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It seems to me a rather big coincidence that statistical physics works so well.

I can see how consistent macroscopic observations can occur just because the microstates that give rise to that behaviour are overwhelmingly more likely than other microstates. And I can see how the word 'likely' here is really just a statement about there being many more ways for particles in a system to share (for instance) energy equally, than for one particle to have most of it, so that given all sorts of complex interactions the system will most often find itself in a more uniform state.

So say a system is in a given microstate at some time $t_1$, at at that time we observe the corresponding macrostate. Now we have phenomenological thermodynamics to tell us how the macroscopic observables will evolve, and what we can expect to observe at time $t_2$. Clearly that means the microstate at $t_1$ evolves in such a way that it gives rise to those observables at $t_2$. But there's many different microstates that could have caused us to observe the original macrostate at $t_1$, so then it must follow that the overwhelming majority of these microstates also evolve to give rise to the macrostates we expect at $t_2$. I don't see a reason why this should necessarily be so.

Is there some mathematics that I'm missing that explains this coincidence? Or is this really just some strange quirk of nature without which macroscopic physics wouldn't even work?

EDIT: It seems I haven't done a great job explaining my confusion. I've found a paper by E.T. Jaynes that touches on these issues in sections 3 and 4. He seems to explain the coincidence using sharply peaked probability distributions, though I don't quite understand how that works, and Jaynes tends to be a little concise in his explanations. It would be great if someone could explain in more detail, or provide some other references.

EDIT: I've removed the introductory part of this post as it created unnecessary controversy because I wasn't being clear. If some comments don't seem to make sense, it's likely because of this.

EDIT: To hopefully put this controversy to rest, quoting myself from the comments:

I think we're arguing on different levels of abstraction. Within 'theories we construct about the universe' I'm making a distinction between two types of use of probability.

The first kind we use to describe flipping a classical coin; we only use probability for this because we lack information to reason about the coin with our other 'theories about the universe' (such as classical mechanics; we don't know all relevant forces in a 'random' coin toss, but we could in principle program a robot to always throw heads).

The second kind of probability we use to describe randomness in our 'theories about the universe' that we cannot explain by means of an underlying theory (such as classical mechanics in the coin toss example) and thus seems fundamental, such as in quantum mechanics.

Again, this applies to the use of probability alone. Since the use of probability in statistical physics seems to be of the first kind (though that's somewhat debatable as it does use quantum physics), I find it strange that everything works out so well, but this may just be a consequence of the Law of Large Numbers.

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    $\begingroup$ "Assuming the basis of statistical physics is not actually probabilistic" - why would you make that assumption? It is entirely probabilistic - or alternatively, if you stay a system can be in any one of N states and none of them are any less likely than any others, and M of those N states are indistinguishable, then the probability of observing the system in that state is $\frac{M}{N}$. If a father has five daughters and one son, and he introduces you at random to one of their children, it's more likely to be a daughter... $\endgroup$ – Floris Feb 3 '16 at 17:04
  • $\begingroup$ Note that distinguishability (or lack thereof) is a thing that can be checked experimentally for fermions. It's not a matter of "knowledge about the system", but a real consequence of quantum mechanics. $\endgroup$ – dmckee Feb 3 '16 at 17:10
  • $\begingroup$ Floris: I completely agree, but this is not really what my question is about. I may not have explained clearly enough. @dmckee: Yes, I realise that. My point with that passage is not to say that indistinguishability isn't a real property, or that it shouldn't affect our description, but rather that our inability to distinguish particles shouldn't affect physics. I fear my own confusion on the subject is hampering my ability to clearly communicate my point. Dimitri's answer below seems more in the direction of what I meant to ask. $\endgroup$ – Timsey Feb 3 '16 at 17:17
  • $\begingroup$ @Timsey My point is that it does affect physics. For that matter it's a key ingredient in making chemistry happen. The Liouville theorem and the ergodic hypothesis are tools for booting this quantum understanding to classical systems without have to fully understand the rules of decoherence, but at the lowest level the inability to tell if that is electron 1 over there while electron 2 is here or vice versa is a core property of physics. $\endgroup$ – dmckee Feb 3 '16 at 17:27
  • $\begingroup$ @dmckee: I think we agree in principle, but that we're using different meanings of 'our inability to tell'. I agree that the indistinguishability is an actual - ontological - property of nature. I'm just saying that actual property is the thing that matters, and not the fact that 'we can't distinguish them'. My point is that the statement 'we can/can't do this or that' is irrelevant if we also know the actual physical property that makes it so we can't do that. In retrospect, this point doesn't even really seem necessary as an introduction to my actual question. $\endgroup$ – Timsey Feb 3 '16 at 18:01
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The whole point of statistical physics is that we don't really care about what the microstate could be, we just know that there's a great collection of similar ones that give rise to the same macrostate. I believe what you're questionning about is closely related to the ergodic hypothesis

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  • $\begingroup$ Interesting. I came across Liouville's theorem earlier and it seemed related. I'll give this a read, thanks. $\endgroup$ – Timsey Feb 3 '16 at 17:08
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Most things along those lines are just simply the central limit theorem / law of large numbers.

For example, when you compress a gas in an insulated container using a piston, its temperature goes up. Why? Because the moving piston accelerates gas molecules that bounce off it.

And why does the temperature always go up by the same amount? Because there are so many grillions of kajillions of atoms in a macroscopic container, that the central limit theorem pretty much guarantees that each millisecond, a similar number of atoms are hitting the piston, and with a similar distribution of speeds.

For very small systems, the law of large numbers does not apply, and indeed you generally do not deterministic temperature changes etc., but rather predictions about the probability distributions for what will happen.

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    $\begingroup$ Thank you, this helps. It seems to agree with Jaynes' claims about sharply peaked distributions. I still don't quite see exactly why this should work for all microstates underlying all macroscopic observables, but I suppose different microstates that give the same macroscopic behaviour are so similar I can hardly expect them to evolve very differently. The huge number of particles probably stops any discontinuity for occurring: any one particle that does something very unprobable is overshadowed by the ~10^24 others. $\endgroup$ – Timsey Feb 3 '16 at 19:15

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