Transverse-fluctuation calibration of a tensioned tether

For a tether with extension $\ell$ and tensile load $F$, a small transverse coordinate costs $F{x}^2/(2\ell )$. Equipartition gives $F=k_{B}T\ell /⟨x^2⟩$ for one Cartesian transverse coordinate. Using the sum of two transverse variances changes the factor of two. Geometry and measurement filtering must match the calibration model.

For an equilibrium overdamped harmonic coordinate with covariance $⟨x(t)x(0)⟩=V{e}^{-|t|/\tau }$, a camera exposure $\Delta$ records the average over that interval. Integrating the covariance twice gives $\mathrm{Var}(\bar{x})=2V{\Delta }^{-2}{\int }_0^{\Delta }(\Delta -u)e^{-u/\tau }du$. The displayed reduction follows with $s=\Delta /\tau$. In the tether calibration $F=k_{B}Th/V$, ignoring exposure averaging overestimates the inferred force. Independent localization-noise variance instead adds to the measured variance and biases it in the opposite direction. Other filters require their actual response functions.