Thermal covariance of an elastic filament ($C(x,y)=k_{B}T{K}^{-1}(x,y)$)

For a positive quadratic bending operator $K$ with a real orthonormal basis of eigenfunctions, the equipartition theorem gives $C(x,y)=k_{B}T{\sum }_n{W}_n(x)W_n(y)/{\mu }_n$, where $K{W}_n={\mu }_n{W}_n$. This is the inverse-operator Green function multiplied by Boltzmann constant and temperature. Unconstrained zero-energy filament modes prevent a normalizable canonical ensemble, and negative modes signal an unstable quadratic model.

For both ends clamped and $K=A{\partial }_x^4$, the diagonal is $C(x,x)=k_{B}T{x}^3(L-x{)}^3/(3A{L}^3)$. This differs from a clamped-free tip variance. The center variance is $k_{B}T{L}^3/(192A)$.

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.

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.

For the frame fixed by zero mean height and zero first height moment, the variance profile is $P(u)=1/105-11u/105+13u^2/35-u^3/3-u^4/3+3u^5/5-u^6/5$. It obeys $P(1-u)=P(u)$, $P(0)=1/105$ and $P(1/2)=1/320$. Integrating a white noise curvature field twice, subtracting its affine projection and integrating the squared kernel proves the profile. These free-end conditions must not be confused with physically clamping the ends.

A zero-energy mode lies in the kernel of the quadratic elastic filament operator. For pure bending, zero curvature makes the displacement affine. Free-free endpoints therefore leave translation and tilt modes, while slope-constrained, force-free endpoints leave only translation. Both clamped-clamped and hinged-hinged endpoints remove these modes. An unrestricted modal amplitude has constant Boltzmann distribution weight, so no normalizable canonical ensemble or finite displacement variance exists. Fixing the rigid degrees of freedom allows the equipartition theorem on the positive complement.