= Force-extension of a three-dimensional freely jointed chain
{title2=$L=Nb\mathscr L(Fb/(k_BT))$}
Independent link orientations make the <canonical partition function> $Z_1^N$, with the one-link integral defining the <Langevin function>. Differentiating its logarithm with respect to force gives $L=Nb\mathscr L(Fb/(k_BT))$. At high extension, $Nb-L\sim Nk_BT/F$, so the force diverges inversely with remaining length deficit. This requires rigid links and excludes axial bond stretching.
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