er, “last remnants” of a low entropy future in which the whole universe shared its thermodynamic arrow. The low entropy future is not meant to arise merely because there’s a Big Crunch; the assumption is that some mechanism responsible for imposing a low entropy on the Big Bang functions in a time-symmetric manner to impose the same restriction on the Big Crunch.

    The whole history of the universe is assumed to be roughly time-symmetric, with some remnants from each end surviving a certain way past the midpoint, but the overwhelming majority of systems being scrambled by that encounter.

    Are you really sure that if you impose identical low-entropy boundary conditions on the universe, there either won’t be clear thermodynamic arrows ever, or that if there are clear arrows in each epoch, they will be completely scrambled at the midpoint of the evolution? Has anything been published which analysed that scenario in detail? As I said, I don’t imagine for a moment that this has any bearing on reality, but it’s still not clear to me why this should be impossible in principle.

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Greg– things like that have definitely been considered, originally by Thomas Gold (and reconsidered by Huw Price), and more recently in some detail by Gell-Mann and Hartle (http://arxiv.org/abs/gr-qc/9304023). As you say, you can have a low-entropy big crunch, just as you can have a low-entropy whatever, just by imposing it. There would be clear arrows near the Bang/Crunch, and something of a mess in between, depending on how things were interacting.

    My interest in the post was less in scenarios like your story (which I enjoyed very much), and more in the clearly fantastical realm of two localized systems, strongly interacting right next to each other, with two opposite-oriented arrows of time.

    Sandy– I have no objections to arrows of time being oriented differently in different parts of spacetime which are essentially disconnected, but my interest was more in readily-interacting systems, as above.

  • JohnDrori

    Isn’t the case of heat transfer between two boxes an example for interacting subsystems in which one loses entropy while the other gains (although overall entropy increases), meaning incompatible arrows of time?
    I’m just an undergraduate physics student so I might not understand what I’m saying.

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  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    John– Sure, but only because the energy also changes in this very non-closed system. My concern (not clearly spelled out, I’ll admit) was with a quasi-autonomous system that could maintain a reversed arrow of time while interacting unpredictably with the outside world. That’s what can’t happen.

  • Ben Button

    Slightly off-topic: I suddenly realised, from reading this, that I don’t understand the “Horizon Problem” which inflation supposedly solves. The Horizon problem says: how come things were in equilibrium at decoupling, when there had not been enough time for things to equilibrate? The assumption being that equilibrium is some kind of special state that has to be reached from something else. But isn’t equilibrium the most probable state? ie, aren’t *deviations* from equilibrium the things that need explaining? So why be surprised? I can see that inflation does all kinds of other good stuff, but I [now!] find it hard to see that there is a problem here in the first place…..

  • Loki

    Sean, as for Big Crunch entropy – isn’t it the case that a black hole is in fact a state of maximum possible entropy for the matter in it?
    I have a feeling i read once (Penrose?) that the final black mega-hole in case we have BC is no way the same singularity as Big Bang because it will have immence entropy opposite to extremely low of BB. And that is kind of good argument against cyclical universe.

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  • John R Ramsden

    As I see it the only way the Big Bang could start with low entropy is if it were somehow regularized or “evened out” by interactions in a finer-grained high entropy substrate, rather in the manner that random air molecules blown over the surface of an ocean can interact with random water molecules in the ocean to produce a (fairly) regular train of waves. (I believe this process of wave formation relies on non-linear effects, to break the principle of superposition and allow currents and undertows to redistribute energy between waves.)

    One could even extend the analogy and identify particles with randomly produced “rogue waves” in which sufficiently more energy had been collected to produce some qualitatively new effect.

    Of course, this assumes that there is that level of fine graining, and it is then natural to assume finer levels continuing indefinitely, which is somewhat at odds with the idea of a fundamental scale below which nothing can exist as a separate entity. But it would be a way out of the impasse.

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Ben– I don’t think that’s the right way to state the horizon problem (although it is very often stated badly). In a universe filled with nothing but matter and radiation, two distant points on the surface of last scattering were never in causal contact. So why are they at (nearly) the same temperature? That’s the horizon problem. Having everything at the same temperature is certainly not the most probable state in this particular kind of background (an expanding universe). There are many more ways for conditions to be very different from place to place. (Think of it this way: if you had several *disconnected* boxes of gas, with randomly chosen densities and energies, you would indeed be very surprised to find them all at the same temperature, even if that would be the eventual high-entropy state once they were all brought into contact.)

    Loki– a black hole is a state of maximum entropy for a given volume, not for a given amount of matter (although you will sometimes hear otherwise). If they were really the maximum entropy states, black holes couldn’t evaporate. But yes, there is every reason to think that a future crunch will be high-entropy, but of course there is also good reason to think that there won’t be any future crunch.

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

    Anyone in or south of the tropics need only go as far as your air conditioner to find an increasing-entropy chamber coupled to a decreasing entropy one. The rest of y’all can look at your refridgerators.

    As long as you’re willing to put a little work in, it’s easy.

  • Count Iblis

    I don’t understand why the “mixed boundary conditions”, where you impose low entropy initial conditions on box 1 and low entropy “final conditions” on box 2, can be made to work. The whole setup can then be made symmetric w.r.t. time reversal and interchanging the two boxes. Then, if there is an arrow of time in box 1, then there will be an arrow of time pointing in the opposite direction in box 2.

    Not to have an arrow of time would mean that the entropy stays close to the high entropy state except very close to the point when the boundary conditions are imposed as Sean suggested. However, you would expect that box 1 will evolve similarly as if it were coupled to some heat bath (at least in the beginning). By making the coupling arbitrarily small, you should be able to let box 1 gradually heat up instead of jumping very fast to a high entropy state.

    It will still true that most of the time both boxes will be in the high entropy state. But if the coupling is weak, you could have some creature in box 1 living for a while wondering why his box is heating up while in box 2 the same creature exists evolving in the opposite time direction.

  • CarlN

    Yes, yes. Very well. After a few Christmas beers (do you have Christmas beer in the US?) let me tell a Christmas story. We know already that things cannot come from things. Things must come from nothing. Old Eternal (God or Reality) would have reached thermal equilibrium many infinities ago and is just a figment of our imagination.

    Now, poor old Nothing is not so resourceful. Things are created as simply as possible. The easiest way to specify an initial condition is to arrange a zero entropy condition. A single quantum state.

    Since this is Christmas we should also think about good old Nothing and remember how it all came to be:

    1. “When” there is nothing there is not anything that could prevent something to be created from nothing. Such “prevention” does not exist “when” nothing exists.

    2. “When” no things exists there are no conditions (like conservation laws) that need to be fulfilled in order for something to be created from nothing. Such conservation laws does not exist “when” nothing exists.

    3. There is no need for causation for something to be created from nothing as such need does not exist “when” nothing exists.

    Also, besides what exists there is Nothing.
    And what does it mean to exist? Do you think you exist? Yes, you exist in a funny way.


  • Ben Button

    Hi Sean, thanks for your reply.

    You say: “Having everything at the same temperature is certainly not the most probable state in this particular kind of background (an expanding universe). There are many more ways for conditions to be very different from place to place.”

    I thought that what counts is: how many ways can you re-arrange things *so that the macroscopic states are indistinguishable*. I would have thought that uniformity would have *more* ways of re-arranging things without changing macroscopic conditions than non-uniformity, no?

    I guess in General Relativity, a very uniform matter distribution would correspond to very smooth geometry, which is a low-entropy state, so a uniform initial state becomes very unlikely when you take that into account — is that what you mean?

    ” (Think of it this way: if you had several *disconnected* boxes of gas, with randomly chosen densities and energies, you would indeed be very surprised to find them all at the same temperature, even if that would be the eventual high-entropy state once they were all brought into contact.)”

    I guess the point here is contained in “randomly chosen”. Suppose God creates a room full of gas and does it in a completely “random” manner. Won’t the room *immediately* be full of gas, uniformly distributed, in equilibrium? After all, that is the most probable state, right? [Again, I’m ignoring gravity here; if God takes that into account, the result will probably be a black hole I guess….]

  • http://blogs.discovermagazine.com/cosmicvariance/sean/ Sean

    Yes, gravity is the whole point here. In a universe with lots of matter and gravity turned on, the highest-entropy state is very inhomogenous, not smooth at all. Think of what would naturally happen in a collapsing universe: it would not smooth itself out.

    But beyond that, the horizon problem is really an issue of causality, not just entropy. The different parts of the universe are all rapidly evolving, but they are evolving in perfect synchrony, despite never having been in causal contact.

  • Ben Button

    Thanks, sorry I should have dug further back in this blog….

    “The different parts of the universe are all rapidly evolving, but they are evolving in perfect synchrony, despite never having been in causal contact.”

    That may not be as bad as it sounds. After all, we have no problem accepting that laws of nature are valid all over spacetime, but there is no suggestion that this was established “causally”. If there is some law of nature that dictates initial smoothness everywhere in space, then that would solve the problem without any violation of causality. So I still find the “horizon problem” rather fishy. Anyway thanks again.

  • http://AgingAdvice.org Josh Mitteldorf

    This is a subject for biology and not for physics.

    > In fact, there is no relationship between the second law of thermodynamics and the fact that most animals senesce.

    > In fact, there is no theoretical reason why an animal could not get more robust, more fertile, and stronger with each passing year.

    > In fact, there are some animals and many more plants that fit the definition of “negative senescence”. Mortality rates decline over time, rather than rising exponentially as in the usual process of senescence.

    There is a review paper on this subject, “The Case for Negative Senescence”, from the laboratory of James Vaupel, who is probably the world’s most prominent demographer:
    Theor Popul Biol. 2004 Jun;65(4):339-51.

  • http://www.americafree.tv Marshall

    Ben Button Says:
    December 25th, 2008 at 8:27 pm

    Slightly off-topic: I suddenly realised, from reading this, that I don’t understand the “Horizon Problem” which inflation supposedly solves.

    Think of it this way. Go out with your handy horn antenna and measure the cosmic microwave background (CMB) background temperature in a random direction. Now measure the temperature in the exact opposite direction. You are seeing photons, greatly redshifted, from the last scattering surface of the so called recombination. If you make some simple assumptions about the topology of the universe (e.g., that spacetime isn’t a torus or a Klein bottle or some other tricky topology), then those photons come from regions of spacetime that had no causal connection with each other at the time of recombination – i.e., there is no way for a star or a nova or a intelligent being in one of those regions to communicate with or influence the other, at least using light or matter to send signals. These two regions only “came through the horizon” and became causally connected with you now, when you got their photons, and are still not causally connected with each other. (You are closer to them than they are to each other.)

    This leads to the Horizon Problem : if these two regions have never been in contact, then how do they “know” to have the same temperature ? You could have weird topologies for the entire universe (where looking in opposite directions means that you are really looking at the same distant objects), but I am not aware of any interesting theories that require or assume weird topologies. A related problem is the smoothness of the universe – gravity will make small initial perturbation grow, and it is hard to see how an initial big bang would give us the smoothness of the microwave background we observe. Note that having weird topologies really don’t do anything to solve this problem.

    In inflation, a small piece of very early space time greatly expanded, at much faster than the speed of light, making everything we see (even the entire cosmic microwave background) to have come from one tiny original patch of spacetime, which was causally connected (and quite smooth, because it was tiny). So, in a sense you have pushed the Horizon problem “up a step” to the theory of inflation (there is no need to postulate faster light travel now, but you have to postulate a rapidly expanding spacetime then), but the theory of inflation seems to hang together well, and it does solve the horizon and smoothness problems.

  • http://arunsmusings.blogspot.com Arun

    Isn’t there a purely geometric way of posing the problem? The direction of time on the worldline of a particle is defined by the worldcones – the volume of one is shrinking and the other is expanding. Spacetime events move from within the future lightcone to the past lightcone. If this is a good definition of past and future, then consider two observers, each of whose past lies in the other’s future. We’re not invoking entropy here nor the reversibility of the laws of physics, nor any physical property of the observer (e.g., memory). What rules this out?

  • http://www.virgilanti.com/journal/ Virge

    “Notice that we can choose conditions that reverse the arrow of time for some individual isolated system. Entropy counts the “typicalness” of the system’s microscopic state, from the point of view of macroscopic observers.”

    A system that you have contrived such that it will change over time to a “low entropy” state must itself be very highly ordered. It’s not “typical”. It’s just that the very special ordering is not as obvious as it is when all your particles are bunched up in one place.

    To me this seems to be a way of deceiving yourself that you’ve created something different. All you’ve done is to create a system that shows the weakness of our method of calculating the macroscopic property we call entropy.

    Sean: “It’s more likely that both entropies will remain high for most of the evolution, and one will suddenly go down at the end.”

    It’s not a statistical thermodynamics measure of entropy that will “suddenly go down”. It’s a sloppy macroscopic measure of entropy that you’re deceiving yourself with.

    Concepts like the Arrow of Time only work as a macroscopic approximation when you don’t have a deceptive intelligence contriving your starting conditions. As soon as you introduce a deliberate deceiver, you’re not talking physics, you’re talking parlor tricks.

    And you’re right that interactions with other systems will break the illusion. A system designed to evolve so that it simulates a reversed Arrow of Time is so fragile that any interaction from outside the system will wreck the illusion.

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Cosmic Variance

Random samplings from a universe of ideas.

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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