  The steady states of the two-species (positive and negative particles)
asymmetric exclusion model of Evans, Foster, Godreche and Mukamel are studied
using Monte Carlo simulations. We show that mean-field theory does not give the
correct phase diagram. On the first-order phase transition line which separates
the CP-symmetric phase from the broken phase, the density profiles can be
understood through an unexpected pattern of shocks. In the broken phase the
free energy functional is not a convex function but looks like a standard
Ginzburg-Landau picture. If a symmetry breaking term is introduced in the
boundaries the Ginzburg-Landau picture remains and one obtains spinodal points.
The spectrum of the hamiltonian associated with the master equation was studied
using numerical diagonalization. There are massless excitations on the
first-order phase transition line with a dynamical critical exponent z=2 as
expected from the existence of shocks and at the spinodal points where we find
$z=1$. It is for the first time that this value which characterizes conformal
invariant equilibrium problems appears in stochastic processes.
