= Rigid-motion-projected thermal covariance of a free filament
{title2=$C(x,y)=\frac{k_BT}{A}\sum_{n\geq1}\frac{W_n(x)W_n(y)}{k_n^4}$}
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> $k_BT/(Ak_n^4)$ for normalized modes. The resulting <covariance> is the bending operator pseudoinverse multiplied by $k_BT$. It depends on how the rigid motion is fixed.
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