  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 $\eta/s = 0.08$.
