Canonical Probabilities by Directly Quantizing Thermodynamics

I’ve had this derivation kicking around a while, and today seemed like as good a day as any to make a fuller write-up of it:

  • B. C. Stacey, “Canonical probabilities by directly quantizing thermodynamics” (PDF).

The idea is that Boltzmann’s rule $p(E_n) \propto e^{-E_n / k_B T}$ pops up really naturally when you ask for a rule that plays nicely with the composing-together of uncorrelated systems. This, in turn, gives a convenient expression to the idea that classical physics is what you get when you handle quantum systems sloppily.