In Section 6.5 I derived the useful relation $F=-kT\ln \left(Z\right)$between the Helmholtz free energy and the ordinary partition function. Use analogous argument to prove that $\varphi =-kT\times \ln \left(\hat{Z}\right)$, where $\hat{Z}$ is the grand partition function and $\varphi$is the grand free energy introduced in Problem 5.23.

Formula for grand potential is:

$\varphi =U-TS-\mu N$

$\varphi =U-TS-\mu N$

Use thermodynamic equation:

$dU=TdS-PdV+\mu dN$

Equation for an infinitesimal change in grand potential is:

$$\begin{gathered} d\varphi =d(U-TS-\mu N) \\ =dU-d\left(TS\right)-d\left(\mu N\right) \\ =dU-TdS-SdT-\mu dN-Nd\mu \end{gathered}$$

Substitute $dU=TdS-PdV+\mu dN$

localid="1647239388505" $$\begin{gathered} d\varphi =TdS-PdV+\mu dN-TdS-SdT-\mu dN-Nd\mu \\ =-SdT-PdV-Nd\mu \end{gathered}$$

Differentiate equation $d\varphi =-SdT-PdV-Nd\mu$with respect to $\mu$

$\frac{\partial \varphi }{\partial {\mu }_{TV}}=-N$

We differentiate function $\tilde{\varphi }=-kT\times \ln (\widehat{Z)}$ with respect to $\mu$.

${\left(\frac{\partial \tilde{\varphi }}{\partial \mu }\right)}_{TV}=-\frac{kT}{\hat{Z}}\left(\frac{\partial \hat{Z}}{\partial \mu }\right)$

Here, $\hat{Z}$ is grand partition function.

As, average number of particles is given by:

localid="1647240173822" $\overline{N}=\frac{kT}{\hat{Z}}\left(\frac{\partial \hat{Z}}{\partial \mu }\right)$

Substitute $\frac{kT}{\hat{Z}}\left(\frac{\partial \hat{Z}}{\partial \mu }\right)=\overline{N}$ in above differential equation,

${\left(\frac{\partial \tilde{\varphi }}{\partial \mu }\right)}_{TV}=-\overline{N}$

For grand canonical function $\varphi and\tilde{\varphi }$at $\mu =0$we get,

$$\begin{gathered} \tilde{\varphi }=-kT\times \ln \left(Z\right) \\ =F \end{gathered}$$

Here, $F=-kT\times \ln \left(Z\right)$is called Helmholtz free energy.

Substitute $\mu =0$in equation $\varphi =U-TS-\mu N$

$\varphi =U-TS$

As, Helmholtz free energy is $F=U-TS$

Substitute $F=U-TS$in equation $\varphi =U-TS$we get,

$\varphi =F$

So, $\varphi$and $\tilde{\varphi }$at same initial conditions have same values. Therefore, they are the same functions. Hence, the grand canonical function is $\varphi =-kT\times \ln \left(\hat{Z}\right)$.