Does random have rules?

[Alex K]

the second law definitely holds only on average for large enough systems, and for finite systems there is a small chance that they are violated. It's just that for macroscopic systems this probability is unmeasurable small.

Typical example: if you have a box full of gas molecules, there is the slim chance 2^-N that at some point in time they will all sit in the left half of the box, which would correspond to a state of much lower entropy than the starting state.

(I'm probably going to get the details all wrong, but hopefully you can decipher the essence.)

It seems like the slim chance of all the gas molecules being in the left half of the box would require a simultaneous observation of all those particles? Maybe some of the thermodynamic laws can be restated to say that such an observation is illegal?

That's largely correct - let's stay in the classical realm for a moment; there, observation is not forbidden. But - the energy you would need to expend to observe these guys to know when to shut the box in the middle to preserve them on one side, outweighs the usable energy you could draw out of this in violation of the second law. But it's a very subtle argument.

JC Maxwell and friends already thought about this in detail:

http://en.wikipedia.org/wiki/Maxwell%27s_demon

If the whole universe was just two particles in their probability wave state, then it seems like the entropy of the universe might be the sum of the variances of the two probability waves? If the two probability waves have a single narrow peak, then the entropy would be lower than if the probability waves were more diffuse?

At some point the particles collide or observe each other and their probability waves collapse? Then there are two new probability waves with new entropy values?

Does thermodynamics work with such a simple system?

I think it's problematic. Maybe they will approach something like a thermal energy distribution if you average over very long times. But generally I'd say thermodynamics doesn't really apply.