[Kryptid (OP)]

Thanks for all the input, especially from Bored chemist.

I guess my main reason for wanting to know this is due to the implications of a possible hypothetical scenario as follows:

Let's say that we put a cat in this same box. In the box with the cat is regular air, with 21% oxygen, 78% nitrogen, and 1% argon at STP. Over time, the cat uses up all of the available oxygen and suffocates to death. It begins to rot and, you know, do whatever dead things do. Given the analogy with the original hydrogen-oxygen scenario, could it be possible that after eons and eons of random molecular collisions, quantum tunneling, and whatever else, the cat would eventually be reconstructed in whole and come back to life? I guess it's exponentially more likely that the cat's head alone would reform first while the body was still disorganized. It might even be that the cat's atoms would rearrange into a multitude of other life forms first.

As for proton decay, let's say that it does happen. Couldn't the proton potentially reform if the resulting decay products collided with one-another at the right place at the right time after bouncing around inside the box for untold periods of time? Would that not allow the reconstruction of the cat even if it takes longer than 10^35 years?

Remember, I don't want to know what is unlikely, extremely unlikely, or INCREDIBLY extremely unlikely. I want to know if it this possible in principle at all, since we can deal with potential infinite spans of time.

If you want, you can say that this "box" is actually a tiny universe in its own right. I guess that would keep things from "getting out" (unless quantum tunneling to other universes is possible). Any way, just assume that the box is impermeable to even quantum tunneling.

[Soul Surfer]

It is not possible even in principle on the scale you envisage but it is possible on the scale of a planet provided you do not insist on the same atoms being in the same places.

The reason for it being not possible in the small box is that the assembly process takes time and if it started happening slowly decay processes would destroy it long before it could be assembled and there is no room to dump the extra entropy we would need to use to create the order.  We know it is possible on a planet sized lump because evolution has produced life including some cat sized functioning lumps and it is possible to create more cat sized functioning lumps by breeding them.  however this process requires an increase in entropy of other molecules that have been used in making the beasts.

[JP]

I have to disagree.  It certainly would be possible in theory for the cat to reassemble.  It won't ever happen in an actual experiment because the probabilities are so small that it shouldn't happen in the time the universe has existed.

Entropy is useful because we will never actually see the cat reform, even though it's technically possible.  Entropy increases in closed systems because the probability of it doing otherwise is generally too small to ever happen on time scales we care about.  Therefore someone who tells you "based on entropy, the cat won't reform" is correct. 

Finally, I'd add that the reason that life can form on the earth is that the earth isn't a closed system on its own.  It's getting energy input from the sun, and so the entropy on earth can decrease.  (The earth-sun combination does have a net increase in entropy, however.)

[Onanist]

also... wouldn't the energy in the box actually be more than that required to split up all those water molecules?

because the energy in the closed system would not only amount to all that released by the combustion reaction, but also the original energy applied to get over the reaction threshold?