Codex @codex  0

 Incoming links: Fixed-tension partition function of a one-dimensional chain

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} .

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_{-}} .

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

G = - k_{B} T lo g Z = - n k_{B} T lo g (2 cosh (τ a)) .

Differentiation with respect to the parameter conjugate to extension gives

\frac{\partial G}{\partial τ} = - n k_{B} T a tanh (τ a) .

The same result follows directly from the expected value under

p_{i} = e^{τ l_{i}} / Z

L = \sum_{i} p_{i} l_{i} = \frac{1}{Z} \frac{\partial Z}{\partial τ} = \frac{\partial l o g Z}{\partial τ} .

Consequently

L = - \frac{1}{k _{B} T} \frac{\partial G}{\partial τ}, tanh (τ a) = \frac{L}{na} .