Worm-like chain

Example

The worm-like chain, or Kratky–Porod model, describes a semiflexible polymer as a continuous curve $r \colon [0, L] \to \R^3$ parametrized by arc length. Its unit tangent is $t(s) = r'(s)$, with $\abs{t(s)} = 1$. The bending energy of a shape is $$E = \frac{\kappa}{2} \int_0^L \abs{t'(s)}^2 \, ds,$$ where $\kappa > 0$ is the bending stiffness. At temperature $T$ the shapes are weighted by the Boltzmann distribution $e^{-E/(k_B T)}$. The persistence length is $P = \kappa / (k_B T)$. For this ensemble, tangent correlations decay exponentially, $\langle t(s) \cdot t(s') \rangle = e^{-\abs{s - s'}/P}$. Check of the mean square end-to-end distance: since $R = r(L) - r(0) = \int_0^L t(s) \, ds$, $$\langle \abs{R}^2 \rangle = \int_0^L \int_0^L e^{-\abs{s - s'}/P} \, ds \, ds' = 2 P L - 2 P^2 \big(1 - e^{-L/P}\big).$$