Thermal roughness of a tension-dominated interface ($⟨h^2⟩=\frac{k_{B}T}{2\pi \gamma }\log (q_{\max }/q_{\min })$)

For a nearly flat interface, the extra surface energy in the small-slope approximation is $(\gamma /2)\int |\nabla h{|}^{2}dA$. Each nonzero Fourier mode therefore has thermal variance $⟨|h_{\mathbf{q}}{|}^2⟩=k_{B}T/(A\gamma q^2)$ in a convention with height reconstructed as the sum of its mode amplitudes. Summing over a two-dimensional annulus of wavevectors givesThe 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.