Gaussian limit of the end-to-end distribution of a freely jointed chain ($\Phi \sim (3/(2\pi N{b}^2){)}^{3/2}{e}^{-3R^2/(2N{b}^2)}$)

For an isotropic fixed-length step $b$, angular integration gives the characteristic function $\hat{\psi }(\mathbf{k})=\sin (kb)/(kb)$. Independence makes the characteristic function of the sum of $N$ steps its $N$th power. Since $\ln [\sin (kb)/(kb)]=-k^2{b}^2/6+O(k^4{b}^4)$, the central scaling $k=O((b\sqrt{N}{)}^{-1})$ gives $\hat{\Phi }\to \exp [-N{b}^2{k}^2/6]$. The inverse Fourier transform isThis 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 $⟨R^2⟩=N{b}^2$. The approximation holds on the central coil scale $R=O(b\sqrt{N})$, not near the fully stretched finite-length boundary $R=Nb$.