A Universe Out of Chaos

How did the universe come to be? We don’t know yet, of course, but we know enough about cosmology, gravitation, and quantum mechanics to put together models that standing a fighting chance of capturing some of the truth.

Stephen Hawking‘s favorite idea is that the universe came out of “nothing” — it arose (although that’s not really the right word) as a quantum fluctuation with literally no pre-existing state. No space, no time, no anything. But there’s another idea that’s at least as plausible: that the universe arose out of something, but that “something” was simply “chaos,” whatever that means in the context of quantum gravity. Space, time, and energy, yes; but no order, no particular arrangement.

It’s an old idea, going back at least to Lucretius, and contemplated by David Hume as well as by Ludwig Boltzmann. None of those guys, of course, knew very much of our modern understanding of cosmology, gravitation, and quantum mechanics. So what would the modern version look like?

That’s the question that Anthony Aguirre, Matt Johnson and I tackled in a paper that just appeared on arxiv. (Both of my collaborators have also been guest-bloggers here at CV.)

Out of equilibrium: understanding cosmological evolution to lower-entropy states
Anthony Aguirre, Sean M. Carroll, Matthew C. Johnson

Despite the importance of the Second Law of Thermodynamics, it is not absolute. Statistical mechanics implies that, given sufficient time, systems near equilibrium will spontaneously fluctuate into lower-entropy states, locally reversing the thermodynamic arrow of time. We study the time development of such fluctuations, especially the very large fluctuations relevant to cosmology. Under fairly general assumptions, the most likely history of a fluctuation out of equilibrium is simply the CPT conjugate of the most likely way a system relaxes back to equilibrium. We use this idea to elucidate the spacetime structure of various fluctuations in (stable and metastable) de Sitter space and thermal anti-de Sitter space.

It was Boltzmann who long ago realized that the Second Law, which says that the entropy of a closed system never decreases, isn’t quite an absolute “law.” It’s just a statement of overwhelming probability: there are so many more ways to be high-entropy (chaotic, disorderly) than to be low-entropy (arranged, orderly) that almost anything a system might do will move it toward higher entropy. But not absolutely anything; we can imagine very, very unlikely events in which entropy actually goes down.

In fact we can do better than just imagine: this has been observed in the lab. The likelihood that entropy will increase rather than decrease goes up as you consider larger and larger systems. So if you want to do an experiment that is likely to observe such a thing, you want to work with just a handful of particles, which is what experimenters succeeded in doing in 2002. But Boltzmann teaches us than any system, no matter how large, will eventually fluctuate into a lower-entropy state if we wait long enough. So what if we wait forever?

It’s possible that we can’t wait forever, of course; maybe the universe spends only a finite time in a lively condition like we see around us, before settling down to a truly stable equilibrium that never fluctuates. But as far as we currently know, it’s equally reasonable to imagine that it does last forever, and that it is always fluctuating. This is a long story, but a universe dominated by a positive cosmological constant (dark energy that never fades away) behaves a lot like a box of gas at a fixed temperature. Our universe seems to be headed in that direction; if it stays there, we will have fluctuations for all eternity.

Which means that empty space will eventually fluctuate into — well, anything at all, really. Including an entire universe.

This basic story has been known for some time. What Anthony and Matt and I have tried to add is a relatively detailed story of how such a fluctuation actually proceeds — what happens along the way from complete chaos (empty space with vacuum energy) to something organized like a universe. Our answer is simple: the most likely way to go from high-entropy chaos to low-entropy order is exactly like the usual way that systems evolve from low entropy to high-, just played backward in time.

Here is an excerpt from the paper:

The key argument we wish to explore in this paper can be illustrated by a simple example. Consider an ice cube in a glass of water. For thought-experiment purposes, imagine that the glass of water is absolutely isolated from the rest of the universe, lasts for an infinitely long time, and we ignore gravity. Conventional thermodynamics predicts that the ice cube will melt, and in a matter of several minutes we will have a somewhat colder glass of water. But if we wait long enough … statistical mechanics predicts that the ice cube will eventually re-form. If we were to see such a miraculous occurrence, the central claim of this paper is that the time evolution of the process of re-formation of the ice cube will, with high probability, be roughly equivalent to the time-reversal of the process by which it originally melted. (For a related popular-level discussion see From Eternity to Here, ch. 10.) The ice cube will not suddenly reappear, but will gradually emerge over a matter of minutes via unmelting. We would observe, therefore, a series of consecutive statistically unlikely events, rather than one instantaneous very unlikely event. The argument for this conclusion is based on conventional statistical mechanics, with the novel ingredient that we impose a future boundary condition — an unmelted ice cube — instead of a more conventional past boundary condition.

Let’s imagine that you want to wait long enough to see something like the Big Bang fluctuate randomly out of empty space. How will it actually transpire? It will not be a sudden WHAM! in which nothingness turns into the Big Bang. Rather, it will be just like the observed history of our universe — just played backward. A collection of long-wavelength photons will gradually come together; radiation will focus on certain locations in space to create white holes; those white holes will spit out gas and dust that will form into stars and planets; radiation will focus on the stars, which will break down heavy elements into lighter ones; eventually all the matter will disperse as it contracts and smooths out to create a giant Big Crunch. Along the way people will un-die, grow younger, and be un-born; omelets will convert into eggs; artists will painstakingly remove paint from their canvases onto brushes.

Now you might think: that’s really unlikely. And so it is! But that’s because fluctuating into the Big Bang is tremendously unlikely. What we argue in the paper is simply that, once you insist that you are going to examine histories of the universe that start with high-entropy empty space and end with a low-entropy Bang, the most likely way to get there is via an incredible sequence of individually unlikely events. Of course, for every one time this actually happens, there will be countless times that it almost happens, but not quite. The point is that we have infinitely long to wait — eventually the thing we’re waiting for will come to pass.

And so what?, you may very rightly ask. Well for one thing, modern cosmologists often imagine enormously long-lived universes, and events like this will be part of them, so they should be understood. More concretely, we are of course all interested in understanding why our actual universe really does have a low-entropy boundary condition at one end of time (the end we conventionally refer to as “the beginning”). There’s nothing in the laws of physics that distinguishes between the crazy story of the fluctuation into the Big Crunch and the perfectly ordinary story of evolving away from the Big Bang; one is the time-reverse of the other, and the fundamental laws of physics don’t pick out a direction of time. So we might wonder whether processes like these help explain the universe in which we actually live.

So far — not really. If anything, our work drives home (yet again!) how really unusual it is to get a universe that passes through such a low-entropy state. So that puzzle is still there. But if we’re ever going to solve it, it will behoove us to understand how entropy works as well as we can. Hopefully this paper is a step in that direction.

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86 Responses to A Universe Out of Chaos

  1. George says:

    Is there a transcendent timeless reality out there from which our universe emerged? Could this explain the existence of our universe? What can we discover about this transcendent reality? Do we have direct experience of this transcendent reality?

  2. Nick says:

    In Tegmark’s Ultimate Ensemble of all possible mathematical structures, what is the spotlight in the darkness shining upon a particular mathematical structure actualizing it in the sea of potential mathematical structures? If all which can exists exist, why are we not all?

  3. Jonathan says:

    “Along the way people will un-die, grow younger, and be un-born”

    and along this path which way will their memories point? To what is (from our perspective) going to happen (this seems somewhat acausal – memories created before the event occurs), or the other way around, in which case they remember their ‘old’ years and look forward to their ‘young’ years – all seems pretty strange to me. If it’s the former then we could just as well be in the devolving universe state, but wouldn’t know because all we remember is what is to come, which we think is our past.

    If indeed we are in the devolving universe state and heading in the direction of what we think of as the big bang, then there would be a very large chance that this fluctuation will stop pretty soon and we’ll start going back in the ‘normal’ direction – it seems that in this scenario you never actually need to reach the big bang: The ice cube can half freeze, and then start melting again.

    If this is the case, then the big bang need never have happened, but we could just have appeared to have come from it.

    My thought is that the fault in the argument is that the progress towards a lower entropy state should take us through the state with complex structures we see around us…ie. the egg coming back together etc. Surely there are simpler paths to a low entropy state than civilisation undoing itself in perfect unison (though this may be precisely what the US administration is attempting ;-).

  4. BrianC says:

    Following up on Jonathan’s comment, and Sean’s statement that artists will ‘painstakingly remove paint from their canvases’ in this scenario:

    It seems to me that these hypothetical artists will believe that they are actually applying paint to their canvases, not removing it. The reason is that their neurons etc. are simply following a time-reversal of what we consider ‘normal’, and thus their consciousness at any instant in time is identical whether cosmological entropy is increasing or decreasing. They will not remember the ‘past’, but only their ‘future’.

    Extrapolating this a bit, it seems like this reasoning implies that observers have *no way* of judging whether the universal ice cube is in the process of melting or freezing. Therefore this theory must be necessarily untestable by conscious beings, and thus it is truly a philosophical rather than a physical effort at understanding the world.

  5. mtravers says:

    As an alternative to Quantum Theory there is a new theory that describes and explains the mysteries of physical reality. While not disrespecting the value of Quantum Mechanics as a tool to explain the role of quanta in our universe. This theory states that there is also a classical explanation for the paradoxes such as EPR and the Wave-Particle Duality. The Theory is called the Theory of Super Relativity and is located at Super Relativity Website. This theory is a philosophical attempt to reconnect the physical universe to realism and deterministic concepts. It explains the mysterious.

  6. Ian Ward says:

    If the universe is cyclical and each cycle is larger than the previous cycle, then moving towards the low entropy of a big crunch is also moving towards the higher entropy of the next cycle.

  7. Neal says:

    Ian, I don’t think that follows. Of all possible histories, any which go through a big crunch are lower entropy than any which increase entropy indefinitely. The universe will tend to shy away from low-entropy states (by the definition of low entropy), so any history which goes through a big crunch is less probable than a history which does not.

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  9. Jasper says:

    I understand the reasoning, but why are you making the point that the universe should always be the same. If i just displace or remove some particles it still kind of looks like our universe. even statistical physics tells us that you should be looking at ensembles of universes. Maybe the amount of our-universe-like universes is just quite large.

  10. Tyler says:

    Looked at from the perspective of a closed system, if we leave an ice cube (low-entropy) in a glass of water at room temperature, in a few minutes it will melt (high-entropy) and cool the water inside. If we then remove a few milliliters of water from the glass (high-entropy) and freeze it back into an ice cube (low-entropy), we have successfully reversed the second law of thermodynamics within the system and can begin the process again. (Also note: freezing is equivalent to ‘unmelting,’ and less of an awkward term). If the above accurately describes a ‘small-scale’ closed system, then it only makes sense that a ‘large-scale’ closed system, such as the universe, works the same way.

    To me, the only issue arises with the concepts of equilibrium, and closed or open system. For example, if two systems are in thermal equilibrium, then their temperatures are the same. Thus if the definition of an open system is that matter may flow in and out of the system boundaries, then when two systems are in thermal equilibrium, it must also be equivalent to say that this equilibrated system is no longer open, but closed because there is no system boundary between them.

    The universe is equivalent to an ice cube melting in a glass of room temperature water, or a lake freezing over in winter and then thawing again in spring; the universe must be a self-regulating process of continual melting and freezing.

    Sean, I look forward to your comments if you have any.

  11. Jonathan says:

    Tyler, the problem with your argument is that in the re-freezing you are not considering the thing that you are using to re-freeze the water. If you include this then the entropy will increase as you put in energy to freeze the water – your machine will use fuel, will heat up, etc. If you don’t include this, then you are not dealing with a closed system.

  12. Bill Davis says:

    Sean (or anyone!) – I’m reading ‘From Eternity to Here’ at the moment and I’ve been niggled by something since Chapter 10. This post has increased my niggle. The post (and some comments) seem to hint at what I’m thinking, but I haven’t seen it stated explicitly (I’m not a physicist, so more than likely I’m missing something). So: if the arrow of time is determined by the second law (and specifically the fact that memory is a coherent concept only under the assumption of a lower entropy past) then how does it make sense to talk about a fluctuation from maximum entropy at all? You cannot describe a fluctuation without a dimension of time (as in the horizontal axis of figure 54 in the book), but the initial part of the fluctuation involves entropy decreasing in the forward direction of time, which can’t happen by definition. In this context does it make any sense to ask how long you need to wait for a maximum entropy universe to experience a fluctuation of the magnitude that would lead to what we observe? Surely in a uniformly high entropy universe there is no such thing as time?

  13. Ian Ward says:

    Reply to (32): Neal: When a star collapses before going supernova, the total entropy of the star system doesn’t decrease. Maybe big crunches are similar to stellar collapses in that space doesn’t collapse and the entropy of the universe doesn’t actually decrease when moving through them.

  14. ian says:

    Interesting, and in some ways similar both to Nietzsche’s eternal recurrence and Asimov’s last question (just to pick two). Of course, those didn’t make it to arxiv.

  15. Jim Harrison says:

    If a universe really were to revert to its initial state via the same stages by which it got to its endpoint, the conscious animals in that universe would not experience themselves living backwards. They would still remember the future, not the past, assuming, of course, that mental states are absolutely determined by physical ones. So how exactly do you propose to distinguish the road up and the road down in your story? Maybe things are running backwards right now.

  16. Sean says:

    As people have noted (and I’ve said many times myself, although not in this post), the “backwards-living” people in the universe we describe wouldn’t think they were living backwards at all. We always remember the direction in which entropy was lower, so their evolution is internally indistinguishable from an ordinary Big Bang.

    The only difference, therefore, is external. We are talking about processes that happen in a universe that lasts forever. Inside that universe, there will inevitably be universe-creating fluctuations like this (as well as an enormously larger number of smaller fluctuations), and then these fluctuations will decay back to equilibrium. It only makes sense to say that the arrow of time is “backwards” in any one region when we’re comparing it to other regions.

    However, the fact that there are many much smaller fluctuations tends to imply that this is not the right story of the universe. (If it were, we would probably live in a much smaller fluctuation.) So either the universe is not eternal, or something else, as we briefly touch on in the paper.

  17. Steve Smith says:

    Stephen Hawking‘s favorite idea is that the universe came out of “nothing” — it arose (although that’s not really the right word) as a quantum fluctuation with literally no pre-existing state. No space, no time, no anything.

    These ideas are receiving a lot more attention in the popular media and press, and I think that a few pointers to the technical ideas that motivate them are necessary. So here’s some scientific background and links on universe ex nihilo theories, a background that isn’t presented widely enough, even at scienceblogs that address the subject specifically.

    Guth’s Inflationary Universe is a must-read, in which Guth explains ex nihilo theories with the colorful statement:

    The question of the origin of the matter in the universe is no longer thought to be beyond the range of science—everything can be created from nothing … it is fair to say that the universe is the ultimate free lunch.

    Guth provides technical reasons for this claim:

    Now we can return to a key question: How is there any hope that the creation of the universe might be described by physical laws consistent with energy conservations? Answer: the energy stored in the gravitational field is represented by a negative number! … The immense energy that we observe in the form of matter can be canceled by a negative contribution of equal magnitude, coming from the gravitational field. There is no limit to the magnitude of energy energy in the gravitational field, and hence no limit to the amount of matter/energy it can cancel. For the reader interested in learning why the energy of a gravitational field is negative, the argument is presented in Appendix A.

    Guth goes on to explain a simple argument for all this that if you grasp, you will understand a fact of gravity that evaded Newton. Unfortunately, Google books doesn’t have Appendix A online.

    Guth’s technical explanation above is what is meant by the nontechnical, poetic description, like Hawking’s: “Because there is a law like gravity, the universe can and will create itself from nothing.”

    Here are some pointers to a quick technical explanation of the creation of a universe from literally nothing subject to the laws of quantum mechanics.

    A technical account of the universe ex nihilo, following Vilenkin, “Creation of universes from nothing”. Physics Letters B Volume 117, Issues 1-2, 4 November 1982, Pages 25–28. Available here.

    1. Observe the Friedmann–Lemaître–Robertson–Walker metric for universal expansion:

    ds² = dt² – a(t)|dx

    This is the space-time geometry with the spatial scale term a(t) describing the growth/contraction of the universe. This is Vilenkin’s equation (2).

    2. Solve the evolution equation:

    a(t) = (1/H)cosh(Ht)

    where H² = (8π/3)Gρ is the Hubble parameter.

    This is Vilenkin’s equation (3). So far, there is no explanation of a universe from nothing because the de Sitter space isn’t nothing, as everyone agrees.

    3. Observe that at t = 0, the physics has the same form as a potential barrier, for which it is known that quantum tunneling is possible. The description of quantum tunneling involves a transformation tit, with i² = –1.

    Now the evolution equation is

    a(t) = (1/H)cos(Ht) [the cosine “cos”, not the hyperbolic cosine “cosh”]

    valid for |t| < π/2/H. This is Vilenkin’s equation (5). Space-Time is simply the 4-sphere, a compact, i.e, bounded space. At the scale a(t) = 0, this space is literally nothing. No space-time, no energy, no particles. Nothing. The interpretation of (5) is quantum tunneling from literally nothing to de Sitter space, the universe as we know it. See Figure 1a in Vilenkin’s paper for a depiction of the creation of the universe from nothing using this explanation.

    Vilenkin says in the paper, “A cosmological model is proposed in which the universe is created by quantum tunneling from literally nothing into a de Sitter space. After the tunneling, the model evolves along the lines of the inflationary scenario. This model does not have a big-bang singularity and does not require any initial or boundary conditions. … In this paper I would like to suggest a new cosmological scenario in which the universe is spontaneously created from literally nothing, and which is free from the difficulties I mentioned in the preceding paragraph. This scenario does not require any changes in the fundamental equations of physics; it only gives a new interpretation to a well-known cosmological solution. … The concept of the universe being created from nothing is a crazy one. To help the reader make peace with this concept, I would like to give an example of a compact instanton in a more familiar setting. …”

    This is what physicists mean by “nothing”. Nonexistent space-time, subject to the laws of quantum mechanics.

    Guth provides a nontechnical explanation:

    Alexander Vilenkin … suggested that the universe was created by quantum processes starting from “literally nothing,” meaning not only the absence of matter, but the absence of space and time as well. This concept of absolute nothingness is hard to understand, because we are accustomed to thinking of space as an immutable background which could not possibly be removed. Just as a fish could not imagine the absence of water, we cannot imagine a situation devoid of space and time. At the risk of trying to illuminate the abstruse with the obscure, I mention that one way to understand absolute nothingness is to imagine a closed universe, which has a finite volume, and then imagine decreasing the volume to zero. In any case, whether one can visualize it or not, Vilenkin showed that the concept of absolute nothingness is at least mathematically well-defined, and can be used as a starting point for theories of creation.

  18. Lou Jost says:

    I have a problem with the often-repeated idea that if the universe lasts an infinite amount of time, every possible state will happen. Doesn’t this assume the cardinality of the set of all possible states is the same as the cardinality of time?

  19. Lou Jost says:

    “The point is that we have infinitely long to wait — eventually the thing we’re waiting for will come to pass.”

    Even if the cardinalities of the set of all possible times and the set of all possible states were the same, it is still not clear to me that, given an infinite amount of time, every state will occur. Consider a hotel with an infinite number of rooms. (Mathematicians spend an inordinal amount of time in this hotel when they think about infinity.) Suppose it has an infinitely long stream of customers. The receptionist can choose to give every new guest an odd-numbered room. Even though there are in infinite number of guests, the hotel will never need to use the even-numbered rooms, as the hotel will never fill up. (I think this example is given in a book called “Is God a mathematician” by Mario Livio.) If the rooms represent states, this suggests that not all states need to be filled, even given an infinite amount of time.

  20. Julien says:

    Sean, I don’t get why we should consider “backwards-living” peoples in the first place. This would implicate a “backward natural selection”, despite this makes very little difference in the total entropy. Am I wrong to think that a sterile universe would be far more likely for the backward direction?

  21. Trevor says:

    @Lou,

    Usually people talk about the idea that in an infinite Universe (in space and time), anything that can happen with non-zero probability will happen. Some things have zero probability of occurring, and those “states” will not happen. So if there is a law of nature that says no occupation of even-numbered rooms, then indeed, they will not be occupied.

    So for example, even if the Universe is infinite, we don’t expect there to be some part of it where, e.g., like electric charges attract.

    But since there is some non-zero probability of you existing and posting a comment here, and there likely is a non-zero probability of you having done the same thing except with one more sentence, it is quite plausible that in an infinite Universe, that alter-Lou exists.

  22. Lou Jost says:

    But Trevor, ANY particular state has a probability of zero, if states have a one-to-one correspondence with the points on a number line. And again, if the state space has higher cardinality than the time dimension, it is actually impossible that all states will be reached, even given an infinite amount of time.

  23. Lou Jost says:

    And Trevor, if you do not accept that the probability is zero of any particular state, I can use your reasoning to show that the hotel example works even if the receptionist does not enforce any rule about room numbers. If the rooms are filled at random and there is no rule about what room a guest receives, you would say that there is a non-zero probability that the first room is not taken (since this is not prohibited by any law in this new example). There are an infinite number of arrangements of guests in which the first room is not taken. Therefore you cannot argue that all rooms must be taken (or all states must occur) just because there are an infinite number of guests (or an infinite amount of time in the universe). I think Sean’s statement that “eventually, the thing we’re waiting for will come to pass” is not true.

  24. Neal says:

    @Lou:

    “Mathematicians spend an inordinal amount of time in this hotel when they think about infinity.”

    I see what you did there.

  25. Rohan says:

    Sweet. Every girl that ever dumped me is inevitably going to desperately claw her way to get me back. Can’t wait! 😀