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Thermal motion is able to produce capillary waves at the molecular scale. At this scale,gravity and hydrodynamics can be neglected, and only the surface tension contribution isrelevant.
Capillary wave theory is a classic account of how thermal fluctuations distort an interface.It starts from some intrinsic surface h {\displaystyle h} that is distorted. Its energy will beproportional to its area:
where the first equality is the area in this representation, and the secondapplies for small values of the derivatives. The constant of proportionality, σ {\displaystyle \sigma } , is the surface tension.
By performing a Fourier analysis treatment, normal modes are easily found. Each contributes an energy proportional to the square of its amplitude; therefore, according to classical statistical mechanics, equipartition holds, and the mean energy of each mode will be k T / 2 {\displaystyle kT/2}. Surprisingly, this result leads to a divergent surface. This divergence is nevertheless very mild: even for displacements on the order of meters the deviation of the surface is comparable to the size of the molecules. Moreover, the introduction of an external field removes the divergence: the action of gravity is sufficient to keep the width fluctuation on the order of one molecular diameter for areas larger than about 1 mm2.