Hydrodynamic tails and a fluctuation bound on the bulk viscosity

Abstract

We study the small frequency behavior of the bulk viscosity spectral function using stochastic fluid dynamics. We obtain a number of model independent results, including the long-time tail of the bulk stress correlation function and the leading nonanalyticity of the spectral function at small frequency. We also establish a lower bound on the bulk viscosity which is weakly dependent on assumptions regarding the range of applicability of fluid dynamics. The bound on the bulk viscosity $\ensuremath{\zeta}$ scales as ${\ensuremath{\zeta}}_{\mathrm{min}}\ensuremath{\sim}{(P\ensuremath{-}\frac{2}{3}\mathcal{E})}^{2}{\ensuremath{\sum}}_{i}{D}_{i}^{\ensuremath{-}2}$, where ${D}_{i}$ are the diffusion constants for energy and momentum and $P\ensuremath{-}\frac{2}{3}\mathcal{E}$, where $P$ is the pressure and $\mathcal{E}$ is the energy density, is a measure of scale breaking. Applied to the cold Fermi gas near unitarity, $|\ensuremath{\lambda}/{a}_{s}|\ensuremath{\gtrsim}1$, where $\ensuremath{\lambda}$ is the thermal de Broglie wavelength and ${a}_{s}$ is the $s$-wave scattering length, this bound implies that the ratio of bulk viscosity to entropy density satisfies $\ensuremath{\zeta}/s\ensuremath{\gtrsim}0.1\ensuremath{\hbar}/{k}_{B}$. Here, $\ensuremath{\hbar}$ is Planck's constant and ${k}_{B}$ is Boltzmann's constant.