# Partition function index i

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

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 $$i$$".
• 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 – khaled014z Nov 8 at 18:30
• @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. – Aaron Stevens Nov 8 at 18:31