Planar freely jointed chain
= Planar freely jointed chain
{title2=$L/(Nb)=I_1(\xi)/I_0(\xi)$}
If each link is restricted to a plane, $Z_1=\int_0^{2\pi}e^{\xi\cos\theta}d\theta=2\pi I_0(\xi)$, with $\xi=Fb/(k_BT)$. Differentiating gives $L/(Nb)=I_1(\xi)/I_0(\xi)$, where $I_n$ are <Modified Bessel functions of the first kind>. The high-force angular Gaussian has $\langle\theta^2\rangle\sim1/\xi$, so $Nb-L\sim Nk_BT/(2F)$, half the three-dimensional coefficient.