Consider a linear chain of \(N\) atoms, with atom \(N\) adjacent to atom 1 (cyclic
boundary condition), as if the chain were bent into a circle. Each atom can be
in any of three states, called \(A, B\), and \(C\), except that an atom in state
\(A\) cannot be adjacent to an atom in state \(C\). Find the entropy per atom of
such a chain as \(N \rightarrow \infty\). (Entropy \(=\) \(k \ln W\), where \(W\) is
the total number of allowed configurations.) Hint: Define the three-
dimensional column vectors \(v^{(j)}\) such that the three elements of \(v^{(j)}\)
are the total number of allowed configurations of the \(j\)-atom chain \(1,2,3,
\ldots, j\), with atom \(j\) in state \(A, B\), or \(C\), respectively. For example,
$$
v^{(1)}-\left(\begin{array}{l}
1 \\
1 \\
1
\end{array}\right), \quad t^{(2)}-\left(\begin{array}{l}
2 \\
3 \\
2
\end{array}\right), \quad v^{(3)}-\left(\begin{array}{l}
5 \\
7 \\
5
\end{array}\right), \ldots
$$
Clearly, \(v^{(s+1)}=M v^{(j)}\), where
$$
M=\left(\begin{array}{lll}
1 & 1 & 0 \\
1 & 1 & 1 \\
0 & 1 & 1
\end{array}\right)
$$
is the so-called transfer matrlx appropriate to this problem. 'Then
$$
W=\operatorname{tr} M^{N} \approx(\text { largest eigenvalue of } M)^{N}
$$
Show thereby that the entropy per atom is \(k \ln (1+\sqrt{2}), \quad\) Suppose
atom \(N\) is not adjacent to atom 1. What is the entropy per atom?
Suppose the constraint forbidding an atom in state \(A\) to be adjacent to an
atom in state \(\mathrm{C}\) is removed. Now what is the entropy per atom?
