Gaussian chain ($⟨e^{i\mathbf{k}\cdot ({\mathbf{R}}_n-{\mathbf{R}}_m)}⟩=e^{-k^2{b}^2|n-m|/6}$)

An ideal Gaussian chain is a polymer model whose independent segment vectors have zero mean and isotropic Gaussian distribution. If $⟨|\mathbf{r}{|}^2⟩=b^2$, each Cartesian variance is $b^2/3$, so a displacement across $\ell$ links has density $(3/(2\pi \ell b^2){)}^{3/2}\exp [-3R^2/(2\ell b^2)]$ and characteristic function $\exp [-k^2{b}^2\ell /6]$. The continuous-contour version uses the same characteristic function for any contour separation. It is the large-scale limit of a freely jointed chain, while allowing unbounded Gaussian segment lengths and omitting interactions between distant parts of the chain.

A continuum Gaussian chain has separation characteristic function $e^{-a|s-t|}$, $a=k^2{b}^2/6$. Integrating over a contour of length $N$ gives the polymer scattering functionwhere $R_g^2=N{b}^2/6$ in this continuum convention and $\mathcal{D}(x)=2(e^{-x}-1+x)/x^2$, with continuous value $\mathcal{D}(0)=1$. Its limits are $\mathcal{D}(x)=1-x/3+x^2/12+\cdots$ and $\mathcal{D}(x)\sim 2/x$ for large $x$.

For a discrete chain with $N$ sites, retain the self terms and let $z=e^{-k^2{b}^2/6}$. The exact finite expression is $g_N=1+(2/N){\sum }_{\ell =1}^{N-1}(N-\ell )z^{\ell }=(1+z)/(1-z)-2z(1-z^N)/[N(1-z{)}^2]$, with $g_N(0)=N$. Taking $N\to \infty$ while $x=N{k}^2{b}^2/6$ stays fixed gives $g_N/N\to \mathcal{D}(x)$. The continuum expression describes the coil scaling range, whereas the discrete expression tends to one as $kb\to \infty$.