= Curvature-noise representation of a free filament
{title2=$\langle\xi(s)\xi(t)\rangle=(k_BT/A)\delta(s-t)$}
The quadratic bending energy makes curvature a <white noise> field in the generalized thermal description. Integrate it twice, then subtract the affine least-squares projection to fix translation and tilt. With $u=x/L$ and $t=s/L$, the dimensionless kernel is $(u-t)_+-(1-t)^2/2-(1-3t^2+2t^3)(u-1/2)$. Its second spatial derivative is the source delta and its mean and first moment are zero. Its squared integral yields the <free-filament variance with fixed translation and tilt>.
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