### Bending energy and persistence length

Persistence length $L_p$ is a basic mechanical property quantifying the bending stiffness of a polymer and is defined as the relaxation of $\langle\cos(\theta)\rangle$ of bond angles of the polymer chain: $$ \langle\cos(\theta(s))\rangle = \exp(-sl/L_p) $$ The problem of calculating $L_p$ becomes calculating $\langle\cos(\theta(s))\rangle$, if we choose the $\theta$ as the angle of adjacent bonds, i.e., $s=1$, we have: $$ \langle\cos(\theta)\rangle = \exp(-l/L_p) $$ in simulations, the stiffness constant is given by a bending energy $U_b$, when $U_b$ is large, where excluded volume effect is negligible, we can calculate $\langle\cos(\theta)\rangle$ as $$\langle\cos(\theta)\rangle=\frac{\int_0^\pi \cos(\theta)\sin(\theta)\exp(-\beta U_b)\mathrm{d}\theta}{\int_0^\pi\sin(\theta)\exp(-\beta U_b)\mathrm{d}\theta}$$ the $\sin(\theta)$ is a geometric weight that when 2 bonds are at an angle of $\theta$, then in 3-D space, the number of bonds is proportion to $\sin(\theta)$. Here is an example