Past exam of the mathematics course of the University of Cambridge / 2023 / ii / Paper 2 / 37A / c

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- Solution c

Solution

Several microstates can have the same extension. If

g (l)

denotes their state degeneracy, grouping equal terms in the state sum gives

Z = \sum_{i} e^{τ l_{i}} = l \sum g (l) e^{τ l} .

For the one-dimensional freely jointed chain, a macrostate with

n_{+}

positive links and

n_{-} = n - n_{+}

negative links has

l = (n_{+} - n_{-}) a, g (l) = \frac{n !}{n _{+} ! n _{-} !} .

Hence

Z = \sum_{n_{+} = 0}^{n} \frac{n !}{n _{+} ! n _{-} !} (e^{τ a})^{n_{+}} (e^{- τ a})^{n_{-}} .

The binomial theorem now gives the fixed-tension partition function of a one-dimensional chain

Z = (e^{τ a} + e^{- τ a})^{n} = (2 cosh (τ a))^{n} .

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