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Within the framework of relativistic fluctuating hydrodynamics we compute thecontribution of thermal fluctuations to the effective infrared shear viscosityof a conformal fluid, focusing on quadratic (in fluctuations), second order (invelocity gradients) terms in the conservation equations. Our approach is basedon the separation of hydrodynamic fields in soft and ultrasoft sectors, inwhich the effective shear viscosity arises due to the action of the soft modeson the evolution of the ultrasoft ones. We find that for a strongly coupledfluid with small shear viscosity--to--entropy ratio $\eta/s$ the contributionof thermal fluctuations to the effective shear viscosity is small butsignificant. Using realistic estimates for the strongly coupled quark--gluonplasma created in heavy ion collisions, we find that for $\eta/s$ close to theAdS/CFT lower bound $1/(4\pi)$ the correction is positive and at most amountsto 10% in the temperature range 200--300 MeV, whereas for larger values $\eta/s\sim 2/(4\pi)$ the correction is negligible. For weakly coupled theories thecorrection is very small even for $\eta/s=0.08$ and can be neglected.

Shear viscosity from thermal fluctuations in relativistic conformal fluid dynamics