  We consider the one-dimensional driven ABC model under particle-conserving
and particle-non-conserving processes. Two limiting cases are studied: (a) the
rates of the non-conserving processes are vanishingly slow compared with the
conserving processes in the thermodynamic limit and (b) the two rates are
comparable. For case (a) we provide a detailed analysis of the phase diagram
and the large deviations function of the overall density, G(r). The phase
diagram of the non-conserving model, derived from G(r), is found to be
different from the conserving one. This difference, which stems from the
non-convexity of G(r), is analogous to ensemble inequivalence found in
equilibrium systems with long-range interactions. An outline of the analysis of
case (a) was given in an earlier letter. For case (b) we show that unlike the
conserving model, the non-conserving model exhibits a moving density profile in
the steady-state with a velocity that remains finite in the thermodynamic
limit. Moreover, in contrast with case (a), the critical lines of the
conserving and non-conserving models do not coincide. These are new features
which are present only when the rates of the conserving and non-conserving
processes are comparable. In addition, we analyze G(r) in case (b) using
macroscopic fluctuations theory. Much of the derivation presented in this paper
is applicable to any driven-diffusive system coupled to an external
particle-bath via a slow dynamics.
