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CAF123
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Homework Statement
Consider a zipper of N links, each of which can either be open or closed with associated energy 0 if closed and ##\epsilon## if open.
a) Suppose the N links are independent, compute the partition function of the system and the average number of open links
b)Now assume that the zipper opens up from one of its ends. To open the ith link, all links between i+1 and N therefore have to be open. Write down the partition function for a zipper with N links.
Homework Equations
boltzmann distribution
The Attempt at a Solution
a) For single link, ##Z = e^{-\beta \epsilon} + 1## and since links independent, ##Z_N = (1+e^{-\beta \epsilon})^N##. Then
$$\langle n \rangle = \frac{1}{Z_N} \sum_{i=1}^N n_i P(n_i) = \frac{N e^{-\beta \epsilon}}{(1+e^{-\beta \epsilon})^N}$$ is it ok?
b)Just looking for a hint on how to go about starting this part, it says the geometric summation may come in helpful. Sequences are like, say for N=4 links, 0000, 0001, 0011,0111 or 1111. Could write the hamiltonian like ##\mathcal H = \sum_{j=i}^{N} \epsilon## where ##j \geq i## are open and ##i \leq N##.
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