= Thermal covariance of an elastic filament
{title2=$C(x,y)=k_BT K^{-1}(x,y)$}
For a positive quadratic bending operator $K$ with a real <orthonormal basis> of <eigenfunctions>, the <equipartition theorem> gives $C(x,y)=k_BT\sum_nW_n(x)W_n(y)/\mu_n$, where $KW_n=\mu_nW_n$. 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 $C(x,x)=k_BT x^3(L-x)^3/(3AL^3)$. This differs from a clamped-free tip <variance>. The center <variance> is $k_BT L^3/(192A)$.
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