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

TL;DR

The paper demonstrates that thermal fluctuations of long-wavelength hydrodynamic modes generate a finite, positive correction to the infrared shear viscosity and induce a nonanalytic, frequency-dependent contribution to the second-order relaxation time $\tau_\pi$, signaling a breakdown of the gradient expansion in relativistic hydrodynamics. The authors compute the fluctuation-induced viscosity $\eta_{\rm new}$ via a Kubo framework using hydrodynamic correlators with a momentum cutoff $p_{\max}$, finding $\eta_{\rm new} \propto p_{\max} \gamma_{\eta} T (\epsilon+P)^2 / \eta_{\rm cl}^2$, which leads to a lower bound on the total viscosity $\eta$. The practical impact for QCD-like matter depends on $\eta_{\rm cl}/s$ and lattice inputs for $s/T^3$ and $\tau_\pi/\gamma_\eta$; for $\eta_{\rm cl}/s \approx 0.16$ there can be a finite window where second-order hydrodynamics remains applicable, while for $\eta_{\rm cl}/s \approx 0.08$ the framework breaks down at relevant frequencies. The work suggests that including hydrodynamic fluctuations in simulations could be important for accurately modeling heavy-ion collisions and the QGP, particularly at lower viscosities.

Abstract

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 $τ_π$ 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 $η/s = 0.16$ but important if $η/s = 0.08$.

Figures (2)

• 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, in red) 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 (\ref{['thebound']}) (left) and applicability of second-order viscous hydrodynamics (\ref{['viscbound2']}) (right). The unknown parameters $\tau_\pi$ and $p_{\rm max}$ were assumed to be $\tau_\pi/\gamma_\eta=3$ and $p_{\rm max}=1/(2 \tau_\pi)$ and $s/T^3$ was evaluated using lattice QCD equations of state from the hotQCD Karsch and Wuppertal-Budapest Fodor collaborations.