= Thermal roughness of a tension-dominated interface
{title2=$\langle h^2\rangle=\frac{k_BT}{2\pi\gamma}\log(q_{\max}/q_{\min})$}
For a nearly flat interface, the extra <surface energy> in the small-slope approximation is $(\gamma/2)\int|\nabla h|^2dA$. Each nonzero <Fourier mode> therefore has thermal variance $\langle|h_{\boldsymbol q}|^2\rangle=k_BT/(A\gamma q^2)$ in a convention with height reconstructed as the sum of its mode amplitudes. Summing over a two-dimensional annulus of wavevectors gives
$$
\langle h^2\rangle=\frac A{(2\pi)^2}\int_{q_{\min}}^{q_{\max}}2\pi q\,dq\frac{k_BT}{A\gamma q^2}=\frac{k_BT}{2\pi\gamma}\log\frac{q_{\max}}{q_{\min}}.
$$
The smallest wavevector is set by the lateral size or pinning, and the largest by the breakdown of the continuum description, commonly a molecular length. A fixed mean height removes the zero mode. <Gravity> supplies an additional restoring term and changes the long-wavelength cutoff when it is retained. Roughness here means a height variance; its square root is the <root mean square> amplitude, even when the mean height fluctuation is zero.
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