Stickiness of sound: An absolute lower limit on viscosity and the breakdown of second-order relativistic hydrodynamics

Hydrodynamics predicts long-lived sound and shear waves. Thermal fluctuations in these waves can lead to the diffusion of momentum density, contributing to the shear viscosity and other transport coefficients. Within viscous hydrodynamics in $3+1$ dimensions, this leads to a positive contribution to the shear viscosity, which is finite but inversely proportional to the microscopic shear viscosity. Therefore the effective infrared viscosity is bounded from below. The contribution to the second-order transport coefficient ${\tau }_{\pi }$ is divergent, which means that second-order relativistic viscous hydrodynamics is inconsistent below some frequency scale. We estimate the importance of each effect for the quark-gluon plasma, finding them to be minor if $\eta /s=0.16$ but important if $\eta /s=0.08$.

Figure 1

A shear wave; fluid moves to the left in a band of fluid near the middle of the figure. Viscosity determines the loss (by diffusion) of the forward motion of this fluid. Sound waves (dotted red lines) leaving the left-moving fluid carry, on average, net left-moving momentum, which is not compensated by sound waves arriving in the left-moving fluid. Hence sound waves contribute to viscosity.

Figure 2

Examples for the viscosity over entropy density bound (4.3) (left panel) and applicability of second-order viscous hydrodynamics (4.9) (right panel). The unknown parameters ${\tau }_{\pi }$ and $p_{\max }$ were assumed to be ${\tau }_{\pi }/{\gamma }_{\eta }=3$ and $p_{\max }=1/(2{\tau }_{\pi })$, and $s/T^3$ was evaluated using lattice QCD equations of state from the hotQCD [44] and Wuppertal-Budapest [43] collaborations.