Hydrodynamic tails and a fluctuation bound on the bulk viscosity

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 non-analyticity 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 $\zeta$ scales as $\zeta_{\it min} \sim (P-\frac{2}{3}{\cal E})^2 \sum_i D_i^{-2}$, where $D_i$ are the diffusion constants for energy and momentum, and $P-\frac{2}{3}{\cal E}$, where $P$ is the pressure and ${\cal E}$ is the energy density, is a measure of scale breaking. Applied to the cold Fermi gas near unitarity, $|\lambda/a_s|\geq 1$ where $\lambda$ is the thermal de Broglie wave length and $a_s$ is the $s$-wave scattering length, this bound implies that the ratio of bulk viscosity to entropy density satisfies $\zeta/s \geq 0.1\hbar/k_B$. Here, $\hbar$ is Planck's constant and $k_B$ is Boltzmann's constant.