Does random have rules?

[watchamadoodle]

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.

Hmmmm. I wonder if we defined a property of probability waves (similar to variance) and call that "QM entropy". Maybe we could tinker with the definition of "QM entropy" until thermodynamics works at the QM level with a small number of particles. Then wouldn't it be neat if we could prove mathematically that this "QM entropy" scales-up to the entropy in steam engines and so forth?
(I like to day dream. )