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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 macros
by karpathy 1y ago
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 1y agoYes, 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 https://en.wikipedia.org/wiki/Ergodic_hypothesis
- vitus 1y agoIt'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.
- tomnicholas1 1y agoIsn't that more about enumerating the microstates? The Pauli exclusion principle just ends up forbidding some of the microstates (forbidding a significant fraction of them if you're in the low-temperature regime).
- vitus 1y agoIt is about enumerating the microstates, but in a way that takes into account how the particles interact with each other (aka making assumptions about the dynamics). If we didn't take into account any interactions, we'd be unable to do anything with statistical mechanics beyond rederiving the ideal gas law.
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- oh_my_goodness 1y agoIn 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.