Trapped phonons

TL;DR

This paper investigates how confined geometries influence the shear viscosity contributed by phonons in superfluids, showing that boundary interactions significantly affect viscosity, especially in optical traps at low temperatures.

Contribution

It introduces a simplified model analyzing phonon boundary interactions and derives an effective shear viscosity proportional to trap size and phonon density.

Findings

Shear viscosity depends on boundary scattering and trap size.

Phonons not in equilibrium exert shear stress on boundaries.

Viscosity decreases with temperature in optical traps.

Abstract

We analyze the effect of restricted geometries on the contribution of Nambu-Goldstone bosons (phonons) to the shear viscosity, $\eta$, of a superfluid. For illustrative purpose we examine a simplified system consisting of a circular boundary of radius $R$, confining a two-dimensional rarefied gas of phonons. Considering the Maxwell-type conditions, we show that phonons that are not in equilibrium with the boundary and that are not specularly reflected exert a shear stress on the boundary. In this case it is possible to define an effective (ballistic) shear viscosity coefficient $\eta \propto \rho_{\rm ph} \chi R$, where $\rho_{\rm ph}$ is the density of phonons and $\chi$ is a parameter which characterizes the type of scattering at the boundary. For an optically trapped superfluid our results corroborate the findings of Refs. \cite{Mannarelli:2012su, Mannarelli:2012eg}, which imply that at very low temperature the shear viscosity correlates with the size of the optical trap and decreases with decreasing temperature.