Thermal bending fluctuations of a clamped filament

A clamped filament of length $L$ and bending modulus $A$ has equilibrium tip variance $k_{B}T{L}^3/(3A)$. Under local transverse viscous drag, each bending mode relaxes exponentially at a rate proportional to its fourth power of wavenumber.

The bending modes of a filament clamped at one end and force-free and torque-free at the other obey $\cos {\alpha }_n\cosh {\alpha }_n=-1$. The slowest mode has ${\alpha }_1\simeq 1.875$.

For the positive roots $q_n$ of $\cos q\cosh q=-1$, ${\sum }_n{q}_n^{-4}=1/12$. The tip-force compliance of a cantilever is $L^3/(3A)$ directly, whereas expansion in normalized clamped--free bending modes and $W_n(L{)}^2=4L^{-1}\int W_n^{2}dx$ gives $4L^3{\sum }_n{q}_n^{-4}/A$. Equating the two responses proves the sum and evaluates the full thermal endpoint variance.

The elastic energy $U=A\int (W^{\prime \prime }{)}^{2}dx/2$ minimized without forcing gives $W^{\prime \prime \prime \prime }=0$. With fixed $\int W^{2}dx$, stationarity of $U-A{k}^4\int W^{2}dx/2$ gives $W^{\prime \prime \prime \prime }=k^{4}W$: a bending eigenmode. A general admissible shape expands in these normal modes. The natural boundary conditions for a free endpoint are $W^{\prime \prime }(L)=W^{\prime \prime \prime }(L)=0$, while the clamp imposes $W(0)=W^{\prime }(0)=0$.

For ${\zeta }_{\perp }{h}_t=-B{h}_{xxxx}$, separated normal modes decay at rates $B{\alpha }_n^4/({\zeta }_{\perp }{L}^4)$. Clamping at zero and vanishing moment and shear at $L$ give $\cos {\alpha }_n\cosh {\alpha }_n=-1$. A mode isThe fourth-order operator is self-adjoint: two integration by parts operations give $\int f{g}^{\prime \prime \prime \prime }=\int f^{\prime \prime }{g}^{\prime \prime }$ under these boundary conditions. Its zero eigenspace is trivial and its eigenfunctions form an orthogonal complete basis. Thus a nonzero projection on the first mode gives long-time decay proportional to ${\varphi }_1(x/L)\exp [-B{\alpha }_1^{4}t/({\zeta }_{\perp }{L}^4)]$, where ${\alpha }_1=1.8751040687\dots$. If this projection vanishes, the lowest excited mode replaces it.

A filament free of moment and shear at both ends has $h_{xx}=h_{xxx}=0$. Its positive biharmonic eigenvalues $q_n^4$ obey $\cos (q_{n}L)\cosh (q_{n}L)=1$, in addition to the two rigid zero modes representing translation and rotation.