Persistence Length

#Physics
The length at which a polymer stays essentially straight

$\displaystyle {\left\langle{\vec{t}(s)\cdot \vec{t}(u)}\right\rangle}=e^{-\lvert s-u\rvert / \xi_{p}}$

• $\displaystyle \vec{t}(s)$ is the tangent vector at a distance $\displaystyle s$ along the polymer
• $\displaystyle \xi_{p}$ is the persistence length and is generally $\displaystyle \frac{a}{2}$ where $\displaystyle a$ is the length of one segment

$\displaystyle {\left\langle{R^{2}}\right\rangle}\approx 2\int_{0}^{L} \int_{0}^{\infty} e^{-x /\xi_{p}} , \mathrm{d}x , \mathrm{d}s=2L\xi_{p}$

• $\displaystyle L$ is the contour length
• Applies for when $\displaystyle L\gg \xi_{p}$

$\displaystyle \xi_{p}=\frac{a}{2}$

• $\displaystyle a$ is the length of one segment
• Can be found as $\displaystyle {\left\langle{R}\right\rangle}=Na^{2}=aL=2L\xi_{p}\Rightarrow$ above equation