[karpathy]

What I never fully understood is that there is some implicit assumption about the dynamics of the system. So what that there are more microstates of some macrostate as far as counting is concerned? We also have to make assumptions about the dynamics, and in particular about some property that encourages mixing.

[tomnicholas1]

Yes, that assumption is called the Ergodic Hypothesis, and generally justified in undergraduate statistical mechanics courses by proving and appealing to Liouville's theorem.

[1] https://en.wikipedia.org/wiki/Ergodic_hypothesis

[vitus]

It's worth noting that there's more than just ergodicity at play, although that's a fundamental requirement. For instance, applying the Pauli Exclusion Principle gives rise to Fermi-Dirac statistics.

[oh_my_goodness]

In equilibrium we don't have to make an assumption about the dynamics or the mixing. We just expect to see the most probable state when we measure.

It's interesting to try to show that the time average equals the ensemble average. It's very cool to think about the dynamics. That stuff must be happening. But those extra ideas aren't necessary for applying the equilibrium theory.