  The three species asymmetric ABC model was initially defined on a ring by
Evans, Kafri, Koduvely, and Mukamel, and the weakly asymmetric version was
later studied by Clincy, Derrida, and Evans. Here the latter model is studied
on a one-dimensional lattice of N sites with closed (zero flux) boundaries. In
this geometry the local particle conserving dynamics satisfies detailed balance
with respect to a canonical Gibbs measure with long range asymmetric pair
interactions. This generalizes results for the ring case, where detailed
balance holds, and in fact the steady state measure is known only for the case
of equal densities of the different species: in the latter case the stationary
states of the system on a ring and on an interval are the same. We prove that
in the N to infinity limit the scaled density profiles are given by (pieces of)
the periodic trajectory of a particle moving in a quartic confining potential.
We further prove uniqueness of the profiles, i.e., the existence of a single
phase, in all regions of the parameter space (of average densities and
temperature) except at low temperature with all densities equal; in this case a
continuum of phases, differing by translation, coexist. The results for the
equal density case apply also to the system on the ring, and there extend
results of Clincy et al.
