\sect{Baruch's C28.} Potts model in 1-dimension (1d) . A set of N atoms, each with $p$ states is arranged on a 1d chain with periodic boundary conditions. The atom at the $n$-th site is in a state $i_n$ that is chosen from the set $\{1,2,...,p\}$. Two neighboring atoms at sites $n$ and $n+1$, respectively, have an interaction energy $-J$ ($J>0$) if they are in the same state, i.e. $i_n=i_{n+1}$, and 0 interaction otherwise. The Hamiltonian is therefore ${\cal H}= -J\sum_{i=1}^N \delta_{i_n,i_{n+1}}$ where $\delta_{i_n,i_{n+1}}$ is the Kronecker symbol, and the boundary conditions are $i_{N+1}=i_1$. \begin{itemize} \item[(a)] Derive the free energy for $p=2$. (Consider here and below the limit $N\rightarrow \infty$.) \item[(b)] Derive the free energy for a general $p$. Hint: Show that the eigenvector of the transfer matrix whose all entries are equal has the largest eigenvalue. \item[(c)] Find the internal energy $E$ at the low and high temperature limits and interpret the results.\\ \end{itemize} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%