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Confusion: Fluctuation Theorem, Poincare Recurrence Theorem

  1. Mar 16, 2015 #1
    Is Poincare' Recurrence Theorem (PCRT) considered a possible explanation for the "low entropy" initial conditions of the universe?

    Is the following a roughly correct paraphrasing of it? For a phase space obeying Liouville's theorem (closed, non-compress-able, non-decompress-able), the probability of the system entering the lowest probability state must be 1 over "some set" of steps (I hate to use the work infinity).

    Is PCRT in conflict with the Fluctuation Theorem (FT)?
    I am confused as to whether the "away from equilibrium" clause in the FT implies that the universe should obey the FT and is therefore in conflict with PCRT, or whether that means FT isn't relevant when thinking about the universe - which is assumed to be a closed system, in equilibrium, as far as we know.

    I found some old threads discussing this but I sure would appreciate some re-illumination - while I am up against this confusion.

    I'm also struggling with the notion of compress-ability or decompress-ability of a phase space. Is there any way for the probability of a given state to change in a Liouville (Hamiltonian?) space?

    If you had one extra degree of freedom like "time" and in one time step added an identical copy of some "space state" to the phase space does that count against that state's probability, does that increase the phase space volume and break Liouville's rule?

    Weird, possibly badly worded question I know, I can understand if it's just too much to try and answer succinctly.
    Last edited: Mar 16, 2015
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  3. Mar 17, 2015 #2


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    I am not a specialist but I have a strong impression that such an explanation would not be satisfactory at all.

    In short, I think the PCRT recurrence might hold here (not positive), so in principle the universe might reverse its course from the big bang and go back to that initial state. But that is so much counter to its natural evolution (expanding forever along an essentially deterministic trajectory), that if this is indeed possible, it will have such a tiny probabilty, that the time needed for this to happen will be completely humongous, something many many orders of magnitude larger that its current age I would guess.

    So saying "we started there because we first got there through recurrence" sounds to me like an atrociously unlikely explanation for this fact - explaining why we started in a low probability state by supposing a much lower probabilty process as a cause.

    Actually, I think this argument is in a way generic, and even if the above is wrong, the following weaker generic version should still apply : for any system subject to PCRT, yes you can reach a low entropy state (low volume in phase space) if you wait long enough - but you need to wait a time about inversely proportional to the volume of that region. So recurrence will always be a very poor explanation for having started there (in a very unlikely place) a given finite time ago.
    Last edited: Mar 17, 2015
  4. Mar 18, 2015 #3


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    It is a possible explanation, but not a very satisfying one. See e.g. about the Boltzmann brain problem.
  5. Mar 18, 2015 #4
    Thanks guys. Wasn't proposing...
    I just came across these as I try to ease my cognitive dissonance w/respect to how the low entropy beginning could be - and continue to struggle with the idea of a closed phase space picture of "the whole universe"... Especially one with a low entropy starting point. I was half expecting you guys to say, yes of course, the universe, a closed phase space, entered the low probability singularity from whence we are are ejected because in infinite time (which is time as far as we know) It eventually had to do that.

    Fluctuation theorem would have been my next rook move, because it seems to say that over infinite time, the probability of the net movement in phase space being negative (coarse to fine) is zero (as is the probability that it never made a single negative move from coarse to fine). This seems consistent with the deterministically statistical gradient of our clock-works, but seems to clearly state that for a closed system in infinite time, like our universe as far as we know, such a beginning as we have is impossible.

    But it sounds like you all are actually sitting here with this paradox right out on the table... And I suppose are as irritated by it as I am. I guess that is a comfort. Maybe FT doesn't apply as I'm imagining it (or I am misunderstanding it).

    Starting the second Susskind "minimum" book (QM) now. Very much appreciated the first and expect to refer to it often. Love the new connection I now have between between symmetry and conservation.
    Last edited: Mar 18, 2015
  6. Mar 18, 2015 #5


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    I would just add that, from what I read here and there about it, the lack of a convincing explanation for the low entropy start of the universe seems to me to be more or less the current state of physics rather than just something common to this thread's participants.

    And I will also offer what is perhaps the start of an explanation for a tiny little fraction of this : unless we are already in a thermal state, wherever we were 14by ago must at least be lower entropy than where we are today, since entropy has been increasing ever since. Perhaps you might say "buy why aren't we in a thermal state then ? " - to which I would venture "because we'd all be dead then".,.

    : )
  7. Mar 18, 2015 #6
    Yes, yes. I grok the fairly useless recursive circularity of the question of ultimate beginnings in time. Just checking to make sure someone hasn't found a way off... the circle... And figure out if everyone else is as freaked out.

    And this is just me trying to shed the conclusion I came to at some point, i don't know when, in the past. That one could at least get some sleep by saying it can't be... a closed system, which is apparently how I got Liouville, Hamilton, Entropy and Expansion screwed up.

    I have high hopes for Gibbs now, to get things sorted out, moreso.
  8. Mar 18, 2015 #7


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    I do not have the technical know
    I would hate to dash your hopes and trouble your sleep, and I wouldn't have the technical knowhow to to so, but I am afraid you might have to prepare yourself for a disappointment - that is, if my best guess about the applicability of Liouville here is correct, which of course it may not be : )

    Interesting topic..,
  9. Mar 18, 2015 #8

    Yep, there it is. It is fascinating to me that one might feel more comfortable saying that this is just the solipsistic ground state, where-as another might be more comfortable saying that the clear implication is that the consciousness is not fundamental, but emergent, from something else real that can be neither uniform random chaos nor consciousness.

    Or the same person might say both depending on whether or not they have had their morning coffee...

    I actually hat philosophy. So I am done now.
    Last edited: Mar 18, 2015
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