Hydrodynamic tails and a fluctuation bound on the bulk viscosity

TL;DR

This paper investigates the small-frequency behavior of bulk viscosity in fluids, deriving model-independent results including a lower bound on bulk viscosity that relates to scale breaking and diffusion constants, with applications to cold Fermi gases.

Contribution

It establishes a weakly assumption-dependent lower bound on bulk viscosity based on hydrodynamic tails and spectral function non-analyticities, applicable to strongly interacting quantum gases.

Findings

Derived the long-time tail of the bulk stress correlation function.

Identified the leading non-analyticity of the spectral function at small frequency.

Established a lower bound on bulk viscosity related to diffusion constants and scale breaking.

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 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.