\sect{Baruch's C24.}
Consider a one dimensional Ising model of spins $\sigma_i=\pm
1,\, i=1,2,3,...,N$ and $\sigma_{N+1}= \sigma_1$. Between each
two spins there is a site for an additional atom, which if present
changes the coupling $J$ to $J(1-\lambda)$. The Hamiltonian is
then \[H = -J \sum_{i=1}^N \sigma_i \sigma_{i+1}(1-\lambda n_i)\]
where $n_i=0$ or $1$ and there are $N' = \sum_{i=1}^N n_i$ atoms ($N' <
N$), i.e. on average $\langle n_i\rangle =N'/N$.
\begin{itemize}
\item[(a)] Evaluate the partition sum by allowing all configurations of
spins and of atoms.
\item[(b)] If the atoms are stationary
impurities one needs to evaluate the free energy F for some random
configuration of the atoms and then average F over all
configurations. (The reasons for this average are given in Ex.
C25). Evaluate the average F. Find the entropy difference of (a)
and (b) and explain its origin.\\
\end{itemize}
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