= End-to-end distance of a worm-like chain
{title2=$\langle R^2\rangle=2\ell_pL-2\ell_p^2(1-e^{-L/\ell_p})$}
For a free three-dimensional <worm-like chain> with contour length $L$ and <persistence length> $\ell_p$, the unit-tangent correlation is $\langle\mathbf t(s)\cdot\mathbf t(s')\rangle=e^{-|s-s'|/\ell_p}$. Since $\mathbf R=\int_0^L\mathbf t(s)\,ds$, integrating this correlation gives
$$
\langle R^2\rangle=2\int_0^L(L-u)e^{-u/\ell_p}\,du
=2\ell_pL-2\ell_p^2(1-e^{-L/\ell_p}).
$$
For $L\gg\ell_p$, the leading term is $2\ell_pL$, the <ideal chain> law with <Kuhn length> $2\ell_p$. For $L\ll\ell_p$, a <Taylor series> gives $\langle R^2\rangle=L^2-L^3/(3\ell_p)+O(L^4/\ell_p^2)$, approaching a rigid rod. This ideal semiflexible-chain model omits long-range <excluded volume>; sufficiently long dilute chains in a <good solvent> can cross over to swollen-chain statistics.
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