  It is well known that systems with long-range interactions may exhibit
different phase diagrams when studied within two different ensembles. In many
of the previously studied examples of ensemble inequivalence, the phase
diagrams differ only when the transition in one of the ensembles is first
order. By contrast, in a recent study of a generalized ABC model, the canonical
and grand-canonical ensembles of the model were shown to differ even when they
both exhibit a continuous transition. Here we show that the order of the
transition where ensemble inequivalence may occur is related to the symmetry
properties of the order parameter associated with the transition. This is done
by analyzing the Landau expansion of a generic model with long-range
interactions. The conclusions drawn from the generic analysis are demonstrated
for the ABC model by explicit calculation of its Landau expansion.
