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Ice crystals are spatially ordered, and in every randomness there is a low possibility of temporarily order. If given enough boiling water, and sufficient time, could local clusters water molecules happen to be in a crystalized state?

This may seem absurd, but I believe it must be possible, imagine dropping an ice cube in boiling water(and water vapor) in a perfect closed system, the ice cube melts while the water keeps boiling (because of the water vapor), then because of Poincare recurrence theorem, there will be an similar ice cube after a sufficiently long but finite time.

EDIT: The temperature is 100 Celsius and 1 atm (not at triple point)

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Do not hold your breath. It is the same answer to "what is the probability that all the air ends up in the upper right corner of the room and we suffocate". –  anna v Jan 12 '14 at 15:29
So is there a small chance it might snow in the upper right hand corner of the room? –  Jitter Jan 12 '14 at 15:48
@Jitter, no it'll snow in the lower left corner, so we'll die w/o even getting a single run down the slope :-) –  Carl Witthoft Jan 12 '14 at 15:57
At the triple point of water, the probability is quite high. –  Olin Lathrop Jan 12 '14 at 15:58
@OlinLathrop true, but I don't think you can convince water to boil at the triple point. –  Carl Witthoft Jan 12 '14 at 17:13

1 Answer 1

Please read this article in the wiki

In statistical mechanics, Boltzmann's equation is a probability equation relating the entropy S of an ideal gas to the quantity W, which is the number of microstates corresponding to a given macrostate :

Boltzman formula

In this formula connecting the statistical probability to entropy arising from thermodynamics one sees that all microstates are counted in. This would include the subset of microstates that are postuled in the question.

To get a probability number, one would have to do all the permutations of molecules at that temperature and pressure to appear in the volume under consideration and take the ratio to all the possible configurations with the rest of the molecules. In normal temperatures and pressures this would be a very very very small number for boiling water, and I am not willing to do the calculations for it. In addition one would have to include the phase transition into the calculations also ( or the binding of water molecules in the two forms).

In a supersaturated vapor, as in a cloud chamber or up high in the atmosphere where the jets leave track, the probability is quite high once a seed appears that the phase transition will happen, but that is another story.

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