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In principle, the combined product of density and curvature should be averaged over the fluctuations of the surface: 2 hθ (z-h) . A separate average is appropriate since each quantity depends on different regions of the frequency of fluctuations. The curvature is dominated by the large-wavelength part of the spectrum while the density is dominated by the low-wavelength components. Again, it is straightforward to show this for a Gaussian interface.
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In principle, the combined product of density and curvature should be averaged over the fluctuations of the surface: 2 hθ (z-h). A separate average is appropriate since each quantity depends on different regions of the frequency of fluctuations. The curvature is dominated by the large-wavelength part of the spectrum while the density is dominated by the low-wavelength components. Again, it is straightforward to show this for a Gaussian interface.
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The neglect of the y coordinate has no influence on the results presented in this paper.
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Note that, at very low velocities, thermally activated events can occur before the spinodal limit considered here. However, this corresponds to velocities below 1 nm/sec, well below the experimental range.
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A detailed analysis shows that the dissipation coefficient γ may itself depend on the position of the atoms. This effect is neglected here. See, e.g., PRBMDO 0163-1829 10.1103/PhysRevB.64.245409
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