Dembski's Buddy (part 2): Murphy's Law and Poincare Recurrence

Jun 20 2007 Published by under Intelligent Design

Part two of our crackpot's babblings are actually more interesting in their way, because they touch on a fascinating mathematical issue, which, unfortunately, Mr. Brookfield is compeletely unable to understand: the Poincare recurrence theorem.

Brookfield argues that the second law of thermodynamics in not really a law, since it's statistical, and that there must therefore be some real law underlying the statistical behavior normally explained by the second law. Here's his version - be prepared to giggle:

"The second law of thermodynamics has a rather different status than that of other laws
of science, such as Newton's law of gravity, for example, because it does not hold
always
, just in the vast majority of cases."

Well, if it is a "law" then it must hold always by definition. If it "does not hold always"
then it is not a law, period. If it is a "pseudo law" then that is fine for pseudo science, but
I am not interested in doing pseudo science. Hawking says that the thermodynamic arrow
is reversible because..

"...The probability of all the gas molecules in a box being found in one half of the box at
a later time is many millions of millions to one, but it can happen."

The type of event that Hawking is referring to here is known as "Poincaré Recurrence"--
named after the French mathematician Henri Poincaré. The result of any such occurrence
will indeed reverse the thermal characteristics of the box contents, violating the internal
thermodynamic arrow. This internal reversal however will not (in my opinion) reverse
the real arrow -- the unrelenting order to disorder movement of the total physical system.

Yes, folks - Brookfield is a real scientist, doing real science; Steven Hawkings and his ilk are all just pseudo-scientists studying psuedo-laws; real scientists like Brookfield throw out hopeless pseudo-laws like the second law of thermodynamics in favor of Murphy's law. And yes, that Murphy's law. Brookfield really tries to argue for the use of Murphy's law as a better statement of the principle of the second law. But we'll get to that later.

Poincare's recurrence is a real thing, and it is a real problem. There are some attempts to explain it, but it's still somewhat of an open issue. Here's the short
version:

Take a fixed set of particles with a fixed amount of energy in a fixed amount of
finite space. With no external influence, given a sufficiently long period of time, those particles will return to a state arbitrarily similar to the initial state.

The proper statement of it is somewhat more complicated than that: it talks in terms of the phase space of the system, and behavior of the system in terms of its dynamics in the phase space. In a dynamic finite system, the phase volume is fixed; and the phase trajectories within that space don't intersect. So if you think about the system over time, it's sweeping out paths through its phase space, and since paths can't intersect, the volume is constantly being decreased. Eventually, it's going to run out of places to go - except for returning to its starting point, and creating a loop.

Yet another statement of this is based on topology: take a topological space, (T,τ), which is second-countable (that is, the base of the space is a countable set) and Hausdorff (a space where you can separate things in terms of neighborhood inclusion). If there is a Borel algebra on (T,τ) (that is, an algebra operating on subsets of T which is closed under complementation and countable unions). If f : T→T is a function representing measure-preserving state transitions, then f has full measure, which means that all points are recurrent: f will trace loops through (T,τ).

So why is this a problem? Suppose that you take a finite space - a box, filled with a collection of gas particles. And you start with it in a very low entropy state - like a state where all of the gas particles are in one corner of the box. Then the process of interactions in the box should result in an increase in entropy as the particles disperse. But the recurrence means that there's a path where the entropy must decrease, to return back to that initial state. (Note: this paragraph originally contained an incredibly stupid error: I wrote high entropy where I should have written low.)

There are a couple of suggestions for why this isn't a problem. The least hand-wavy explanation that I've is that you can't isolate a subspace from its environment - so environment noise breaks things so that paths through the phase space do intersect, which violates one of the premises of the proof of the theorem. So by this argument, finite sub-spaces of the universe aren't subject to it; they're perturbed by the environment. And the entire space of the universe - even if it's fixed, it's expanding - so it's not a constant volume, and so it isn't subject to recurrence. If the universe were finite and not expanding, or expanding with a maximum limit, then the universe as a whole could be cyclic.

That explanation might be satisfactory; I don't find that explanation to be entirely convincing given my level of knowledge, but since I have pretty much no expertise in dynamical systems, I really don't know enough to judge it. We're talking about some very difficult advanced math here: to really criticize it, you need to fully understand the proof - which would in the best possible case, take me at least several weeks to months of studying dynamical systems to be able to really understand. (And I stress the "at least" above. I don't know enough to know how much I don't know about this, so that "weeks to months" is a lower bound.) What is clear is that it's not clear how Poincare recurrence applies to the physical world. We don't have complete phase-space models of real (not completely isolated) systems that are precise enough to let us figure out exactly whether or not Poincare recurrence applies, or what the timescale of the recurrence is.

Anyway - now you know a bit of what the Poincare recurrence is, and why it's a problem. In fact, you know more about it that Brookfield. But that's OK, because he's not really interested in understanding it. He thinks he already knows everything he needs to - not a diffeq in sight, but he's come to his conclusions that the Poincare recurrence theorem is completely incompatible with the second law of thermodynamics, and that this means that 2LOT is wrong.

Such obvious inconsistency causes me to believe that Hawking's Second "Law" of
Thermodynamics, with its statistical formulation, is not a real law but merely a good
approximation to a genuinely real 100% valid physical law.

When ID theorists speak of the Second Law of Thermodynamics my feeling is that they
are almost always referring to its design (order) implications -- the real arrow and not its
isolated thermal implications. Thus, we really need a name for this new, profound and all
powerful cosmic law, lurking just behind the Second "Law" of Thermodynamics.

I had originally been afraid to bring this "new law" idea forward due to the likelihood of
its name turning out to be "Brookfield's First Law of Irreversible Cosmic Catastrophe"
or equivalent. I then realized, however, to my enormous relief that we already have a
possible second name "Murphy's Law."

Brookfield gets one thing right: when IDists talk about 2LOT, they're always talking about its implications in terms of order and chaos, not its actual thermodynamic meaning. The problem is, Brookfield thinks that they're right to do that. But
thermodynamics doesn't talk about order and chaos: it talks about entropy. Entropy can sometimes be informally described as chaos, but that's just a metaphor. The 2LOT is about thermodynamic behavior and a specific measure
of a key thermodynamic property called entropy - it's not about order versus chaos, or design versus randomness.

But the second part of that is where it becomes truly laughable. Yes, the infamous Murphy's law - "If anything can possibly go wrong, it will" is Brookfield's model for a "correct" replacement for what he calls the second pseudo-law.

Luckily we now have Murphy's law that states "if something can go wrong, it will." Or
more scientifically "If left to its own devices, the universe is doomed!"

Other Murphylian statements might include;

"The physical universe is on a collision course with itself!"

"The universe's matter is just mindlessly crashing around inside its space!"

Let us now re-examine Hawking's box in the light of our new "Murphylian" knowledge.
Now what could possibly go wrong? Well, for starters, during "Poincaré recurrence"
there exists an absolute pressure differential between the part of the box that contains no
particles (zero pressure) and the part of the box that contains all of the particles (all
available pressure concentrated). If this pressure difference (or movement) either
damages or destroys the box then this is certainly in keeping with Murphy's Law.

His argument against the Poincare recurrence is: "Yeah, well, if it happened, it would break things", and since breaking things is in keeping with Murphy's law, then he concludes that a Poincare recurrence doesn't break his new and improved second law.

During "recurrence," the physical box is put in the most uncomfortable situation of being
two sizes at once. If the structural damage incurred by this anti-thermal spike is sufficient
to keep "Murphy's Arrow" on the "straight and narrow," then Murphy's Law is the more
scientific (realistic) of the two. Also, if the physical system in question (the box, the
typewriter, the monkey, etc.) has sustained any damage between recurrences then these
local "recurrences" are illusory in terms of the system as a whole. Remember the inner
walls of the box are constantly being hit by flying particles. During Hawking's (particular
type of recurrence) one side of the box is spared, but the other side is hit with double the
intensity.

The Poincaré "recurrence" example is not telling us that physical systems can be ordered
-- by accident -- by chance -- by randomness (the opposite of order). It is telling us
instead that any such internal statistical analyses are incomplete and that such narrow
assumptions can only lead us to illogical "order by randomness" "light by darkness" type
conclusions.

The orderly box is being constantly assaulted from inside by energized particles at all
levels from microscopic to macroscopic. Given enough time, its destruction is inevitable.
As long as every configuration state, including Poincaré "recurrence," is doing its equal
part to bring about the eventual destruction of the box, then Murphy's Arrow is perfectly
straight.

So while the internal system's state during recurrence is indeed a violation of
"Maxwellian Distribution" (in which all particles "should" be spread evenly over the
internal system space) it is completely consistent with *Murphylian Distribution (in
which the system's self-destructive potential (stress) is spread evenly over all of the
system's available configuration states).

And there, folks, is the crowning glory of his work. To refute a beautiful mathematical
proof of an amazing and profound theorem with unknown implications on the real world, all it takes is an argument that "the box would break", combined with the knowledge that order never emerges from randomness. Not a diffeq for miles to be scene - not an equation of any time, not a bit of mathematical analysis, not a refutation of any part of the proof of the theorem. All that stuff is unnecessary in the light of the brilliance of a mind like
Brookfield! Pinheads like Hawking might need that math stuff - but Brookfield can
disprove it with a wave of the hands, and a snap of the fingers: "Poincare recurrence of a collection of gas molecules in a box would make the box break", and poof! no more problem.

Brookfield of course then needs to explain to us why none of those bozo scientists were able to see something as glaringly obvious as his argument:

So Steven Hawking (and apparently the entire scientific community) have missed out on
discovering "Devolution" by mistakenly assuming the following;

#1. An orderly finite box (the universe is not a box{it is an infinite non-recurring
system})

#2. A perfectly strong box (no box is perfectly strong).

#3. A perfect "bounce" (no bounce is a perfect bounce).

In the real world, boxes are not perfectly strong, nor perfectly elastic and the physical
universe was never a box in the first place.

So, if ones "order to disorder" model of the universe, is that of an order-adulterated gas
inside box,

...but the real universe is not a box and its contents are subsequently not thus constrained,

...then such a model cannot accurately represent the universe, nor properly display a
cosmic law of order and disorder -- of constraint and absence of constraint.

Now... Read that, and compare it to an explanation of Poincare's recurrence and the
explanations of why we don't know how/if it applies to the real universe. Brookfield has
re-invented a shabby non-mathematical version of one of the possible explanations
why Poincares recurrence theorem doesn't really work in the real world. One of the
extremely well-known explanations. An explanation which no one who studies
thermodynamics could possibly have missed. But he thinks that the entire scientific
community has missed this obvious fact - and he thinks that his "collapsing box"
explanation takes care of the problem in terms of his "new" law.

And that's the final piece of patheticness here. The second law of thermodynamics
does not say "Disorder always increases." It does not not say
"The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium.". What it really says is: ∫(∂:Q/T)≥0, where Q and T are well-defined terms about heat energy and time; and that Q is defined for the macro-state of a system in terms of a combination of the microstates of the components of that system. (That second part is what Brookfield objects to: the use of the statistical combination of microstates to explain the thermodynamic behavior of macrostates.) The fact that it's really a precisely defined mathematical equation is important: it's what makes it useful, and what makes it a law. It means that it's a precise statement, with measurable implications, and which can be used to make precise predictions. Brookfield's "replacement" for the supposedly flawed second law is not a physical law; it's not a scientific statement of any kind. It's a mishmash built on undefined terms that produces the result that he wants, while not being able to actually precisely predict or describe anything.

34 responses so far

  • Justin says:

    Having just completed a Theory of Computation course, I'm struck by your explanation of the Recurrence problem in terms of a finite space that eventually runs out. We studied Linear Bounded Automata, and how the Halting problem can be solved within that realm. Somehow, the phase space which must return to its starting point and make a loop reminded me of that.

  • Anonymous says:

    I usually only lurk here, but I have to nitpick because I've seen you do this before - it's 'Stephen Hawking' not 'Steven Hawkings'.

  • gg says:

    MCC: "And you start with it in a very high entropy state - like a state where all of the gas particles are in one corner of the box. Then the process of interactions in the box should result in an increase in entropy as the particles disperse. But the recurrence means that there's a path where the entropy must increase, to return back to that initial state."
    It's been a while since I dabbled in stat mech, but shouldn't 'very high' be 'very low' and 'entropy must increase' be 'entropy must decrease'?

  • Let me refer back to an 1895 paper by the immortal
    Boltzmann, which has recently attracted attention in
    the controversy over so-called "Boltzmann Brains":
    Boltzmann's original paper (pdf). The reference is
    Nature 51, 413 (1895), as tracked down by Alex
    Vilenkin. Don Page copied it from a crumbling
    leather-bound volume in his local library, and the
    copy was scanned in by Andy Albrecht. The discussion
    is just a few paragraphs at the very end of a short
    paper.
    "I will conclude this paper with an idea of my old
    assistant, Dr. Schuetz.
    "We assume that the whole universe is, and rests
    for ever, in thermal equilibrium. The probability that
    one (only one) part of the universe is in a certain
    state, is the smaller the further this state is from
    thermal equilibrium; but this probability is greater,
    the greater is the universe itself. If we assume the
    universe great enough, we can make the probability of
    one relatively small part being in any given state
    (however far from the state of thermal equilibrium),
    as great as we please. We can also make the
    probability great that, though the whole universe is
    in thermal equilibrium, our world is in its present
    state. It may be said that the world is so far from
    thermal equilibrium that we cannot imagine the
    improbability of such a state. But can we imagine, on
    the other side, how small a part of the whole universe
    this world is? Assuming the universe great enough, the
    probability that such a small part of it as our world
    should be in its present state, is no longer small.
    "If this assumption were correct, our world would
    return more and more to thermal equilibrium; but
    because the whole universe is so great, it might be
    probable that at some future time some other world
    might deviate as far from thermal equilibrium as our
    world does at present. Then the afore-mentioned
    H-curve would form a representation of what takes
    place in the universe. The summits of the curve would
    represent the worlds where visible motion and life
    exist."

  • MaxPolun says:

    That last bit underlines pretty well one of the main problems with creationists (and cranks in general), they think that the informal phrasings of scientific and/or mathematical law is just as good as the formal statements. This is completely wrong.
    Also: I don't know how applicable it is to situations like the one you are describing here, but Poincaré recurrence is used all the time in analyzing chaotic systems.

  • gg says:

    #4: Really nice quotation, Jonathan! I'm reminded of a time when some students complained to me about the 'old books' that were references for the class I was TA-ing for. I pointed out to them that if you really want to understand a subject on the deepest level, you should look back at the works of the great thinkers who spent their entire careers on the subject, rather than just reading the more user-friendly modern textbooks.

  • Orri says:

    Just a couple of nitpicks:
    "So why is this a problem? Suppose that you take a finite space - a box, filled with a collection of gas particles. And you start with it in a very high entropy state - like a state where all of the gas particles are in one corner of the box. Then the process of interactions in the box should result in an increase in entropy as the particles disperse. But the recurrence means that there's a path where the entropy must increase, to return back to that initial state."
    The last sentence should read "But the recurrence means that there's a path where entropy must decrease, to return back to that initial state."
    Also:
    "And that's the final piece of patheticness here. The second law of thermodynamics does not say "Disorder always increases." It does not not say "The entropy of an isolated system not in equilibrium will tend to increase over time, approaching a maximum value at equilibrium."
    The second sentence contains "not" twice when the context implies you only mean it once (in your actual post, the first of the two is italicized).
    - Orri

  • IanR says:

    Well, in addition to Murphy's Law there's always Russell's Law....it's impossible to distinguish a creationist from a parody of a creationist.

  • Orri:
    It is probably a sign of my having been steeped in a degenerate culture that my first thought upon seeing MarkCC's double not was the following.
    "I'm not not licking toads!"
    Sigh.
    An important point to note about Poincaré recurrence is that the timescales involved are stupefyingly long. A standard exercise in graduate-level statistical physics books (Kerson Huang's springs to mind) involves calculating the time one would wait for a Poincaré recurrence of some reasonable system. The answer, give or take a zillion, is always "many times the present age of the Universe."
    There seems to be a whole lot of confusion about this corner of statistical mechanics (which recently poked its head up again at Cosmic Variance). Chris Hillman wrote back in 1999,

    I can't speak for the statistical mechanicians, but I -can- say that it is definitely the case that the Poincare and Kac recurrence theorems do -not- imply anything like "periodicity". The theorems say that in a (discrete time) ergodic dynamical system, -if- you wait long enough (Kac's theorem tells you how long you can expect to wait), eventually you'll return to within any small neighborhood of your original position in the phase space, but of course because of the "exponential divergence of initially nearby trajectories" in a chaotic system, this doesn't imply anything like "periodicity" in the sense of repeating history.
    See for instance Petersen, Ergodic Theory, Cambridge U Press, 1983, for the statement and proofs of these theorems --- they are actually pretty easy to prove, so you don't need to read very much in this book to follow along, provided you have had some exposure to measure theory.

  • Ken Watts says:

    "Well, in addition to Murphy's Law there's always Russell's Law....it's impossible to distinguish a creationist from a parody of a creationist."
    That's so true.
    I recently did a parody interview of a creationist ("Unintelligent Design") on my site, and it was freaky how hard it was to come up with anything that they haven't actually said already.

  • Waterbreath says:

    Entropy can sometimes be informally described as chaos, but that's just a metaphor.

    In my experience, this is the downfall of a large number of crackpots. When people move outside their own areas of expertise, they of course rely on metaphor to glean a bit of understanding. The problem is when they think the incomplete understanding the metaphor imparts is actually complete. They see a metaphor, and decide that what's really going on must be exactly like that. Which leads them to make all sorts of spurious conclusions with no scientific basis at all.
    It's why people get so off track when you tell them, for example, that photons act like waves. Because now they're not thinking of photons, or electromagnetism, anymore. Now they are thinking of water, or sound, or vibrating guitar strings, and they don't realize that these are all different variations on a mathematical theme. They instead draw a one-to-one parallel, and assume that if it's true for water, it must be true for light. And so, when you later have to tell these people that photons also act like particles sometimes, their brains break. And amazingly enough a certain portion of people, rather than persevere in understanding, or give up and leave science to the scientists, decide that the picture they already have in their head must be correct and complete, and the new information contradicts or is unintuitive, so that must be wrong.
    I'm convinced there's no hope in arguing with such people.

  • Oleg Tchernyshyov says:

    Mark,
    If this thread were a movie, I'd call it Shooting fish in a barrel 2.
    Cheers,

  • Edward says:

    To keep in the spirit the brilliant Brookfield, perhaps it would be informative to ask him what happens if the box he used to disprove Poincaré recurrence is made out of pure ADAMANTIUM!
    Given his clearly well-informed background, he will know that adamantium is an unbreakable metal in the comic books.
    Since an adamantium box cannot break, Brookfield's "proof" has a fatal flaw. Since this reduce's "Brookfield's First Law of Irreversible Cosmic Catastrophe" (né Murphy's law) to the comic book proof it is, using a comic book substance... I feel a long pretentions name for a novel "law" that can be attributed to me coming on!
    Joking aside, this is and enjoyable post - it is both amusing and sad to see the muddled thinking of folks like Brookfield highlighted and interesting to see the actual science highlighted.

  • Coin says:

    There are a couple of suggestions for why [Poincare Recurrence] isn't a problem. The least hand-wavy explanation that I've is that you can't isolate a subspace from its environment - so environment noise breaks things so that paths through the phase space do intersect, which violates one of the premises of the proof of the theorem. So by this argument, finite sub-spaces of the universe aren't subject to it; they're perturbed by the environment. And the entire space of the universe - even if it's fixed, it's expanding - so it's not a constant volume, and so it isn't subject to recurrence. If the universe were finite and not expanding, or expanding with a maximum limit, then the universe as a whole could be cyclic.
    Well... okay. You mind if I poke at this for a moment?
    Let's say that we have a hypothetical universe where the space of the universe is expanding with a maximum limit. Like, let's say that this universe is just like ours, with a big bang and relativity and everything, except instead of dark energy, they have something going on where the expansion of space is everywhere decelerating with time (I can never remember if this is equivalent to a positive or negative cosmological constant). So this universe expands, but only up to a certain limit, at which point it turns around and contracts until there is a big crunch.
    Does this universe have to follow the Poincare recurrence theorem?
    And if, since this is always how it works in science fiction, the big crunch immediately triggers another big bang, which results in another expanding but decelerating universe, which results in another big crunch, ad nauseum... if T is the time of one big bang and T+1 is the time of the next, then what can we say about the entropy immediately after time T versus the entropy immediately after time T+1? Is it guaranteed to be less, because of SLoT? Is it not guaranteed to be less, because of poincare? Or is there some reason you just can't meaningfully ask this question?

  • Coin says:

    less, because of SLoT
    ...Err... I mean, of course, "greater".

  • Ken Watts says:

    "Now they are thinking of water, or sound, or vibrating guitar strings, and they don't realize that these are all different variations on a mathematical theme. They instead draw a one-to-one parallel, and assume that if it's true for water, it must be true for light."
    My dissertattion topic was metaphor theory. I studied for my degree (true confessions time) at a seminary, and while the people I was actually working with understood what I was doing, I also had to do some required papers with theology professors.
    I tried to explain the difference between a light wave and a water wave to one of those guys, and he got very agitated, pulled out a dictionary, and triumphantly read the definition of "wave" to me.
    As far as he was concerned, we had reached rock bottom. There was nothing left to discuss.
    I suppose, he was right in a way. I certainly had no desire to continue the conversation...

  • jh says:

    All the particles in the corner of the box is a LOW entropy state. Forcing the particles to be near the corner reduces the number of configurations (and hence, the entropy).

  • Xanthir, FCD says:

    Coin:

    Let's say that we have a hypothetical universe where the space of the universe is expanding with a maximum limit. Like, let's say that this universe is just like ours, with a big bang and relativity and everything, except instead of dark energy, they have something going on where the expansion of space is everywhere decelerating with time (I can never remember if this is equivalent to a positive or negative cosmological constant). So this universe expands, but only up to a certain limit, at which point it turns around and contracts until there is a big crunch.
    Does this universe have to follow the Poincare recurrence theorem?

    I don't think so. For one, the universe may return to the Big Crunch well before the time when it is expected that the recurrence will occur. The recurrence seems to depend on the space in question being finite in volume but unbounded in time. The universe you described fulfills the former but not the latter.

    And if, since this is always how it works in science fiction, the big crunch immediately triggers another big bang, which results in another expanding but decelerating universe, which results in another big crunch, ad nauseum... if T is the time of one big bang and T+1 is the time of the next, then what can we say about the entropy immediately after time T versus the entropy immediately after time T+1? Is it guaranteed to be less, because of SLoT? Is it not guaranteed to be less, because of poincare? Or is there some reason you just can't meaningfully ask this question?

    This is an interesting case. At first you want to say yes, but some thought makes you turn to no (what if the universe gets bigger with every Bang?). More thought makes you unsure. Note that I am explicitly ignoring the case of the singularity itself, which could be considered to be identical to all other singularities.
    My best argument goes like this. Even if every Bang results in the universe unfolding in a new way (which is possible even when scaled to infinity), you must be able to find a ball of space arbitrarily close to the 0-point of the singularity which occurs in every explosion. Since this ball occurs an infinite amount of times, but is finite in volume, it is subject to recurrence.
    However, it's possible that whatever parameters govern the shape of the unfolding universe can change without bound, and have an affect on the state of the system. Then you'd be unable to find a ball with identical state.
    I think it all hinges on every variable being bounded in some way. In the examples given by Mark most variables are bounded by the conservation laws, and volume is bounded by definition. This theoretical universe, though, has an unbounded variable, which can prevent recurrence.
    Though the official recurrence stuff is definitely more advanced than I'm talking about here, the simple case of "Will it become cyclic?" is easy to reason about. Assuming a deterministic universe, a space is guaranteed to go from state x-1 to x to x+1 as it advances from time t-1 to t to t+1. The transition from one state to the next is guaranteed by the static laws.
    Now, imagine that, at some time tf the universe returns to the same state it was in at time t. The rules of the universe say that state x always evolves into state x+1, so you'll go from xf-1/tf-1 to x/tf to x+1/tf+1.
    That is, you are forced to enter a loop if you ever return to a previous state. It's impossible to just touch state x and then go on to some state other than x+1. Thus, intersection is impossible. The only thing you can do is join with a previous worldline and start cycling endlessly.
    In a finite static volume (to be simplistic), there are only a finite number of states you can occupy. Given enough time, you'll eventually run out of unique states, and be forced to return to a previous state.

  • MarkH says:

    Everybody gets Murphy's law wrong.
    What Edward Murphy really said, was that if there is more than one possible way to do something, eventually, some one will do it the wrong way.
    It's more about idiot-proofing things. In this case, we'll need to work hard to IDiot proof science, or these misunderstandings will inevitably continue.

  • Torbjörn Larsson, OM says:

    we'll need to work hard to IDiot proof science, or these misunderstandings will inevitably continue.

    But that isn't sufficient - as the saying goes, you can make equipment foolproof but never student proof.
    [At least that was I heard as a course assistant. Funny thing, if a research lab 'borrowed' equipment between courses, it was still likely to turn up busted and in need of assistant care and repair...]
    Anyway, in your spirit we could probably criticize Hawking (as I did already in part 1) for being less than precise in his popular texts. When he says that the law "does not hold always" it is perhaps more precise to note that the classical formulation holds (with the usual caveats).
    But the corresponding statistical formulation is then about ensemble averages, which I believe is often forgotten. What Poincaré recurrence shows is that there is a potential that the probability that a single system violates the law temporarily. Which isn't more surprising than when the uncertainty principle violates for example energy conservation temporarily.
    Not that dogmatic individuals would admit constraints or violations of laws.

    such a model cannot accurately represent the universe

    Ha! It is Brookfield that insists on that, no doubt influenced by his sporadic reading of Penrose and Hawking.
    AFAIK we can't in general find nicely foliated solutions of GR that admits a measure of total energy and thus studying total entropy rigorously. I believe there are some exceptions in cosmology, such as a closed de Sitter universe/bubble universe. But in general it seems to me laws of energy and entropy are only guaranteed locally.

    all it takes is an argument that "the box would break", combined with the knowledge that order never emerges from randomness.

    Continuing being nitpicky, Brookfield doesn't know that order doesn't emerge (except in the context of his dogmatic world view), he assumes so. Without the slightest shred of evidence, I might add.

  • Torbjörn Larsson, OM says:

    Let me refer back to an 1895 paper by the immortal Boltzmann, which has recently attracted attention in the controversy over so-called "Boltzmann Brains"

    Sean Carroll at Cosmic Variance is a staunch critic/user of Boltzmann's Brains in his proposals of time symmetric cosmologies. He has discussed his latest ideas here:

    Boltzmann's H-theorem, while interesting and important, is even worse. It makes an assumption that is not true (molecular chaos) to reach a conclusion that is not true (the entropy is certain, not just likely, to increase toward the future -- and also to the past).
    The nice thing about stat mech is that almost any distribution function will work to derive the Second Law, as long as you don't put some constraints on the future state. That's why textbook stat mech does a perfectly good job without talking about the Big Bang. But if you want to describe why the Second Law actually works in the real world in which we actually live, cosmology inevitably comes into play. [Bold added.]

    His original article on Boltzmann's Brains on Cosmic Variance is here, though I believe he made more.

    Can't resist a preview from the third installment of the article.

    Yes, and Brookfield isn't aware that "sparticles" is a term already used for corresponding supersymmetric particles. Or he knows it, and are trying to exchange yet another part of science for his pseudoscience.

  • Since Cosmic Variance seems to be exhibiting, ahem, variable accessibility, I'll offer a link to Mehran Kardar's statistical mechanics lecture notes. See lectures 8 and 9 for a treatment of the Boltzmann Equation and the H-theorem which should, I think, clarify the different assumptions at work.

  • Fj_ says:

    Wow, in that article on Boltzmann's Brains from prev comment there is quite an insightful statement:
    "So if we are explaining our low-entropy universe by appealing to the anthropic criterion that it must be possible for intelligent life to exist, quite a strong prediction follows: we should find ourselves in the minimum possible entropy fluctuation consistent with life's existence."
    I suppose one can somehow ponder upon the actual rules governing microstate evolution, to prove (in a sense) that quantum fluctuation unwrapping into a single life-sustaining solar system involves much more drastic decline in entropy than one leading to big bang, that produces a vast number of star systems, of which one is habitable. However, that's not the point.
    Think about Fermi's paradox -- it will be explained then! "Minimum possible entropy fluctuation consistent with life's existence" contains exactly one life-sustaining planet (of which all that other galaxies are considered being a byproduct)!

  • grumpyoldman says:

    Everybody gets Murphy's law wrong.

    Seems to me that Murphy couldn't get his own law right!

  • Lepht says:

    Let us now re-examine Hawking's box in the light of our new "Murphylian" knowledge.

    in the knowledge of what Murphy actually said, here we have meta-Murphy - someone is doing Murphy's Law of someone doing it wrong, wrong.
    i'm actually kind of impressed.
    Lepht

  • What hasn't been discussed (unless I missed it in my quick reading) is the question of why creationists consider the Poincare recurrence theorem relevant. If the Poincare recurrence time for the Earth is trillions of years (it's actually much longer than that), then what is its relevance to the question the creationists really care about, which is how living creatures came into existence on Earth?
    Presumably they are imagining a nonexistent dialog with mainstream scientists:

    Creationist: It is impossible for nonliving matter to turn into living matter, because systems only get less ordered with time, never more ordered.

    Evolutionist: Not true! By Poincare's recurrence theorem, if we wait long enough, nonliving matter will turn into living matter.

    That's the only way that I can see how Poincare recurrence is relevant to creation/evolution, if the creationist think that evolution relies on Poincare recurrence in some way. Of course, evolution doesn't rely on violations of the second law of thermodynamics.

  • Henryk Plötz says:

    Everybody gets Murphy's law wrong.
    What Edward Murphy really said, was that if there is more than one possible way to do something, eventually, some one will do it the wrong way.

    Ah, thank you. I just wanted to mention that. What still needs to be said is that "Anything that can go wrong, will." is not Murphy's law but Finagle's law. See the jargon file entry on Finagle's Law for details.
    --
    Henryk Plötz
    Grüße aus Berlin

  • Flex says:

    Daryl McCullough wrote, "What hasn't been discussed (unless I missed it in my quick reading) is the question of why creationists consider the Poincare recurrence theorem relevant."
    I think creationists, as I do, have little understanding of the subtleties of mathematics. I don't claim understanding, which is why I enjoy reading this blog so much because Mark Chu-Carroll has been able to give me insights into some of the subtleties.
    But in the creationist quest to have the appearance of rationality, like a trained dog, they alert and point at anything which appears to support one of their pet notions.
    In this case, the notion is that science doesn't have an accurate model of the universe so their untestable belief is superior.
    I see their chain of thought as follows:
    Creationist: The 2nd law of thermodynamics means evolution couldn't happen.
    Rationalist: The 2nd law of thermodynamics applies to closed systems while the earth, because of the energy supplied by the sun, cannot be considered a closed system.
    Creationist: Curses! Foiled again! Hmm, let's do a search of mathematics to see if the 2nd law of thermodynamics really says what they claim.
    Creationist: Whoa! Will you look at that! The Poincare recurrance appears to violate the 2nd law of thermodynamics! I don't understand the math, or the proof, but I understand that there is an unexplained discrepancy here! This means that the 2nd law of thermodynamics could be wrong! The model science uses is broken! Everybody look! Science's model of the universe is broken which means their model is wrong! So our model is right! Our model isn't broken! We win! We win! We win!
    Rationalist: Breathtaking inanity.

  • Mark C. Chu-Carroll says:

    Darryl:
    The only interest of the creationists/IDists in Poincare recurrence is because
    they see it as a disproof of naturalism.
    To the IDists, there are exactly two possibilities for how the world works: either it's science (which to them means evolution), or it's creation. *Any* argument that shows that conventional science is wrong is, automatically,
    a proof that creationism is right.
    So the whole Poincare recurrence thing is just yet another "Look, science is wrong, so we must be right!" argument. They don't really care what it means.

  • Anonymous says:

    Coin:

    Let's say that we have a hypothetical universe where the space of the universe is expanding with a maximum limit. Like, let's say that this universe is just like ours, with a big bang and relativity and everything, except instead of dark energy, they have something going on where the expansion of space is everywhere decelerating with time (I can never remember if this is equivalent to a positive or negative cosmological constant). So this universe expands, but only up to a certain limit, at which point it turns around and contracts until there is a big crunch.
    Does this universe have to follow the Poincare recurrence theorem?

    I think it does - and in fact, the "big crunch" would seem to satisfy the
    conditions of a poincare recurrence: the singularity-crunch leading to a new bang would be an arbitrarily similar recurrence of the original bang.

    And if, since this is always how it works in science fiction, the big crunch immediately triggers another big bang, which results in another expanding but decelerating universe, which results in another big crunch, ad nauseum... if T is the time of one big bang and T+1 is the time of the next, then what can we say about the entropy immediately after time T versus the entropy immediately after time T+1? Is it guaranteed to be less, because of SLoT? Is it not guaranteed to be less, because of poincare? Or is there some reason you just can't meaningfully ask this question?

    I don't think that 2LOT can be applied to a singularity; I don't think that
    a singularity has a meaningful measure of entropy. (What's the heat of an infinitely small, infinitely dense point which is all of space?)

  • http://arxiv.org/pdf/0706.4032
    Title: How much information about a dynamical system do its recurrences contain?
    Authors: Geoffrey Robinson, Marco Thiel
    Comments: 9 pages, 0 figures
    Subjects: Dynamical Systems (math.DS)
    We show that, under suitable assumptions, Poincare recurrences of a dynamical system determine its topology in phase space. Therefore, dynamical systems with the same recurrences are topologically equivalent.

  • Torbjörn Larsson, OM says:

    Jonathan:
    Interesting find; recurrences seems to play the same role as singularities in characterizing some systems.
    Regarding the subject of the thread, going through some of its references at high speed, it seems recurrence times can be constrained. And the constraint is characterized by a metric entropy measure. I was intrigued by the measure theoretic relation to the usual formulation of probabilistic entropy measure.
    Though I don't know what to make of it in Brookfield's use on cosmology. The constraint can be made when one can make a countable partition. Fine in our case, and it indicates almost everywhere exponentially large (in terms of dominant Lyapunov exponent, or Hausdorff dimension for a fractal system) return times. But what happens in our expanding system, where the entropy goes up (but the entropy density goes down)? I dunno.
    Btw, I find it humoristic that the paper claims that reconstruction by recurrences is important for algorithms elaborating protein structures. Brookfield's failure of understanding (if not exactly rejection) of recurrence is a problem for him all over when he tries to discuss creationism as an alternative for science.

  • Francois Leyvraz Waltz says:

    Just to try and give an idea of how long a Poincare time is: the probability of a
    recurrence arising in a short time (say a year) is the ratio of that to the Poincare
    recurrence time. To get an idea of that probability, start from the probability of
    being dealt a royal flush in poker *immediately*. Then think about that happening,
    say, ten thousand times in a row. Then think of *that* happening simultaneously to
    everyone on Planet earth, still as a result of pure chance: that begins to be a
    recurrence probability for a rather small system (less than the millionth part
    of a gram of substance).
    These are rough estimates, but I believe they give a hint of the fact that Poincare
    recurrences should not really be taken seriously, just as the celebrated monkeys
    writing Shakespeare by chance.

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