= Fixed-volume capillary spectrum of a cylindrical membrane
{title2=$\langle u_q^2\rangle$}
For a tension-dominated cylindrical <lipid vesicle> of fixed length $L$ and volume, a real axisymmetric sine amplitude $u_q$ has stiffness $K_q=\pi\sigma L[(qR_0)^2-1]/R_0$. Fixed volume requires a mean-radius shift $-u_q^2/(4R_0)$. Stable modes have $\langle u_q^2\rangle=k_BT/K_q$ by the <equipartition theorem>. Modes with $qR_0<1$ have the <Rayleigh–Plateau instability>, rather than a negative equilibrium variance. Bending modifies very short wavelengths and near-marginal modes.
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