Overdamped relaxation of a free filament ($h={\sum }_{j\in \{\mathrm{tr},\mathrm{tilt}\}}{c}_j{W}_j+{\sum }_{n\ge 1}{c}_n{W}_n{e}^{-A{k}_n^{4}t/\zeta }$)

The overdamped filament bending equation diagonalizes in the complete free-free bending spectrum. Positive modes decay at rates $A{k}_n^4/\zeta$, while translation and tilt remain fixed. The long-time shape is the affine least-squares projection of the initial height. Free-end boundary values conserve the total height integral and its first spatial moment. For square-integrable initial data the expansion gives the smoothing semigroup solution.