= One-dimensional entropic chain force-extension relation
{title2=$\mathcal F=(k_BT/a)\operatorname{artanh}(L/(Na))$}
For $N$ independent links pointing left or right with length $a$, fixed signed extension $L$ gives $N_R=(N+L/a)/2$ and $N_L=(N-L/a)/2$. The multiplicity is $\binom N{N_L}$ when these counts are nonnegative integers. Leading <Stirling's approximation> gives $\mathcal F=(k_BT/(2a))\ln(N_R/N_L)$ as the externally applied <tension> conjugate to $L$. Its small-extension limit is $k_BT L/(Na^2)$. At fixed tension, the exact one-link <partition function> is $2\cosh(\beta\mathcal Fa)$ and the mean extension is $Na\tanh(\beta\mathcal Fa)$, confirming the same inverse relation without using Stirling at the full-extension endpoint.
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