= Gaussian limit of the end-to-end distribution of a freely jointed chain
{c}
{title2=$\Phi\sim(3/(2\pi Nb^2))^{3/2}e^{-3R^2/(2Nb^2)}$}
For an isotropic fixed-length step $b$, angular integration gives the <characteristic function> $\widehat\psi(\mathbf k)=\sin(kb)/(kb)$. Independence makes the <characteristic function> of the sum of $N$ steps its $N$th power. Since $\ln[\sin(kb)/(kb)]=-k^2b^2/6+O(k^4b^4)$, the central scaling $k=O((b\sqrt N)^{-1})$ gives $\widehat\Phi\to\exp[-Nb^2k^2/6]$. The inverse <Fourier transform> is
$$
\Phi(\mathbf R,N)\sim\left(\frac3{2\pi Nb^2}\right)^{3/2}\exp\left[-\frac{3R^2}{2Nb^2}\right].
$$
This is a density with respect to three-dimensional vector volume, not a radial density; the radial density is $4\pi R^2\Phi$. It has $\langle R^2\rangle=Nb^2$. The approximation holds on the central coil scale $R=O(b\sqrt N)$, not near the fully stretched finite-length boundary $R=Nb$.
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