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Heat death, quantum uncertainty and Feynman's Path Integral

  1. Jan 24, 2012 #1
    Hi all,

    as a complete noob, I must first ask that people understand that I have only a layman's understanding of cosmology. However, after watching a few of Brian Cox's lectures on entropy and the heat death of the universe, I had a rather interesting thought (although as I am not a cosmologist, this maybe just uneducated ramblings).

    So here we go: If we take the state of the universe after the last black hole has evaporated due to Hawking radiation, all we should have left is a vast soup of particles, all at a few millionths of a degree above absolute zero. In every website and book have been able to read, it usually states that this situation will then continue until the end of time (although some state that time itself is effectively stopped, as there will be no more changes and therefore it would be impossible to measure time). It occurred to me that no one seems to have considered Feynman's path integral, which calculates the probability of a particle being in a particular place at a particular time. One of the parameters of this integral is time, and if this parameter is set to near infinity (as we would have in the case of the heat death) then I think there may be a possibility that the most improbable events could eventually occur. Would this not mean that given enough time all the particles in the heat death universe could eventually converge to one single location simply due to quantum effects alone, thus effectively reversing entropy? Could this generate a situation similar to the first few microseconds after the big bang?

    So can someone please shoot some holes in this idea?
  2. jcsd
  3. Jan 24, 2012 #2


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    Hello Zedhex and welcome to PF! Excellent question. It is certainly possible, and expected, that given enough time a homogeneous bath of particles in thermal equilibrium will exhibit extreme departures from homogeneity, due to either quantum or thermal effects. Such large fluctuations have been considered as possible explanations for the seemingly low entropy initial state of the universe -- i.e., for creating conditions for the big bang. Leonard Susskind has mused about this very question and offered the following analogy: suppose we deposit a drop of ink into a bathtub filled with water. After sufficient time has elapsed, the ink will be uniformly dispersed throughout: this is the analog of the thermal end-state of the universe. Now, if we wait for a sufficiently long time, it is distinctly possible that all the ink reassemble into a form identical to that of the initially deposited drop: this is the analog of the big bang.

    The potential shortfall with this interpretation is something called the Boltzmann Brain problem, which essentially states that indeed fluctuations of the type above will occur if given sufficient time. But many, many more fluctuations will occur which prepare a state with slightly more entropy than a fluctuation that prepares a state with such low entropy as our big bang. In other words, it's significantly more likely that we would inhabit a smaller, less complex universe than the one we live in. The "Brian" in the term Boltzmann Brain refers to the whimsical (or terrifying, depending on your view) idea that it's in fact more likely that the big bang created a single, fully functioning brain than it did our universe. Reasoning like this put significant pressure on any possible anthropic understanding for the origin (and subsequent evolution) of the universe. See Carroll's blog post on all this if you're interested in a much more detailed account: http://blogs.discovermagazine.com/cosmicvariance/2006/08/01/boltzmanns-anthropic-brain/
  4. Jan 25, 2012 #3
    An interesting post, but I think the Boltzmann Brain argument is a bit of a fudge. With enough time, it is perfectly possible that all sorts of odd low entropy states may occur (one is reminded of the Hitchhikers Guide Galaxy episode in which a whale suddenly appears in orbit around the planet), but that does not mean that even lower entropy states are not possible, just much less likely. I think that part of the problem is the sheer unimaginable scope of deep time for our human brains. The most critical part of this seems to be that with enough time, low entropy states, although highly improbable in any specific timeframe, become inevitable within a long enough period of time. The fact that the period of time required is exponentially larger than the current age of the universe is simply irrelevant. A second factor derives from figure 1 in the link that you sent, showing that if we have an entropy perturbation which temporarily moves our universe back to an earlier state, it is statistically impossible to differentiate between point A and point B in the graph. Thus it may be possible that what we perceive as a big bang was nothing more than a quantum variation in a previous heat death universe, and we are living in one of an infinite number of cycles which return to a high entropy state. Thus the universe, when seen in a deep time context, would spend the vast majority of its time in a high entropy state with very very occasional blips of low entropy.
  5. Jan 25, 2012 #4
    The problem with infinity is that we don't yet know how to make sense of probability. Infinities also enter from the eternal inflationary scenario, and it is called the "measure problem". See the excellent article here for more details [and how the measure problem relates to the arrow of time as well as an attempt to make things finite again via horizon complementarity]
  6. Jan 25, 2012 #5


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    Sure, and the Boltzmann Brain argument doesn't disagree -- of course all configurations are possible given enough time. The formal problem that the argument poses is one to the anthropic principle: while huge departures from homogeneity will occur, slightly less huge departures will occur way more often. In other words, it is much, much more likely that we would live in a universe with only a single galaxy, or a single star, than the one we find ourselves in. Sure, you can argue that we just got lucky and happen to be the result of an incredibly, insanely unlikely fluctuation. But that doesn't explain anything: it's still just as fine tuned as postulating that the universe simply started out in a very low entropy state.
  7. Jan 26, 2012 #6
    Aha, you may have a point there. But wouldn't it also be true that the only high entropy states that lasted long enough for intelligence to emerge would have to have something like the complexity of our own universe. So if we looked at a graph of entropy against time we would have immensely long periods of stasis punctuated by short bursts of high entropy, the overwhelming majority of which would be unable to support such features as stars, galaxies and planets. These episodes would dissipate fairly rapidly. The fact that we happen to live in one of these unusual bursts of extremely high entropy would indeed be dumb luck, unless Yenchin is right (apologies I haven't read your link yet - I will in a moment) and probabilistic arguments are completely unusable in the case of near infinite time.
  8. Jan 26, 2012 #7


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    I think you mean short bursts of "low" entropy.
  9. Jan 26, 2012 #8
    Isn't the Boltzmann Brain problem relying on assumption of how likely it is for a being with out kind if intelligence to evolve. I don't think we really know enough to be able to get a good handle on that probability yet.

    When I consider this problem, I look at it backwards instead, it makes me consider that life with our intelligence may be so unlikely that it needs a fluctuation bigger then our visible universe to exist for this long to be likely to evolve. That is a bit depressing. It might mean we are very lonely.

    However, it also may be possible that certain things go wrong in smaller fluctuation. Maybe it isn't likely to survive for so long or something.

  10. Jan 30, 2012 #9
    yes I think I got things a little bit confused there - corrected:

    "But wouldn't it also be true that the only low entropy states that lasted long enough for intelligence to emerge would have to have something like the complexity of our own universe. So if we looked at a graph of entropy against time we would have immensely long periods of stasis punctuated by short bursts of low entropy, the overwhelming majority of which would be unable to support such features as stars, galaxies and planets. These episodes would dissipate fairly rapidly. The fact that we happen to live in one of these unusual bursts of extremely low entropy would indeed be dumb luck"
  11. Jan 30, 2012 #10


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    Why do you think so? What does the evolution of intelligence on Earth have to do with the rest of the universe? Life apparently needs at most a rock and an energy source; and maybe less. It is apparent that our universe is quite a bit more roomy than we need.
    Yes, unless one can successfully argue for an anthropic need for such low initial entropy. But, as I pointed out above, it seems that life can make due with much less, and so anthropic arguments fail. Dumb luck is one answer, but it more probably indicates that our picture of the low entropy universe emerging as a fluctuation is not the whole story.
  12. Jan 30, 2012 #11


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    This is very true and we should all be careful with making presumptions about the likelihood of life in the universe. I hope that we'll soon have a better understanding of that.
  13. Feb 1, 2012 #12
    I would have thought that that would be self-evident - in order for life to appear we would first need a stable universe, which implies that we would need enough order (i.e low entropy) to allow stars and planets to appear. Life on Earth has taken around 4.5 billion years to produce intelligent life. It would be nice if we had a few other examples to compare that to, but as we don't then it is rather hard to estimate how much time is required (on average) for intelligent life to appear. However I would suggest that it is fairly certain that your rock and energy source would have to be around for quite a long time. In addition to that, if we examine the chemistry of life on our planet it requires quite a few elements that can only be created by dying stars (basically, anything heavier than carbon), therefore if our life forms on earth are representative then our stable universe would have to exist for at least twice the time required for intelligent life to develop on a single rock.
  14. Feb 1, 2012 #13


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    Agreed that stability is key. But a flat universe would afford this kind of stability without the hundreds of billions of galaxies it possesses. Given stability, we could make due with a lot less.
  15. Feb 3, 2012 #14
    Maybe there is some reason why a smaller fluctuation then the one we are currently observing isn't very likely to be stable for so long so we need one this big where life might be common.

    I think the argument is that if we are living in a random fluctuation, then we would most likely to find our selves in the most probably kind of fluctuation, which would be the smallest and shortest that is likely for beings like us to evolve in. I commented earlier that if this is true, then it might mean that life like us is extremely unlikely, but considering this further now, I realised that I was only considering the size of the universe and not the time.

    How much longer will our universe be suitable for live like us to evolve? I'm not an expert, but I would guess we wouldn't expect galaxies and star compositions to change that much for a long time, compared to the current age of the universe. Which means we evolved very early on in the life of the universe we see. I guess that this would not be the most likely expectation. Maybe there is some argument why it is harder for life like us to evolve as galaxies ages? I certainly don't know enough to answer these questions.

    In the end though, you never know, as mentioned earlier by somebody, we could just be in an unlikely fluctuation, just like it is very unlikely to be living at the point in time where the moon is just the right size to eclips the sun so closely.
  16. Feb 4, 2012 #15
    The heat death of the universe follows from misapplying entropy and thermodynamics to the 'cosmos'.

    So far as I know no black hole has been detected an experts continue to use the term «candidate to black hole». Even Hawking has started to doubt about the existence of black holes.

    I would not take Feynman's path integral very seriously, specially Feynman view that particles go forward and backward in time. His views are based in a misinterpretation of Dirac and Klein-Gordon equations and, for instance, in modern quantum field theory all particles move forward in time.

    Moreover, Feynman formalism is for particles whose entropy is zero and constant. The formalism does not give the dissipative equations of motion associated to increase in entropy. Therefore it makes no sense to apply it to a system where entropy increases.
    Last edited: Feb 4, 2012
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