 Rigid-motion-projected thermal covariance of a free filament

Free bending has two zero-energy filament modes, so it does not define an unrestricted normalizable thermal height distribution. Fix the mean displacement and linear tilt, or project out that affine kernel. On the positive complement, the equipartition theorem gives independent Gaussian mode variances $k_{B}T/(A{k}_n^4)$ for normalized modes. The resulting covariance is the bending operator pseudoinverse multiplied by $k_{B}T$. It depends on how the rigid motion is fixed.