Before asking the question, I should clarify that I am a college student majoring in mathematics and CS, so my understanding on physics is quite shaky. So please understand if this question turns out to be ill-defined.
I was reading a book on quantum mechanics and entropy. In the book, the author says that by the second law of thermodynamics, microstates of a system evolve to produce maximum Boltzmann entropy. Consequently, it turns out that the microstates eventually form an ensemble of Boltzmann distribution (aka canonical ensemble), i.e.
$$P_n = \frac{1}{Z} e^{-\epsilon_n/k_B T}$$
The author then notes that the only information contained in the above distribution is the energy of the states (since temperature is also determined by energy). This is paradoxical, as it seems to imply that all the other information about the initial state has been lost.
To explain how this is possible, the author introduces ergodic hypothesis. And to explain how ergodic hypothesis can hold, the author employs chaos theory and Heisenberg's uncertainty principle. So to summarize:
- Boltzmann distribution is only dependent on the energy of the states.
- Such information loss can be explained by chaos theory and uncertainty principle.
- Thus, quantum mechanics and chaos theory is what drives the second law of thermodynamics.
But I can't agree with this conclusion, since I'm pretty sure that statistical thermodynamics is independent from quantum thermodynamics. Already I have found many comments on PSE saying that the second law of thermodynamics doesn't prove QM. On the other hand, the author is a working theoretical physicist and I don't think the book contains any errors. So in the end, what is the relationship between Boltzmann distribution and the uncertainty principle?