  We investigate stationary nonequilibrium states of systems of particles
moving according to Hamiltonian dynamics with specified potentials. The systems
are driven away from equilibrium by Maxwell demon ``reflection rules'' at the
walls. These deterministic rules conserve energy but not phase space volume,
and the resulting global dynamics may or may not be time reversible (or even
invertible). Using rules designed to simulate moving walls we can obtain a
stationary shear flow. Assuming that for macroscopic systems this flow
satisfies the Navier-Stokes equations, we compare the hydrodynamic entropy
production with the average rate of phase space volume compression. We find
that they are equal when the velocity distribution of particles incident on the
walls is a local Maxwellian. An argument for a general equality of this kind,
based on the assumption of local thermodynamic equilibrium, is given. Molecular
dynamic simulations of hard disks in a channel produce a steady shear flow with
the predicted behavior.
