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I am currently following an introductory class on statistical mechanics. This course starts with general notions of statistics (as a reminder), goes over explaining why one would want a statistical description of the world by taking a look into the classical and quantum cases. Based on these various examples, the course starts rather classically by introducing the micro-canonical ensemble. There, one defines entropy as

$S = -k_BT\log\Omega$

Further down the book students are introduced with the canonical ensemble. Entropy is defined in this new ensemble as

$S = -k_B \Sigma_{n,j} P_{n,j}\log P_{nj}$

The author kickstarts the proof of this statement with the following line:

$S = k_B\log Z + \frac{k_BT}{Z}\left(\frac{\partial Z}{\partial T}\right)_{V,N_1,...,N_C}$

where $c$ is the number of different particle species in the system.

The proof goes on from there as follows:

$S = k_B\log Z + \frac{k_BT}{Z}\Sigma_{n,j}\frac{E_n}{k_BT^2}e^{-\frac{E_n}{k_BT}} = k_B\log Z - k_B\Sigma_{n,j}P_{n,j}\log\left(P_{n,j}Z\right)$

My question lies on the first line of the proof. Why is it valid that the author can kickstart the proof with the line hereabove?

Let me conclude this question by wishing a wonderful 2024 to whomever is reading this. Cheers!

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  • $\begingroup$ I cannot follow the argument. The Gibbs-Shannon entropy is defined as in your second equation. Maximizing it under various constraints gives the well-known probabilities and equations for entropy. BTW: You should always number your equations. Also use $$ environment for in-line math. $\endgroup$
    – Tobias Fünke
    Commented Jan 16 at 20:42
  • $\begingroup$ When "entropy is defined", there is nothing to prove, it's a definition. Actually that formula defines information entropy of a probability distribution. Showing that its value for canonical distribution is or corresponds to thermodynamic entropy can be subject of a proof/disproof. $\endgroup$
    – Ján Lalinský
    Commented Jan 16 at 22:25

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The author may have missed a few lines that define the canonical ensemble. In its essence, the canonical ensemble describes the thermodynamic potential $F = -k_B T lnZ$, and then already uses it to derive the entropy value $S = -\frac{\partial F}{\partial T}$

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Remember that in the canonical ensemble $p_k=\frac{E^{-\beta E_k}}{Z}$ where $Z=\sum_{m} e^{-\beta E_m}$, so the probability as the author defined it for the first time is

$ S=-k_{B}\sum_{k}\frac{E^{-\beta E_k}}{\sum_{m} e^{-\beta E_m}}\log{\frac{E^{-\beta E_k}}{\sum_{m} e^{-\beta E_m}}}=k_{B}\sum_{k}\frac{E^{-\beta E_k}}{\sum_{m} e^{-\beta E_m}}(\log{Z}-\log{e^{-\beta E_k}}) $$

$=k_{B}(\sum_{k}\frac{E^{-\beta E_k}}{Z}\log{Z}-\frac{E^{-\beta E_k}}{Z}\log{e^{-\beta E_k}})=k_{B}\frac{Z}{Z}\log{Z}-k_{B}\sum_{k}p_k E_k=K_B\log{Z}-\beta \langle E\rangle $$

this is more or less the way from probability to thermodinamic entropy

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