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I'm confused on how the index in the partition function represents the microstate

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In the derivation, we calculated the number of microstates

$ \Omega = \frac{N!}{\Pi_{i} n_{i}!}$

and I think this index $i$ represents the number of energy level

But in the derivation, it turned out we had to maximize

$ - \sum_{i} N P_{i} log(P_{i})$

Which I believe throughout the derivation, the index $i$ is the same, though it makes sense it should represent the number of microstates, but that last expression came from $ \Omega$ , so how can both $i$'s be the number of energy levels and the number of microstates at the same time? I'm confused.

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Your first equation should be a sum over all microstates. If $i$ represents the microstate then the equation is fine. If $i$ represents the energies, then that form is assuming there is no degeneracy in the energy for different microstates. In general, if there are $n_i$ number of microstates with energy $E_i$, then your partition function will be $$Z=\sum_in_i\,e^{-\beta E_i}$$ where here $i$ is over all energies.

However, you have defined $i$ to be for the microstates, so there isn't any issue here. I guess the confusion would then be relieved by realizing that $E_i$ means "energy of microstate $i$" rather than "energy level $i$".

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  • $\begingroup$ So, I kind of got it, but what if the degeneracy $n_{i}$ is equal to $1$ and then $Z$ becomes $Z=\sum_in_i\,e^{-\beta E_i} = Z=\sum_i\,e^{-\beta E_i}$ where $i$ in this context is over the energy levels, how can I differentiate between both forms now? The $Z=\sum_i\,e^{-\beta E_i}$ with $i$ representing microstates is the same as the $Z=\sum_in_i\,e^{-\beta E_i}$ ( $n_{i} = 1)$ with $i$ representing energy levels $\endgroup$ – khaled014z Nov 8 at 18:30
  • $\begingroup$ @khaled014z If $n_i=1$ for all $i$, then they are the same form. In that case summing over all energies or all microstates will give the same results, as you have a one-to-one correspondence between microstates and energy levels. $\endgroup$ – Aaron Stevens Nov 8 at 18:31

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