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That pretty much says it.

Suppose I have some powder of $NaCl$. It is kept in contact with itself in vacuum. You are free to remove all the disturbances that bother you.

Is that true that, well, there exist a ($\mbox{very}^\mbox{very}$ large) amount of time $T$ that for every moment $t>T$ you will have a single beautiful crystal with 99% probability.

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In infinite time surely all (but few) of the atoms will fly away because of fluctuations. So what is your question actually? Whether the powder freezes due to radiative cooling, disregarding evaporation completely? Is that all? – Marek Mar 1 '11 at 13:19
Well, but due to Poincare recurrence they will also come arbitrarily close again. – Daniel Grumiller Mar 1 '11 at 15:05
@Daniel: that theorem only holds for compact phase spaces. I see no such assumption in the statement of the problem. – Marek Mar 1 '11 at 15:31
Well, we have only access to a finite region of space if there really is a cosmological constant, so for all "practical" purposes (like the question posed above) we live in a box. See also my comment below my answer and the link there. – Daniel Grumiller Mar 1 '11 at 23:12
up vote 5 down vote accepted

If you wait long enough, it'll become a bunch of iron (assuming it's confined so that the atoms can't evaporate off as Marek pointed out). After all, an iron nucleus is more stable than any other nucleus. The probability of the other nuclei tunnelling together to form a big hunk of iron is absurdly low, but it's not strictly zero. If you wait long enough, it'll happen.

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I think that I should accept this answer... – Kostya Mar 4 '11 at 15:45

Technically, if you have infinite time to wait, then yes. This is based on the standard "if it isn't forbidden, it is compulsory" idea of QM. However, as other answers have correctly pointed out, there are other potential quantum fluctuations that are also available and may be more probable. Combined with the unlikelihood of the desired quantum fluctuation, in any practical sense, the answer is no because the universe will end before this occurs.

Also, with regard to the last line of your question, which is slightly different, the answer is a resounding no. Although you can conceivably get a single crystal at some very large time T, you will not retain a single crystal for all time t>T because another fluctuation will destroy it eventually.

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Not exactly. Due to QM there is always some chance (maybe very very tiny) that the state will become another not-forbidden. But here we have a thermodynamical problem, with time going to infinity. Hence probability may not be small, but going to zero with time. – Piotr Migdal Mar 1 '11 at 17:41
@Piotr I feel like your objection is correct, but my own ignorance is preventing me from connecting things properly. As said in the original answer, other non-forbidden quantum fluctuations can occur. But they can also de-occur, i.e. reverse, no? So that, while fluctuations that take us away from the state we want occur, other fluctuations that bring us back also occur. The net result would be a long-time random walk to the desired state, i.e. there is no point at which it becomes impossible to get to the specified final state. Where does thermodynamics come in? – Mitchell Mar 2 '11 at 16:37
Thermodynamics in the sense that you investigate macroscopic state of object in the limit $t\rightarrow \infty$ and with given statistical conditions (pressure, temperature, ...). But is only a word. You statement of (re)occurrence would be true in finite space. In the infinite space you can easily imagine counterexample: two particles qoing in the opposite directions. As chance of tunneling is $\approx \exp(-c x)$, (and $x = v t$) the total probability they tunnel to each other in any finite time is $\int \exp(-cx) dt \propto \exp(-c x_0) \ll 1$, where $x_0$ is the initial distance. – Piotr Migdal Mar 2 '11 at 21:06

If you wait very, very long then your NaCL will be part of a black hole, for this will be the only entities in the Universe in the far future, at least according to our current understanding.

Then, eventually, these black holes evaporate and your NaCl will be part of Hawking radiation.

In the ensuing thermal ensemble you will have random fluctuations. If you wait long enough then, indeed, you have a probability arbitrarily close to 1 that one of these random fluctuations will be that your original NaCL powder forms a perfect crystal, or that some monkeys type the work of Shakespeare ;-)

So in short, the answer seems to be 'yes'.

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The first two paragraphs are correct, according to our best theoretical understanding. Not the third, though: in the expanding Universe, the temperature approaches zero rapidly enough that a fluctuation to a lower-entropy state in the distant future won't occur. – Ted Bunn Mar 1 '11 at 17:49
I disagree, of course. Also the third paragraph is correct, given our current understanding of the Universe, in particular given the positive cosmological constant. You may want to check this talk by Sean Carroll…, starting at minute 7. – Daniel Grumiller Mar 1 '11 at 23:09
Another comment: the temperature in the Universe will never reach zero if we have a positive cosmological constant. There will always be thermal radiation from the deSitter horizon. – Daniel Grumiller Mar 1 '11 at 23:21

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