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

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 $η/s$ as a function of the temperature. The minimum provides a bound on $η/s$ which is independent of the conjectured bound in string theory, $η/s \geq \hbar/(4πk_B)$, where $s$ is the entropy density. For the dilute Fermi gas at unitarity we find $η/s\gsim 0.2\hbar$. 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 $ω$ diverges as $1/\sqrtω$, and that the shear viscosity in two dimensions diverges as $\log(1/ ω)$.