(a) Show that the partition function of an Ising lattice can be written as

\[
Q_{N}(B, T)=\sum_{N_{+}, N_{+-}}^{\prime} g_{N}\left(N_{+}, N_{+-}\right) \exp \left\{-\beta H_{N}\left(N_{+}, N_{+-}\right)\right\},
\]

where

\[
H_{N}\left(N_{+}, N_{+-}\right)=-J\left(\frac{1}{2} q N-2 N_{+-}\right)-\mu B\left(2 N_{+}-N\right)
\]

while other symbols have their usual meanings; compare these results to equations (12.3.19) and (12.3.20).

(b) Next, determine the combinatorial factor \(g_{N}\left(N_{+}, N_{+-}\right)\)for an Ising chain \((q=2)\) and show that, asymptotically,

\[
\begin{aligned}
\ln g_{N}\left(N_{+}, N_{+-}\right) \approx & N_{+} \ln N_{+}+\left(N-N_{+}\right) \ln \left(N-N_{+}\right) \\
& -\left(N_{+}-\frac{1}{2} N_{+-}\right) \ln \left(N_{+}-\frac{1}{2} N_{+-}\right) \\
& -\left(N-N_{+}-\frac{1}{2} N_{+-}\right) \ln \left(N-N_{+}-\frac{1}{2} N_{+-}\right) \\
& -2\left(\frac{1}{2} N_{+-}\right) \ln \left(\frac{1}{2} N_{+-}\right) .
\end{aligned}
\]

Now, assuming that \(\ln Q_{N} \approx\) (the logarithm of the largest term in the sum \(\sum^{\prime}\) ), evaluate the Helmholtz free energy \(A(B, T)\) of the system and show that this leads to precisely the same results as the ones quoted in the preceding problem as well as the ones obtained in Section 13.2.