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Consider the simplest such system: a gas-filled chamber.

I understand that, were all the gas particles concentrated in one corner, the entropy of the chamber will be lower than the entropy of the same chamber but homogenized.

If so, the two states do not actually represent the same macro-state, hence we will never expect particles in a homogeneous gas chamber to spontaneously segregate.

This can be verified experimentally by trying to separate the gas into distinct regions, one of high pressure and another one with vacuum. The more we try to segregate, the harder it gets showing that this separation will never happen spontaneously.

However, such a chamber is often [erroneously?] represented as pebbles on a board: https://www.youtube.com/watch?v=kfffy12uQ7g But pebbles remain in the segregated state indefinitely, whereas gas particles always homogenize spontaneously and always resist segregation as discussed.

Therefore, can there EVER be any spontaneous fluctuations that lowers the entropy (segregate the content) of such an isolated system?

Nonlin.org
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  • Have you read the answers to the questions here, here, here, here, – Chemomechanics Aug 10 '20 at 21:02
  • here, here, and here? – Chemomechanics Aug 10 '20 at 21:02
  • Thanks. I have seen others, but not these. However, I just am reading a number of those answers and they seem experimentally false. True or False: gas molecules are NOT like pebbles on a board since pebbles stay put in a corner or whatever configuration whereas gas molecules do not? Why do gas molecules actually resist being segregated in a corner of the isolated system when they do NOT resist being in any of the homogeneous states? Just a matter of probabilities? It seems not. – Nonlin.org Aug 11 '20 at 19:30
  • No. The answer seems wrong. Going back to the gas in a corner or even an asymmetrical state after a homogeneous state seems not just improbable, but experimentally false and against the arrow of time. Please read my other latest comments below. By the way, it would be nice to have one unified conversation instead of three different ones with three different people. – Nonlin.org Aug 12 '20 at 09:22

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It sounds like you're asking if it's possible for all the random movements of particles through pure statistical chance to to result in one of the infinitesmal combinations of particle positions where the particles are segregated instead of one of the infinitely greater combinations where the particles are not segregated? Yes, it's mathematically possible but you will never see it in your lifetime, or anyone else's, or even the universe's lifetime. I am not a physicist so I don't know for sure how that plays into entropy calculations but think it is considered a spontaneous increase in entropy.

This is at the core of the theory that after the heat death of the universe, over infinite time, things by random chance would happen to find themselves back close together again such that entropy is reduced which would result result in live universe, and over even infinite time, an identical universe indistinguishable from the current one with the same chain of events as the current one (as well as many other infinite very different, and very similar variations along the way).

I don't expect this to be demonstrable, even on small scales unless you do so over time spans so vast that it makes the ultimate life time of our universe looks like literally nothing.

The closest thing is probably a Go board where you repeatedly empty the board and randomly fill the board over and over again until you end up with a combination where all the white stones are on one side and all the black stones are on the other side. Even that would still take nearly forever, even via computer since there are so few combinations where the stones are segregated compared to those where they are not. And this is just 361 moving pieces with very limited degrees of freedom compared to a real system.

DKNguyen
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  • Thanks, but "pure statistical chance" does not explain why gas molecules resist being pushed all into one corner of the isolated system. Because of this, the Go board analogy seems to be experimentally false. Agree? In addition, statistically, we should also see measurable fluctuations post homogenization. But I don't think we ever do. Here I am not going for the extreme, but a mix that becomes just less than perfect after a while.Think milk and coffee. Stare at the mug your whole life and you will never ever see anything other than a perfectly homogeneous mix (except organic degradation) Why? – Nonlin.org Aug 11 '20 at 19:37
  • I think I already covered that. It's pretty intuitive to see that there far far more combinations that result in homogenity than segregation. The number of combinations that result in visibly imperfect mixes also far exceeds those for segregation. Now, if the the number of visibly imperfect is far fewer than those that result in homogenity that would explain why you don't see anything other than homogenity over long periods of time. What do you mean by the Go board seems experimentally false? – DKNguyen Aug 11 '20 at 20:15
  • I get your point about combinations. However, as explained, there are two major problems. First, unlike Go pieces that stay put indefinitely, gas molecules DO NOT. This is major! Second, like I said, I am NOT going for extreme so let's compare garden variety fluctuations. Say in a symmetrical chamber sectioned in half, fluctuations going from 50-50 distribution of gas molecules to 49-51 or 45-55 or 40-50, etc. Statistically, there's not a big difference between 50-50 to 49-51. But do we EVER see even 49-51 spontaneous distribution post homogeneous? I think not! This is also major! – Nonlin.org Aug 12 '20 at 08:48
  • I think you're getting hung up on the Go pieces not moving. That part is taken cared of by you as you refill the board. As for the 49-51 thing, isn't that just a matter of granularity and the extremely large number of molecules in any given sample on human scales? – DKNguyen Aug 12 '20 at 13:19
  • With the Go pieces it's supposed to be an isolated system without external intervention. As such, the Go board behaves FUNDAMENTALLY different than a real gas chamber. Regarding the 49-51 thing, no, it's not a matter of granularity. This is somewhat like the Double-slit experiment where we see a clear and expected pattern with large number of particles. With gas, we expect some sort of distribution of x-(1-x)% chambers (given many chambers or one chamber sampled over time) from the very likely 50-50 to the most unlikely 100-0 with 49-51 being slightly lower than 50-50. This is not what we see! – Nonlin.org Aug 13 '20 at 07:56
  • Obviously, the pattern will be given by something like this: https://www.chegg.com/homework-help/questions-and-answers/4-ideal-gas-containing-n-molecules-box-volume-v-box-two-equal-parts-volume-v-2-weight-numb-q43308678. In other words, if fluctuations indeed occur, we should be able to sample and see them. But we don't see any fluctuations as far as I know. Why? – Nonlin.org Aug 13 '20 at 08:19
  • @Nonlin.org There are limits to measurability so if your sample size (say, one drop) contains a nearly uncountable number of particles then 49-51 would still be wildly improbable since which means that many things in the drop had to have happened in just the right way to get 49-51, as opposed to if you were able to obtain and measure a sampler size of just 100 molecules where it only takes 1 molecule being swapped out to achieve 49-51. Whereas the same thing among many trillions would be unmeasurable. That is what I mean by granularity and sample size. – DKNguyen Aug 13 '20 at 13:16
  • I suppose that makes sense. However, why is the gas actively resisting compression in a corner whereas the Go pieces do not? Doesn't that mean that "all molecules in a corner" is no longer a micro-state of the macro-state? Isn't that a fundamental difference between theory and the observable? Therefore the question remains: "can entropy ever decrease in an isolated system?" Again, if I look at the formula, the probability of entropy decreasing is rapidly asymptotic to zero for any reasonable gas filled isolated system. – Nonlin.org Aug 14 '20 at 19:10
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    @Nonlin.org I think it can (on tiny scales, at least, with the probability exponentially decreasing as the system size increases). The issue, to me, seems to be on of measurability and the scales we operate at. If your system was 10 molecules (and you could test for it) I don't see why you wouldn't eventually see some imbalance. But scale that up to trillions upon trillions along, then our inability to obtain and such infinitsemal samples along with our inability to measure something with such precision and you're forced to work at a scale where probabilities are unobservably low – DKNguyen Aug 14 '20 at 19:12
  • OK thanks. Good point. – Nonlin.org Aug 14 '20 at 19:16
  • Sorry, but upon further reflection I am still doubtful of something not observed experimentally. It may be that 10 molecules isolated systems are impossible or that atomic forces would prevent an imbalance of 1 molecule which is 10% if looked another way. The most telling argument I see against decreasing entropy is that Go pieces are not a good model for what happens since they stay put as opposed to the gas molecules that - experimentally - push back with enormous force. – Nonlin.org Aug 15 '20 at 05:17
  • @Nonlin.org I think you are getting too hung up on Go pieces not moving on their own. Gas molecules pushing back each other is simulated by Go pieces being in designated positions on the board that do not overlap, and the motion of gas particles which shuffles them altogether is simulated whenever you move the pieces around on the board. Models are models (obviously) and so don't cover everything and neglect some things depending on their intended usage. In this case, I think you are homing in on a difference between the model and the real thing that are unimportant to the topic at hand. – DKNguyen Aug 15 '20 at 05:20
  • Let's think some more and watch for new developments. Great discussion. Thanks. – Nonlin.org Aug 15 '20 at 05:59
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Since there is no such law that molecules may not congregate in a corner it could happen but that does not mean that a single fluctuation itself has lower entropy than, say, a more likely one such as in the uniform spatial distribution. Statistical entropy is not about a single arrangement of molecules, rather it is about the "size" i.e., number of similar arrangements.

Following along the same lines, one could say that equilibrium thermostatic entropy is about the constraints that maintain equilibrium. If you allow a change in the equilibrium by changing the constraints then a new equilibrium will establish itself with a new entropy that is larger than the starting one. This formulation of the 2nd law is essentially identical to the usual one.

If you have a cylinder with a dividing wall and only on one side you have gas then punching a hole will let the gas through it and occupy the hitherto empty half. When the gas occupies the larger volume it will have larger entropy but the entropy increase is, therefore, not caused by the molecules now moving in a larger volume but by the act of punching a hole and thereby changing the constraints on the gas.

hyportnex
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  • I just found this link that seems to disagree with you: https://www.chegg.com/homework-help/questions-and-answers/4-ideal-gas-containing-n-molecules-box-volume-v-box-two-equal-parts-volume-v-2-weight-numb-q43308678 . It seems any unbalanced configuration has a lower entropy than a homogeneous mix. True? – Nonlin.org Aug 11 '20 at 19:48
  • I do not understand what disagreement you are talking about. Please, elaborate. – hyportnex Aug 11 '20 at 21:04
  • Yes, you would be right with your punched hole example. However, there seems to be a problem with your assumption: "Since there is no such law that molecules may not congregate in a corner it could happen". This doesn't seem to be true. Remember that once released, the gas molecules will never return to a corner except to a limited extent and with enormous external effort. Therefore, those particular states seem unavailable once the gas has expanded. Isn't this the link between entropy and the arrow of time? Per below, we never seem to see any fluctuations at all once the gas is homogenized. – Nonlin.org Aug 12 '20 at 09:12