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My book uses the argument that the multiplicities of a few macrostates in a macroscopic object take up an extraordinarily large share of all possible microstates, such that even over the entire lifetime of the universe, if each microstate had an equal chance of being accessed, fluctuations away from equilibrium would never occur.

My question to this statistical proof is this: In the real world, is there really an infinitesimal but nonzero chance that macroscopic systems could access some of the more unlikely macrostates (e.g. heat transfer from a cold object to a hot object)?

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    $\begingroup$ I think you are essentially asking whether the ergodic hypothesis (en.wikipedia.org/wiki/Ergodic_hypothesis) is true. I'm not competent to comment, but you might try this book chapter by Thorne and Blandford, especially pages 20 - 27. (pma.caltech.edu/Courses/ph136/yr2008/0803.1.K.pdf). $\endgroup$ Nov 11, 2010 at 6:41
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    $\begingroup$ This is a question about the validity of equilibrium stat mech. In that framework, of course, big fluctuations away from equilibrium and therefore interesting events (such as life) never happen. Its a good thing then that most interesting systems in the real world are far from equilibrium! $\endgroup$
    – user346
    Feb 13, 2011 at 7:11
  • $\begingroup$ Related:en.wikipedia.org/wiki/Fluctuation_theorem $\endgroup$
    – Tyberius
    Apr 12, 2019 at 18:07

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I guess so - I mean, as far as I know, there's no law of physics that strictly prohibits those "exotic" states from being realized. As long as the state exists and can be reached by some path from the "center" of the state space where the likely states are, there should be a nonzero (not even infinitesimal, really) probability of accessing it. But for a typical system, that probability is really, really, really small. So small that it's impossible to intuitively comprehend just how unlikely such an event is.

The thing is, a lot of people aren't used to dealing with even moderately large or small numbers. If you confront them with a probability like $10^{-10^{23}}$, they often fail to put the smallness of that value in perspective, and instead focus on the fact that it's not strictly equal to zero. From there they may start coming up with all sorts of nonsensical ideas about walking through walls and spontaneous combustion (the weird kind) and the like. So physicists usually find it easier to just say the probability is zero - and in fact, for any purpose other than a rigorous mathematical proof, it might as well be.

(Sorry about the rant, I know most people are actually relatively sensible about these things, but it bothers me that the crazy ones seem to get all the attention despite being wrong.)

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    $\begingroup$ I just want to add that such a small probability is much smaller than the probability that we get the model wrong, by any measure you can reasonably assume. In other words, if someone tries to seriously compute the probability of seeing what @wrongusername calls an "exotic macrostate", this probability would be dominated by the small possibility that physics has been totally wrong over the last two centuries. $\endgroup$ Nov 16, 2010 at 13:53
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    $\begingroup$ There are two statistics fallacies in this answer: you have to add over all the different folks and walls, not just one. You then have to add over all the different independent types of rare events: not just walking through a wall, but also walking on water, also having carbon-based life evolve, also etc... this fallacy also comes up in arguments directed against the theory of Natural Selection. So at least please remove the ranting part, which is really otiose. $\endgroup$ Jun 5, 2013 at 15:48
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    $\begingroup$ No, that doesn't apply to what I'm saying here because even the combined probability of any sort of entropically forbidden macroscopic event happening is incredibly small. It's not unreasonable to assume the probability of having carbon-based life evolve is orders of orders of magnitude larger than that of walking through a wall or walking on water. $\endgroup$
    – David Z
    Jun 5, 2013 at 20:31
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    $\begingroup$ Yes, several orders of magnitude. But as far as I know, no one has done better than that. No halfway decent calculation has ever been done, and a million-dollar prize hangs on it. Several orders of magnitude: does getting it up to $10^{-10^{19}}$ really help? $\endgroup$ Jun 6, 2013 at 12:48
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    $\begingroup$ us.net/life and tbiomed.com/content/6/1/27 are offering the million dollar prize, but they are falling into the same statistical fallacy you are. No one, as far as I know, has any idea of how to estimate the number of different independent types of "entropically forbidden macroscopic events", in your case, or "evolution of life", in their case. $\endgroup$ Jun 6, 2013 at 12:53
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You can get a quantitative estimation of the relative probability of 2 macrostates of different entropies $S$ from the $S=k\ln\Omega$ formula, where $k=1.38\times 10^{-23}\ \mathrm{J}/\mathrm{K}$ is the Boltzmann constant.

We have $P\propto\Omega\propto e^{-S/k}$ . This means that $\frac{P_1}{P_2}=e^{-\frac{S_1-S_2}{k}}$.The presence of $k$ as a denominator of the exponent ensures that the probability is small as soon as the entropy difference is bigger than a few times $k$, and completely negligible when the entropy difference is as small as $1000k\sim10^{-20}$ J/K. When you allow $S_1-S_2$ to take any reasonable macroscopic value , the you have the insanely small probabilities David Zaslavsky spoke about.

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The figures being batted around here are correct for the probability of this particular macro-event's occurring here and now. But there is a statistical fallacy involved in drawing from this the conclusions that are being drawn.

This well-known fallacy, but I don't know if it has a name, let me baptise it "the cash-register fallacy", is as follows: you have just dumped a week's worth of groceries on the cashier, they ring it all up, it totals to $77.11 and they say, "wow, look at that. What are the odds of that happening?" Well, the odds of this particular striking lucky number's happening were 1/10,000. But there are so many other striking numbers that would have produced the same impression, that when you add them all up.... it is not so unusual. In fact I have had to listen to cashiers say the same thing to me five times now, and have stopped buying groceries.

The relevant probability that needs to be estimated is the probability for "a striking macroscopic event" to occur, and to estimate this, we need to count how many (independent) types of such events there are, just as above someone once estimated how many striking five digit numbers there were. No physicist has ever performed this estimate. No one has any idea how to do it, and I suspect it could affect the conclusion.

Something similar to this fallacy has been present in the history of debates about whether random Natural Selection can indeed be the real motor of evolution. Around 1900, opponents of Darwin's theory of random Natural Selection used the same general line of reasoning as some of the professional physicists contributing to this site. Sir Ronald Fisher exposed the fallacy involved. Currently, some scientists (they might be associated with the so-called "Intelligent Design" agenda, but I cannot tell for sure) are offering a million dollar prize for anyone who can show that the probabilities for random mutation acting with Natural Selection's producing six detailed biochemical processes which are key to life as we know it, over the time-span in which the Universe has been in existence, acting at a certain rate of operations per second, are at all plausible.

They are falling into the same statistical fallacy as is present in the other post here. The relevant probability that needs to be estimated is not the probability that these six particular mechanisms could have been produced by chance, but the probability that any of God only knows how many possible alternatives that would work, even though they did not in fact happen, and produce "life", could have arisen through chance.

Unless and until someone can estimate how many different independent alternatives there are, no assertion can be made about the probability of life's having arisen through chance alone. And similarly here, none of the posters have the right to make an assertion about the probability, during the total career of the Universe so far, of a macroscopic violation of entropy's occurrence. Here, though, I think it might be feasible to estimate how many different independent types of violations should be counted.

Falling into this fallacy is related to the failure to undertand the difference between a micro-state and a macro-state, a misunderstanding endemic to students of Stat Mech and Thermo. All micro-states have the same probability as each other: nearly zero! The only physically relevant probabilities are the probabilities of the different macro-states. Here, we have to figure out which is the relevant macro-state. The wrong choice will lead to a calculation lacking any significance. The wrong choice has been made in these posts, and by the anti-Darwinian people behind the offer of the prize, and the results of the calculations are physically meaningless.

Up to here is what interested me the most. But there is one more point to be made: the Universe is not in fact in a state of equilibrium: this is obvious to the naked eye. It seems as though the mixing process has not yet gone on long enough. So none of the laws of thermodynamics even apply to the Universe as a whole. So the extrapolation from the probability here and now to the probability over the life-time of the Universe is also invalid.

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  • $\begingroup$ The fallacy that you call the "cash register" fallacy is related to the question of are there two people in this room with the same birthday. The probably for a room of 30 people > 50% because it's any two people in the room. Maybe Jane and John had the same b-day and that probability is 1 in 365. But you didn't ask that. You asked if any two people had the same b-day. $\endgroup$
    – Joe C
    Oct 6, 2022 at 17:38
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A question marked duplicate to this question framed the issue another way: “Why is it only ‘almost’ always true that entropy is non-decreasing,” which is the way Wikipedia states the second law of thermodynamics? I found that way of framing the issue even more helpful and interesting, as it leads to a comparison between mathematics and physics.

The word “almost” actually has a rigorous definition in Measure Theory which is part of Mathematics. Mathematical Probability Theory uses Measure Theory a lot (Think of measuring the probability of events!) and inherits the notion of “almost.” When you say an event is “almost sure” to happen, you convey very precise information, namely that the event occurs with probability $100\%$. A trivial example of an almost sure event is getting a number between $1$ and $6$ from rolling a dice. Well, there’s literally no other outcome that can happen. You simply cannot get a $7$ or a $-1$.

Why then bother to use the phrase “almost sure?” What about just saying “sure?” Consider the following example. Suppose you have an ideal random number generator that outputs a random real number between $0$ and $1$ with uniform probability. There are infinitely many real numbers as possible outcomes to choose from, so each single real number, say $0.31415926...$, only gets to share $0\%$ probability to occur, and the probability to not get that given single number is $100\%$. Yet that is not to say that our number $0.31415926...$ simply cannot occur. There’s nothing wrong if it happens to occur, even if a zero probability event happening sounds paradoxical. And indeed, if this number doesn’t occur, some other number has to occur anyway, and then the paradox looms with that number.

That’s why we want to say we are “almost sure” about the event of not getting a particular number. We are pretty much sure about the event because the probability is as large as $100\%$, but we still need to state “almost” due to the technicality that the event still can fail to occur. Simply saying “sure” is inappropriate.

While pedantic, this is more important than you may think, e.g. when you are sampling events and inferring the probability; you cannot immediately conclude that the probability is non-zero if you observe an event just once.

Physicists are less pedantic than mathematicians and presumably use the term “almost” more loosely. Physicists are happy to ignore errors that have negligible observable effects. As other answers noted, the expected time before seeing an instance of decreasing entropy can be significantly longer than the life time of the universe, and so is negligible. Hence they call the event of increasing entropy “almost certain,” although it’s not even “almost,” as there’s a tiny but non-zero chance for the entropy to decrease.

In the end, the confusion evinces the adoption of reasonable approximations in physics, and shows how mistaking “almost almost true” conclusions for true conclusions can cause difficulty in believing them.

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Entropy is mathematically defined on distributions of a variable: in physics only the entropy of microscopic degrees of freedom is considered (except for the obligatory section present in virtually each textbook of statistical mechanics: the sharpness of the 'spreading' of a macroscopic variable). Where as the entropy of microscopic variables are observed to increase with time, the entropy of the distribution of a macroscopic variable tends to decrease with time i.e. become sharper. In the context of dynamical systems (although unstable equilibrium points may exist where the entropy actually increases over time)...

So in the context of physics entropy [implicitly of microscopic degrees of freedom] tends to increase with time while entropy of macroscopic variables tend to decrease with time (for example consider the horizontal position of an ensemble of marbles you have randomly thrown -original entropy in horizontal x position is high- in a parabolic potential, after a while each marble comes to rest in the bottom - so final entropy in horizontal x position is low since the distribution after a some time t of the positions in each ensemble becomes more sharply peaked around the bottom of the pit).

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  • $\begingroup$ or in short: microscopic entropy increases, macroscopic entropy decreases. BTW since life rests on making things more certain (i.e. the probability of having 2 legs should be much higher than other amounts of legs) macroscopic decrease of entropy is necessary, and hence microscopic increase of entropy is necessary... $\endgroup$
    – propaganda
    Jan 25, 2012 at 11:20
  • $\begingroup$ Wait... I am pretty sure that entropy always increases, particularly at macroscopic levels. And entropy is essentially a logarithmic measure of the multiplicity, not a measure of the uncertainty in quantum mechanics. $\endgroup$ Jan 25, 2012 at 20:52
  • $\begingroup$ The logarithmic measure of multiplicity is but one of many equivalent definitions of entropy: when multiplicities are unavailable S[X]=-\int P(X=x)*ln(P(X=x)) dx. Apply to microscopic probability distribution and you get traditional entropy which tends to increase over time, but calculate entropy on macroscopic probability distribution and this will tend to decrease over time... $\endgroup$
    – propaganda
    Jan 26, 2012 at 3:02
  • $\begingroup$ (his question doesnt seem to refer to probability of strange events where wavefunction is small but nonzero in quantum mechanics, but to small probabilities in statistical mechanics) $\endgroup$
    – propaganda
    Jan 26, 2012 at 3:04
  • $\begingroup$ Oh shoot, I'm sorry then. I saw you talking about variables getting sharper, and linking entropy to sharpness, which I haven't heard of being used in stat mech before, only in QM. Any further recommended readings on macroscopic entropy decrease? $\endgroup$ Jan 26, 2012 at 14:55

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