### 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 of harmonic bending energy:

$$U_b(\theta)=-\frac{1}{2}k \theta^2$$

where $k$ is the stiffness constant in unit of $k_BT$, the $\langle\cos(\theta)\rangle$ is simply

$$\langle\cos(\theta)\rangle=\frac{\operatorname{erf}\left(\frac{\sqrt{2} {\left(\pi k + 2 i\right)}}{2 \, \sqrt{k}}\right) e^{\left(\frac{1}{2 \, k}\right)} - \operatorname{erf}\left(\frac{\sqrt{2} {\left(\pi k - 2 i\right)}}{2 \, \sqrt{k}}\right) e^{\left(\frac{1}{2 \, k}\right)} - 2 \, \operatorname{erf}\left(\frac{i \, \sqrt{2}}{\sqrt{k}}\right) e^{\left(\frac{1}{2 \, k}\right)}}{2 \, {\left(\operatorname{erf}\left(\frac{\sqrt{2} {\left(\pi k + i\right)}}{2 \, \sqrt{k}}\right) e^{\frac{2}{k}} - \operatorname{erf}\left(\frac{\sqrt{2} {\left(\pi k - i\right)}}{2 \, \sqrt{k}}\right) e^{\frac{2}{k}} - 2 \, \operatorname{erf}\left(\frac{i \, \sqrt{2}}{2 \, \sqrt{k}}\right) e^{\frac{2}{k}}\right)}}$$

For large $k$ ($k\ge 6$), $\langle\cos(\theta)\rangle\simeq L(k)$, with $L(k)$ being the Langevin function.