 Fixed-area capillary spectrum of a circular boundary

For a circular domain of radius $R_0$ with line tension $\gamma$, the stiffness of a real polar sine or cosine amplitude with integer angular harmonic $m$ is $K_m=\pi \gamma (m^2-1)/R_0$, when the enclosed area is fixed. The equipartition theorem gives $⟨u_m^2⟩=k_{B}T{R}_0/[\pi \gamma (m^2-1)]$ for $m\ge 2$. A positive-frequency complex Fourier mode coefficient has half this variance. The uniform mode is forbidden and $m=1$ is translation.