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 ${\alpha }_1\simeq 1.875$.

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.