  A generalization of the ABC model, a one-dimensional model of a driven system
of three particle species with local dynamics, is introduced, in which the
model evolves under either (i) density-conserving or (ii) nonconserving
dynamics. For equal average densities of the three species, both dynamical
models are demonstrated to exhibit detailed balance with respect to a
Hamiltonian with long-range interactions. The model is found to exhibit two
distinct phase diagrams, corresponding to the canonical (density-conserving)
and grand canonical (density nonconserving) ensembles, as expected in
long-range interacting systems. The implication of this result to
nonequilibrium steady states, such as those of the ABC model with unequal
average densities, are briefly discussed.
