Entropy of indistinguishable particles $S$ is

$$S = k\ln W,$$

but in some places notation is different:

$$S = k\ln Q + \frac{U}{T}$$

Can you explain, please?


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The only time I've ever seen that second notation is when referring to the substitution of the Helmholtz energy, as A = U-TS and therefore A = -kTlnQ. If you want an in depth explanation of how the thing is derived I suggest looking at Chapter 10 of Biomolecular Thermodynamics: From Theory to Application by Douglas E. Barrick.

Here are some photos of the key parts:

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The second formula you quote follows directly from the first. Starting with $W=N!/(n_1!n_2!\cdots)$ for $N$ distinguishable particles spread over set of $i$ energy levels with energy $\epsilon_i$, then $\ln(W)=\ln(N!)-\sum\ln(n_1!)$. Using the Sterling formula for factorials the entropy becomes, after some fiddling around and simplifying, $\displaystyle S=-Nk_B\sum \frac{n_i}{N}\ln\left(\frac{n_i}{N} \right)=-Nk_B\sum p_i\ln(p_i)$ where $p_i=n_i/N$ is the chance that the $i^{th}$ state is occupied. This fraction is also $p_i=e^{-\epsilon_i/k_BT}/Z$ where $Z$ is the partition function and $Z=\sum_ie^{-\epsilon_i/k_BT}$ then $\displaystyle \ln(p_i)=-\frac{\epsilon_i}{k_BT}-\ln(Z)$ and substituting gives

$$S=-Nk_B \left(-\frac{1}{k_BT}\sum(p_i\epsilon_i)-\ln(Z\sum p_i)\right)$$

which becomes $\displaystyle S=\frac{U}{T}+Nk_B\ln(Z)$

after evaluating the terms $\displaystyle \sum p_i=1$, the probability over all states is one and the total energy $\sum p_i\epsilon_i $ is $\displaystyle \sum p_i\epsilon_i=\frac{U}{N}$ where $U$ is the internal energy.

This approach is only valid for solids because in some way the atoms in a solid can be identified. In a gas this is not possible and the number of microstates must be divided by $N!$ because after exchanging atoms/molecules the state of the gas still it indistinguishable. For a perfect gas it is found that $\displaystyle S=\frac{U}{T}+Nk_B\ln(Z) -Nk_B(\ln(Z)-1)$


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