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\sect{Ising model of adsorption sites}

Consider a ring along which $M$ absorption sites are arranged.
The number of particles that can be absorbed at site~$i$ is $n_{i}=0,1$.
Between every two absorption sites a spin $\sigma_{i}=\pm1$ is located.
The ring is surrounded by gas in temperature $T$ and chemical potential~$\mu$.
The absorption energy is ${\epsilon>0}$ if the two adjacent spins are 
in the same direction, and~$-\epsilon$ otherwise.

\begin{enumerate}

\item Write an expression for the energy $E[\sigma_{i},n_{i}]$ of a given configuration.

\item Calculate the partition function $\mathcal{Z}(\beta,\mu)$ 
using the transfer matrix method. Write what is $T_{\sigma_{i},\sigma_{i+1}}$ 
in this problem.

\item Find the Helmholtz function $F(T,\mu)$ assuming ${M\gg1}$.

\item Write an expression for the average number of adsorbed particles ${N = \sum_i \langle n_{i} \rangle}$
as a function of $(\beta,\mu)$.

\item Write an expression for the correlation length $\xi$ that characterizes
arrangement of the spins in the system.

\end{enumerate}