%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%% \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}