Fluctuating hydrodynamics and capillary waves

Abstract

We derive an equation of motion for the instantaneous position of a planar liquid surface using fluctuating hydrodynamics. For the fluid constrained to be in a nonequilibrium steady state by a small temperature gradient, we then obtain the dynamic structure factor ${S}_{\mathrm{ss}}(q,\ensuremath{\omega})$, which represents the spectrum of thermal ripplons (capillary waves) on the liquid surface. Our analysis is done for both classical and quantum fluids. We find that there is an asymmetry in the heights of the two ripplon peaks in ${S}_{\mathrm{ss}}(q,\ensuremath{\omega})$ due to broken time-reversal symmetry. Our numerical estimates of the effect for liquid helium II, suggest that it could be measurable. We have undertaken a moment analysis of ${S}_{\mathrm{ss}}(q,\ensuremath{\omega})$; the first frequency moment is long ranged ($\ensuremath{\sim}\frac{1}{{q}^{2}}$) in the steady state, though it vanishes in equilibrium. From the zeroth frequency moment we recover the familiar result for the infrared logarithmic divergence of the mean-square displacement of the interface; the analysis explicitly involves the coupling of the surface to the bulk fluid. We modify the so-called capillary-wave model by including the surface-bulk coupling, and thereby make it consistent with the capillary-wave dispersion.