  The lattice Boltzmann algorithm efficiently simulates the Navier Stokes
equation of isothermal fluid flow, but ignores thermal fluctuations of the
fluid, important in mesoscopic flows. We show how to adapt the algorithm to
include noise, satisfying a fluctuation-dissipation theorem (FDT) directly at
lattice level: this gives correct fluctuations for mass and momentum densities,
and for stresses, at all wavevectors $k$. Unlike previous work, which recovers
FDT only as $k\to 0$, our algorithm offers full statistical mechanical
consistency in mesoscale simulations of, e.g., fluctuating colloidal
hydrodynamics.
