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 is relevant.
Capillary wave theory (CWT) is a classic account of how thermal fluctuations distort an interface. It starts from some intrinsic surface
where the first equality is the area in this (the Monge) representation, and the second applies for small values of the derivatives (surfaces not too rough). The constant of proportionality,
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