Friday, February 01, 2008

George Wing & Poincaré recurrences

Sean Carroll has received the following mail from a kid:
I don't know if you exist but I do! I do not agree with your article and I do not believe that "Mumbo Jumbo" if you do... Well! It's a disturbing thought but I know how to deal with it! I will not let the world disappear under my nose but if you do, I can't say I’m sorry!


a ten-year-old who knows a little more than some people!
George Wing

P.S.: Some people have a little too much time.
Grammar has been corrected for the sake of clarity.

Now, George's arguments are somewhat less complete and less rigorous than what I would like :-) but he is, of course, right. The predictions by Sean Carroll and others for Boltzmann's brains and reincarnation in otherwise ordinary cosmologies are totally preposterous. In this text, we will look at the nature and correct interpretation of Poincaré recurrences.

Disturbing implications of a cosmological constant?

We may choose the 2002 paper by Dyson, Kleban, Susskind to be my main target. The readers of this blog must know that I immensely admire Lenny Susskind and I also proportionately admire his younger colleagues. ;-) But I have always viewed this paper (and dozens of similar papers) to be a joke and I still look at them in the same way.

But you may want to know: how big a joke it is? It is a big joke, indeed. It has collected 114 citations in less than 6 years. Not bad for a joke.

The content of the paper

They argue in the following way. A positive cosmological constant has been observed. It's so horrifying! It means that we effectively live in an excited de Sitter space. Now, one causal patch of it arguably contains all the degrees of freedom of such a space. The degrees of freedom in other regions are neither independent nor commuting, by the complementarity principle. That means that a de Sitter space literally behaves as a finite entropy system. Events repeat in a de Sitter space much like they repeat in the box.

If you wait for a certain exponential time of order exp(entropy), you will get the same state back, with an arbitrary accuracy. So the entropy eventually decreases again to its present value. A lot of other things will happen, many Boltzmann's brains will be born. They also use the MWAPL whose silliness has been discussed many times on this blog to argue that any cosmology based on de Sitter space predicts that we should be Boltzmann's brains ourselves, after all.

If you don't know, MWAPL is the Main Wrong Anthropic Proportionality Law - a logical fallacy in which the confused thinker assigns the same probability to an ensemble of individual microstates that are not (and usually cannot be) in equilibrium or the same importance to different observers whose equality is not (and usually cannot be) enforced by any conceivable physical mechanism.

I have really dedicated too much spacetime on this blog to debunking of the typicality fallacy. The Dyson+Kleban+Susskind paper also offers you a lot of the usual nonsense about the cosmological origin of entropy. Below, I want to look at related but inequivalent fallacies associated with the Poincaré recurrences that haven't been discussed yet.

Being careful about Poincaré recurrences

What are Poincaré recurrences? Well, it is a phenomenon - a somewhat academic one - that was pointed out by Henri Poincaré in 1890.

Consider gas in a box with a certain value of energy and other conserved quantities (in quantum physics: determined plus minus a small epsilon, so that we don't end up with one eigenstate only). In classical physics, its state is described by a point in the phase space (labeled by coordinates x_i and p_i). Quantum mechanically, it is encoded in a state in the Hilbert space. The dimension of the Hilbert space is essentially the same number as the volume of the phase space expressed as a multiple of a power of Planck's constant. Both of these numbers are large. They are essentially equal to exp(entropy). That's a huge number because entropy itself is greater than 10^{26} or so for macroscopic systems.

The classical evolution is a flow in the phase space. Because the relevant phase space is a compact manifold, it is not hard to believe that you must eventually return, much like Christopher Columbus, close to the point where you started, with an arbitrary accuracy: there is just not enough space for you to avoid this point for too long. One can prove this assertion rigorously, given certain assumptions. However, you will need a hyperastronomical time of order exp(entropy) - a very large number because, once again, the entropy itself is 10^{26} or more - times a less spectacular factor that increases if you want to restore the initial state more accurately.

In quantum mechanics, the evolution following the Schrödinger equation will approximately restore the initial state after a comparable time. By an approximate restoration of the quantum state, I mean that the inner product of the initial and final state will be equal to 1-epsilon where epsilon can be chosen arbitrarily small.

Misinterpreting the theorem

OK, if you return to the same point of the phase space - for example a point where all oxygen atoms are located in the left half of the bedroom - is it correct to say that the entropy will have to drop sometime in the future, in order to return to the initial state that we started with and that could have a low entropy?

The answer is, of course, No. It is No even in ordinary physics of gas in a box. This explains why all the cosmological papers predicting Boltzmann's brains, recurrences, reincarnation, and other (meta)physical phenomena are just wrong. They are not wrong because of some advanced subtleties of quantum cosmology. They are wrong because their authors misunderstand even statistical physics of ordinary gas or other simple, non-gravitational physical systems. I am convinced that virtually every good physicist who understands statistical physics agrees with me.

There's a lot of crap of this kind in the literature but the high volume shouldn't make you hesitate before you completely disregard this stuff because in this case, the proportionality law between volume and relevance breaks down, too.

This comment reminds me of the funny story of Arnesen and Bancroft, two stupid polar explorers who went to the Arctic region in a swimming dress in order to prove global warming. One of them got a frostbite in those -100 degrees and both of them were hungry. The other woman went to search for some food. She returned, saying: "I have two news: bad news and good news. The bad news is that I have only found polar bear manure. The good news is that there's a lot of it over there." :-)

Microscopic vs macroscopic description

We must be very careful to distinguish different types of description of physics. Below we will explain that Poincaré recurrences are only relevant for the exact, microscopic description of a physical system. When we describe a physical system microscopically, we really need to know the initial state completely accurately. When we know it accurately, we can say that it will return to the same point after the recurrence time.

But if we describe the initial state accurately, we can't talk about its entropy and we can't describe it by macroscopic words such as an "egg". Microstates can't be subjects of macroscopic assertions. Microstates have no emotions, if you wish.

If we want to say that the initial state has a certain nonzero entropy or that it is an "egg", it inevitably means that we only describe the state approximately. We only have incomplete information about the initial state. Most typically, we only specify certain macroscopic degrees of freedom - and even these degrees of freedom are specified with a nonzero error margin - and we ignore most microscopic details of the system i.e. allow them to have an arbitrary form.

When we talk about entropy, my statement is a tautology because the entropy is defined with respect to particular ensembles of microstates. If you say that a system has a certain entropy, it means that you only talk about the ensembles and in the same sentence, you simply cannot distinguish the individual microstates in the ensemble from each other.

On the other hand, if we describe the system in the macroscopic language, including the words "entropy" and "egg", we won't be able to deduce anything macroscopic from the Poincaré recurrence theorem. Why? Because the initial state - that we describe as an "egg" or another state with a nonzero "entropy" - includes many microscopic states and each of these microscopic states will evolve into a completely different state after the precisely chosen recurrence time.

Looking at an egg for too long

For example, can we derive the following assertion from the Poincaré recurrence theorem?
After time T (a precise figure comparable to the recurrence time), an egg on the table will evolve into the same egg on the table in your closed lab (with a small error).
The answer is, of course, No. The egg in the sentence above always represents a collection of microstates i.e. incomplete information about the physical system. And it is simply not true that the microstates in the collection will return to the same initial state after a universal time. Individual states get restored but the precise time is different for each of them. For a fixed T, only an exponentially (with an exponential exponent) accurately defined initial state will evolve into the same state with an acceptable accuracy.

The previous sentence actually implies that the recurrence of an egg after time T is exponentially (with an exponential exponent) unlikely because the individual microstates must be treated as comparably important in the ensemble and only a tiny fraction of them confirms the assertion about the recurrence after time T.

If you modify the sentence and allow T to be anything - or if you say that the entropy of a finite closed system will drop sometime in the future - your sentence will be, of course, correct. But such a correct sentence has no consequences for physical measurements at a particular year T in the future where T is gigantic. If you wanted to check the sentence experimentally, you would need gadgets that can survive for these gigantic time intervals. In de Sitter space, one can pretty much show that such (recurrence-time) long-lived gadgets are impossible which is what we really mean if we say that unusual physical phenomena at the recurrence time are "unphysical".

As you can see, recurrence is a very unlikely phenomenon i.e. its contribution to all predicted probabilities of physical phenomena that we care about are tiny. Whether we are freaky observers or results of an ordered evolution is an example of a physically meaningful question that many of us actually care about. If you think for a little while, the previous two sentences imply that the existence of recurrence cannot have any measurable impact on the question whether we are results of evolution of Boltzmann's brains as long as you think carefully. The only way how to magnify the tiny contribution and get a finite or large one is to use some form of the Majorities in spacetime fallacy i.e. to incorrectly multiply probabilities by the "time of the Universe".

Let me return a bit and write one more example of a macroscopic description.

For example, George Wing whose mail started this text is determined by a certain incomplete package of macroscopic information. In this description that makes the term "George Wing" meaningful, the second law of thermodynamics always holds, even after the recurrence time. So George Wing is right that he won't allow the world to disappear or transform into chaotic Boltzmann's brains. A particular microstate in the ensemble may "look" like George Wing but it is not the same thing. A particular microstate without any context is a boring point in the phase space (or a vector in the Hilbert space) that carries no entropy or other macroscopic quantities and qualitative properties.

Some people think about the world in the old-fashioned deterministic way - imagining that it is "objectively" described by a "real" pure state vector at each moment. Because of non-realism of quantum mechanics, it is a wrong interpretation of the state vectors. But even if you adopted this perspective - and in classical physics, you are kind of allowed to imagine that the Universe is a particular point of the phase space at each moment - you are still not allowed to pretend that microstates and their ensembles are the same thing. The Poincaré recurrence time only applies to the exact microscopic states and these individual microscopic states have no qualitative properties.

On other other hand, in the macroscopic language, there is no recurrence because the nearby microstates won't act coherently.


So if you want to talk about a precise microscopic initial state, you won't be allowed to identify this precise microscopic state with an "egg". If we summarize, all conclusions in dozens or hundreds of preposterous papers that argue that a cosmological model implies that we are Boltzmann's brains, reincarnated animals etc. are based on a particular kind of sloppy thinking, namely the inability of the authors to distinguish a microscopic description of a physical system from the macroscopic or approximate one i.e. their inability to distinguish an ensemble of microstates from its individual elements.

When you are careful about these differences, you will never be able to prove any paradoxical conclusions, regardless of any details about your cosmological model. De Sitter space is very likely to be the right description of our Universe in the far future but our existing theories of de Sitter space only predict that we should be Boltzmann's brains if you assume incorrect proportionality laws or if you confuse microstates with their ensembles. There is no paradox here to solve and there is no constraint that Boltzmann's brains could ever impose upon cosmologies. In particular, you cannot rule out de Sitter space by similar metaphysical thinking. I don't really know whether someone really thinks that de Sitter space - the most likely cosmology describing the future of our Universe - can be killed by the kindergarten argument above but many people surely seem to believe it.

Those who still don't get the basic point about the key differences between the microscopic and macroscopic descriptions of physics are simply being dumb (and some of them are dumb without the word "being").

And that's the memo.

Bonus: most of us are freaky?

Finally, I want to address a particular assertion that is repeated exp(entropy) times by Dyson, Kleban, Susskind, Carroll, and dozens of others. They often say something like this:
Most of the microstates included in the macrostate describing us come from a high-entropy state in the past. Consequently, according to the theory, we are likely to be Boltzmann's brains.
You can see that this sentence is self-contradictory nonsense based, once again, on a confusion between macroscopic and microscopic properties.

A consistent physicist can never end up with such an assertion. If he uses a microscopic description of the brain, he will assume that the present state is known accurately and the states in the past can therefore be reconstructed accurately, too. Precise microscopic states cannot be assigned any entropy (or it is zero, if you wish) and the sentence above is wrong because you will never encounter any high entropy in such a microscopic approach.

On the other hand, if a consistent physicist analyzes the sentence in a macroscopic framework, he will identify another problem with the assertion that we have already discussed: wrong priors and wrong algorithms of retrodiction. While it is true that most predecessors of microstates including in a present macrostate belong to high-entropy ensembles, we cannot conclude that it means that the probability of the low entropy initial state is exponentially tiny.

In reality, the correct way to say something about the initial state requires a retrodiction, a form of Bayesian inference, and such a retrodiction follows a different procedure than predictions because it requires you to insert some priors. The realistic priors assign comparable probabilities to macroscopically distinct classes of states. If you wish, the realistic priors for low-entropy initial microstates are hugely exponentially larger than those for high-entropy initial microstates.

Every observation of the macroscopic world that has ever been done is evidence for the previous sentence.

The opposite assertion that the priors should be comparable for individual initial microstates (which would imply that the entropy was high in the past) cannot be justified by any correct independent argument - it is really a dogma - and one can actually easily see that such an assertion is wrong - it is a wrong dogma. One can only assume that individual microstates in an ensemble are equally (or at least comparably) likely in situations where such a state is a result of a previous (sufficiently long) process of thermalization - something that uniformly covers a region of the phase space.

This is clearly not the case of the very beginning of the Universe which is why the "equal priors for microstates" is an incorrect assumption and its conclusion, "a high entropy of the early Universe", is wrong as well, regardless of any detailed cosmological scenarios.


  1. So what if, when calculated the probability of recurrence is exponentially small. That still means that it is non-zero and in an infinite amount of time the recurrence is therefore guaranteed to happen.

  2. Fine but it's just an empty academic statement that doesn't influence any prediction we would ever want to predict, whether it is about experiments in 2012, in the 21st century, or in the next billions of years.

    It's the rate, and not some integrated overall probability over some inaccessible and hypothetical infinite periods of time, that influences events at each moment (and events in the following billions of years). The rate is expo-exponentially tiny so it either adds expo-exponentially small corrections to observable quantities (which we can't observe because they are too small) or large corrections with an expo-exponentially small probability (which is so small that we may assume we won't be affected by it).

  3. Sean Carroll did not say that Boltzmann brains are real. He said the opposite. While the little boy was right about the brains, he also misunderstood Carroll. Read the article again.

  4. I didn't write he wrote the BB were real. I wrote what he wrote, namely the bullshit that the BB are predicted by some totally ordinary cosmological models.

    A simple reason why I was totally accurate and you're just a fucked-up dishonest asshole is that all my claims were based on quotes.

  5. OMG! You still don't get it. Carroll does not agree with those predictions! Are you totally daft as well as unnecessarily rude? You've made the exact same mistake as the little boy that you quoted.

  6. And what's worse is that you think Boltmann's model is "totally ordinary". Jeezus.

  7. If you think I'm intimidated by an idiot who makes the same mistake that a little boy made, you're wrong. Boltzmann's was a "totally ordinary cosmological model"? You've lost your mind.