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I'm working through Kittel's Thermal Physics, 2nd Edition and am stuck on part b of the 7th problem, chapter 3. I know this is a physics question, but my issue is mostly mathematical.

"A zipper has N links; each link has a state in which it is closed with energy 0, and a state in which it is open with energy $ \epsilon $. We require, however, that the zipper can only unzip from the left end, and that the link number s can only open if all links to the left (1,2,...,s-1) are already open."

I've completed part a) by showing that the partition function can be summed in the form: $$ Z =\frac {1 -e^{-\frac{\epsilon}{\tau}(N+1)}}{1-e^{-\frac{\epsilon}{\tau}}} $$

Part b) is asking to find the average number of links open, in the limit $\epsilon >> \tau$.

My issue is simplifying $\log{Z}$.

$$ \log{Z}=\log({\frac {1 -e^{-\frac{\epsilon}{\tau}(N+1)}}{1-e^{-\frac{\epsilon}{\tau}}}})=\log({1 -e^{-\frac{\epsilon}{\tau}(N+1)}})-\log({1-e^{-\frac{\epsilon}{\tau}}})$$

I'm not quite sure how to simplify these logs at all using $\epsilon >>\tau$. Besides noting that this means $e^{-\epsilon/\tau}<<1.$ Wouldn't this just make $\log{Z} = 0 $ ? The book is saying it should be $\log{Z}=e^{-\epsilon/\tau}$ though.

Much appreciated.

1 Answers 1

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Since $\alpha =e^{-\epsilon/\tau}<<1$ we can approximately write the logarithms as $log{(1-{\alpha}^{N+1}}) - log(1-{\alpha}) \approx - {\alpha}^{N+1} + \alpha$. If also $N>>1$ we can neglect first term $-{\alpha}^{N+1}$ and get $\log{Z}=e^{-\epsilon/\tau}$

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    How exactly does the first approximation work? I'm not understanding that. – 2017-06-22
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    What is the "first approximation"? This is simply the first term in the expansion of the logarithm in the Taylor series with respect to the small parameter $\alpha$ near unity – 2017-06-22