= End-to-end distance of a polymer chain
{title2=$R_{\mathrm{rms}}=\sqrt{\langle|\mathbf r_N-\mathbf r_0|^2\rangle}$}
The end-to-end vector joins the two ends of a <polymer> chain. In an isotropic unforced <polymer solution>, its vector <expected value> is zero; a useful measure of chain size is its root-mean-square length. For an <ideal chain> with $N$ independent isotropic steps of length $b$, the cross terms in the square of the step sum have zero <expected value>, so $R_{\mathrm{rms}}^2=Nb^2$. For a long <Gaussian chain>, its <radius of gyration> obeys $R_g^2=R_{\mathrm{rms}}^2/6$. If $N$ counts chemical <repeat units> instead of independent <Kuhn length> segments, the statistical length used in $R_{\mathrm{rms}}^2=Nb^2$ must be calibrated to that choice of $N$; it is not automatically the physical bond length or the <Kuhn length> itself.
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