Transverse tangent correlation of a stretched worm-like chain ($C(r)=(k_{B}T/\sqrt{Af})e^{-|r|/\xi },\xi =\sqrt{A/f}$)

The bulk tangent covariance is twice the inverse Fourier transform of $k_{B}T/(A{q}^2+f)$, one term for each transverse direction. Equivalently the inverse of $-A{\partial }_r^2+f$ is $e^{-|r|/\xi }/(2\sqrt{Af})$. This gives the displayed correlation and identifies a force-dependent correlation length, different from the zero-force persistence length. Finite chains require endpoint conditions or a periodic convention.

Periodizing the bulk transverse tangent correlation of a stretched worm-like chain gives $C_L(r)={\sum }_{j\in \mathbb{Z}}C(r+jL)$, where $d_L(r)$ is the distance to zero on the periodic interval. Summing the geometric series yields the displayed result. At zero separation its finite-length factor is $\coth (L/(2\xi ))$; for $L\gg \xi$ the bulk result is recovered. Periodic tangent fluctuations are a mathematical boundary convention, not a claim that a pulled physical chain forms a closed loop.