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

TL;DR

This paper investigates hydrodynamic fluctuations in a non-relativistic fluid, revealing a temperature-dependent minimum in shear viscosity to entropy density ratio, establishing a bound specific to the dilute Fermi gas at unitarity.

Contribution

It demonstrates that hydrodynamic fluctuations induce a non-universal minimum in shear viscosity for the unitary Fermi gas, providing a bound independent of string theory conjectures.

Findings

Minimum $rac{ ext{shear viscosity}}{ ext{entropy density}}$ ratio found at finite temperature.

Bound on $rac{ ext{shear viscosity}}{ ext{entropy density}}$ is approximately 0.2$rac{ ext{ extbar}}{ ext{ extbar}}$h.

Viscous relaxation time diverges as $1/ oot{2} ext{ extasciicircum}{}{ ext{ extasciicircum}}{ ext{ extasciicircum}}$ of frequency.

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, $\eta/s \geq \hbar/(4\pi k_B)$, where $s$ is the entropy density. For the dilute Fermi gas at unitarity we find $\eta/s\gsim 0.2\hbar$. 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 $\omega$ diverges as $1/\sqrt{\omega}$, and that the shear viscosity in two dimensions diverges as $\log(1/ \omega)$.