[NOTES/SM-04018] Boltzmann Entropy from Canonical Partition Function

Mean energy
The internal energy, or the mean energy, of an ideal gas is given by
\begin{equation}
U = -\frac{\partial \log Z}{\partial \beta} = 3N\frac{kT}{2}.
\end{equation}

Entropy
The entropy is given by
\begin{equation}
S -k \ln Z + \beta U. \label{EQ02}
\end{equation}
where \(U\) is the mean energy. For macroscopic systems the degeneracy \(\Omega(E)\) of energy levels \(E\) is sharply peaked around the mean energy \(\bar E\). Therefore we replace \(U\) with \(\bar E\). Also only one term \(E=\bar E\) in the sum in \EqRef{EQ01} over all levels contributes to the partition function giving
\begin{equation}
Z =k\log \Omega(\bar E)-k \beta \bar E
\end{equation}

The expression \eqref{EQ02} for the entropy, therefore, reduces to the Boltzmann expression

\begin{equation}
S = k \log \Omega(\bar E).
\end{equation}