  We consider the two dimensional (2D) classical lattice Coulomb gas as a model
for magnetic field induced vortices in 2D superconducting networks. Two
different dynamical rules are introduced to investigate driven diffusive steady
states far from equilibrium as a function of temperature and driving force. The
resulting steady states differ dramatically depending on which dynamical rule
is used. We show that the commonly used driven diffusive Metropolis Monte Carlo
dynamics contains unphysical intrinsic randomness that destroys the spatial
ordering present in equilibrium (the vortex lattice) over most of the driven
phase diagram. A continuous time Monte Carlo (CTMC) is then developed, which
results in spatially ordered driven states at low temperature in finite sized
systems. We show that CTMC is the natural discretization of continuum Langevin
dynamics, and argue that it gives the correct physical behavior when the
discrete grid represents the minima of a periodic potential. We use detailed
finite size scaling methods to analyze the spatial structure of the steady
states. We find that finite size effects can be subtle and that very long
simulation times can be needed to arrive at the correct steady state. For
particles moving on a triangular grid, we find that the ordered moving state is
a transversely pinned smectic that becomes unstable to an anisotropic liquid on
sufficiently large length scales. For particles moving on a square grid, the
moving state is a similar smectic at large drives, but we find evidence for a
possible moving solid at lower drives. We find that the driven liquid on the
square grid has long range hexatic order, and we explain this as a specifically
non-equilibrium effect. We show that, in the liquid, fluctuations are diffusive
in both the transverse and longitudinal directions.
