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We consider the two dimensional (2D) classical lattice Coulomb gas as a modelfor magnetic field induced vortices in 2D superconducting networks. Twodifferent dynamical rules are introduced to investigate driven diffusive steadystates far from equilibrium as a function of temperature and driving force. Theresulting steady states differ dramatically depending on which dynamical ruleis used. We show that the commonly used driven diffusive Metropolis Monte Carlodynamics contains unphysical intrinsic randomness that destroys the spatialordering present in equilibrium (the vortex lattice) over most of the drivenphase diagram. A continuous time Monte Carlo (CTMC) is then developed, whichresults in spatially ordered driven states at low temperature in finite sizedsystems. We show that CTMC is the natural discretization of continuum Langevindynamics, and argue that it gives the correct physical behavior when thediscrete grid represents the minima of a periodic potential. We use detailedfinite size scaling methods to analyze the spatial structure of the steadystates. We find that finite size effects can be subtle and that very longsimulation times can be needed to arrive at the correct steady state. Forparticles moving on a triangular grid, we find that the ordered moving state isa transversely pinned smectic that becomes unstable to an anisotropic liquid onsufficiently large length scales. For particles moving on a square grid, themoving state is a similar smectic at large drives, but we find evidence for apossible moving solid at lower drives. We find that the driven liquid on thesquare grid has long range hexatic order, and we explain this as a specificallynon-equilibrium effect. We show that, in the liquid, fluctuations are diffusivein both the transverse and longitudinal directions.Comment: 29 pages, 35 figure

Continuous-time Monte Carlo and spatial ordering in driven lattice gases: Application to driven vortices in periodic superconducting networks