Does random have rules?

[Anomalocaris]

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. )

what does scale up mean? Solving all the correct probabilistic wave function would result in 2nd law of thermal dynamics as a natural outcome?

That seem to me to imply probability waves are non-local and/or systematically evolve by themselves over time.