  Dynamical phase transitions are crucial features of the fluctuations of
statistical systems, corresponding to boundaries between qualitatively
different mechanisms of maintaining unlikely values of dynamical observables
over long periods of time. They manifest themselves in the form of
non-analyticities in the large deviation function of those observables. In this
paper, we look at bulk-driven exclusion processes with open boundaries. It is
known that the standard asymmetric simple exclusion process exhibits a
dynamical phase transition in the large deviations of the current of particles
flowing through it. That phase transition has been described thanks to specific
calculation methods relying on the model being exactly solvable, but more
general methods have also been used to describe the extreme large deviations of
that current, far from the phase transition. We extend those methods to a large
class of models based on the ASEP, where we add arbitrary spatial
inhomogeneities in the rates and short-range potentials between the particles.
We show that, as for the regular ASEP, the large deviation function of the
current scales differently with the size of the system if one considers very
high or very low currents, pointing to the existence of a dynamical phase
transition between those two regimes: high current large deviations are
extensive in the system size, and the typical states associated to them are
Coulomb gases, which are correlated ; low current large deviations do not
depend on the system size, and the typical states associated to them are
anti-shocks, consistently with a hydrodynamic behaviour. Finally, we illustrate
our results numerically on a simple example, and we interpret the transition in
terms of the current pushing beyond its maximal hydrodynamic value, as well as
relate it to the appearance of Tracy-Widom distributions in the relaxation
statistics of such models.
