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A one dimensional chain is made up of N identical elements, each of length l. The angle between successive elements can be either 0◦ or 180◦ , but there is no difference in internal energy between these two possibilities. For the sake of counting, one can think of each element as either pointing to the right (+) or to the left (-). Then one has N = n+ + n− and the length of the chain L = l(n+ − n−), where n+ is the number of elements pointing to right and n− is the number of elements pointing to left. (i) Use the microcanonical ensemble to find the entropy S as a function of N and n+ (ii) Find an expression for the tension F in the chain as a function of T, N, and n+ (iii) Rearrange the expression from (ii) to give the length L as a function of N, T, and F (iv) Use the result for the high temperature behavior from (iii) to find an expression for the thermal expansion coefficient α ≡ 1 L
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