Fixed-tension partition function of a one-dimensional chain ( $Z = (2 cosh (τ a))^{n}$ )

Weighting each configuration by

e^{τ l}

gives

Z = (e^{τ a} + e^{- τ a})^{n} = (2 cosh (τ a))^{n}, L = \partial_{τ} lo g Z = na tanh (τ a) .

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- Entropic Hooke law for a one-dimensional chain Fixed-tension partition function of a one-dimensional chain

Entropic Hooke law for a one-dimensional chain

The physical tension is

f = k_{B} T τ

. At small extension,

f \sim \frac{k _{B} T}{n a ^{2}} L .

At fixed tension the extension is

L = na tanh (f a / (k_{B} T))

and decreases as temperature rises.

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

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