  We present and study lattice and off-lattice microscopic models in which
particles interact via a local anisotropic rule. The rule induces preferential
hopping along one direction, so that a net current sets in if allowed by
boundary conditions. This may be viewed as an oversimplification of the
situation concerning certain traffic and flow problems. The emphasis in our
study is on the influence of dynamic details on the resulting (non-equilibrium)
steady state. In particular, we shall discuss on the similarities and
differences between a lattice model and its continuous counterpart, namely, a
Lennard-Jones analogue in which the particles' coordinates vary continuously.
Our study, which involves a large series of computer simulations, in particular
reveals that spatial discretization will often modify the resulting
morphological properties and even induce a different phase diagram and
criticality.
