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6. MW 6-11. Consider a linear chain of N atoms, with atom N adjacent to atom 1 (cyclic boundary conditions), 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 -+ co. (Entropy = k log W, where W is the total number of allowed configurations.) Hint: Define the three-dimensional column vectors v such that the three elements of v) are the total number of allowed configurations of the j-atom chain 1, 2,3, . .., j, with atom j in state A, B, or C, respectively. For example, 1(2) = 10 00 10 1 (3) = Clearly, v(+1) = Mu() where M = 0 1 is the so-called transfer matrix appropriate to this problem. Then W = Tr(MN) ~N-+co (largest eigenvalue of M)N Show thereby that the entropy per atom is blog(1 + v2). Suppose that 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 C is removed. Now what is the entropy per atom