[NOTES/SM-06005] Entropy and Energy in Terms of Grand Canonical Partition Function

The grand canonical partition function is is s function of \(V,T, \mu\). For convenience, this dependence will be understood and not shown explicitly.
Entropy
To find entropy we use
\begin{eqnarray}
S &=&-k\sum_{N=0}^\infty \sum_{E_k} P(E_k,N) \ln P(E_k,N)\Label{EQ09}\\
&=& - k\sum_{N=0}^\infty \sum_{E_k} P(E_k,N) \big(- \ln \Zca +\beta(\mu N -E)
\big)\Label{EQ10}
\end{eqnarray}
Thus we have
\begin{eqnarray}
\boxed{S = k[\ln \Zca + \beta \bar{U} - \beta \mu \bar{N}]}.\Label{EQ11}
\end{eqnarray}

Mean Energy and Mean Number
We will relate different thermodynamic functions to the derivatives of logarithm of the partition function $\Zca$ with respect to $\beta$ and $\mu$. These derivatives are easily computed. Note the definitions of $\bar{E} and \bar{N}$ are, as usual, as follows
\begin{eqnarray}
\bar{N} = \sum_{N=0}^\infty\sum_{E_k} N P(E_k, N); \qquad
\bar{U} = \sum_{N=0}^\infty\sum_{E_k} E_k P(E_k,N)
\end{eqnarray}

\begin{eqnarray}
\dd[\Zca]{\mu}
&=& \sum_{N=0}^\infty\sum_{E_k} \beta Ne^{\beta(\mu N -E_k)} = \beta
\sum_{N=0}^\infty\sum_{E_k} Ne^{\beta(\mu N -E_k)}\Label{EQ03}\\
&=& \Zca \beta\bar{N} \Label{EQ04B}\\
\dd[\Zca]{\beta}
&=& \sum_{N=0}^\infty\sum_{E_k} (\mu N - E_k)e^{\beta(\mu N -E_k)},\Label{EQ04}
\\
&=& \Zca(\mu \bar{N} - \bar{E}) =\Zca (\mu \bar{N} -U) .\Label{EQ05}
\end{eqnarray}
Hence we have
\begin{eqnarray}
\bar{N} &=& \frac{1}{\beta}\frac{1}{\Zca}\dd[\Zca]{\mu}= kT \Big(\dd[\ln
\Zca]{\mu} \Big)\Label{EQ06}\\
\mu \bar{N} -\bar{U}
&=& \frac{1}{\Zca}\dd[\Zca]{\beta} = \dd[\ln \Zca]{\beta}\Label{EQ07}
\end{eqnarray}
Thus the average internal energy $\bar{U}$ is given by
\begin{eqnarray}
\qquad \bar{U}
&=& \mu \bar{N} -\dd{\beta}(\ln \Zca) = kT \mu \dd{\mu}(\ln \Zca)
-\dd{\beta}(\ln \Zca).\Label{EQ08}
\end{eqnarray}