  Theory and simulation of Brownian colloids suspended in an implicit solvent,
with the hydrodynamics of the fluid accounted for by effective interactions
between the colloids, are shown to yield a marked and hitherto unobserved
discrepancy between the viscosity calculated from the average shear stress
under an imposed shear rate in the Stokesian regime and the viscosity extracted
by the Green-Kubo formalism from the auto-correlations of thermal stress
fluctuations in quiescent equilibrium. We show that agreement between both
methods is recovered by accounting for the fluctuating Brownian stresses on the
colloids, complementing and related to the traditional fluctuating Brownian
forces and torques through an extended fluctuation-dissipation theorem based on
the hydrodynamic grand resistance matrix. Time-averaging of the fluctuating
terms gives rise to novel non-fluctuating stresslets. Brownian Dynamics
simulations of spheroidal particles illustrate the necessity of these
fluctuating and non-fluctuating contributions to obtaining consistent
viscosities.
