We show that a Galilean invariant version of fluid dynamics can be derived by the methods of statistical dynamics using Maxwell's balance equations. The basic equation is non-local, and might replace the Boltzmann equation if the latter turns out not to have global smooth solutions in general. As an approximation, a local form of the equations of motion is derived. It turns out to be a version of the Navier-Stokes system, obeying the Stokes relation, and with the viscosity coefficient rising as Θ1/2 with temperature Θ. The new feature is the presence of the Dufour effect for a gas of a single component. This ensures that the principal symbol of the parabolic system is non-singular.