Hydrodynamic fluctuations and the minimum shear viscosity of the dilute Fermi gas at unitarity

Abstract

We study hydrodynamic fluctuations in a non-relativistic fluid. We show that in three dimensions fluctuations lead to a minimum in the shear viscosity to entropy density ratio $\eta /s$ as a function of the temperature. The minimum provides a bound on $\eta /s$ which is independent of the conjectured bound in string theory, $\eta /s\ge \hslash /(4\pi k_B)$, where $s$ is the entropy density. For the dilute Fermi gas at unitarity we find $\eta /s\text{gsim}0.2\hslash$. This bound is not universal -- it depends on thermodynamic properties of the unitary Fermi gas, and on empirical information about the range of validity of hydrodynamics. We also find that the viscous relaxation time of a hydrodynamic mode with frequency $\omega$ diverges as $1/\sqrt{\omega }$, and that the shear viscosity in two dimensions diverges as $\log (1/\omega )$.

Comments: 26 pages, 5 figures; final version to appear in Phys Rev A