  In this paper, we draw attention to the problem of phase transitions in
systems with locally affine microcanonical entropy, in which partial
equivalence of (microcanonical and canonical) ensembles is observed. We focus
on a very simple spin model, that was shown to be an equilibrium statistical
mechanics representation of the biased random walk. The model exhibits
interesting discontinuous phase transitions that are simultaneously observed in
the microcanonical, canonical, and grand canonical ensemble, although in each
of these ensembles the transition occurs in a slightly different way. The
differences are related to fluctuations accompanying the discontinuous change
of the number of positive spins. In the microcanonical ensemble, there is no
fluctuation at all. In the canonical ensemble, one observes power-law
fluctuations, which are, however, size-dependent and disappear in the
thermodynamic limit. Finally, in the grand canonical ensemble, the
discontinuous transition is of mixed-order (hybrid) kind with diverging
(critical-like) fluctuations. In general, this paper consists of many small
results, which together make up an interesting example of phase transitions
that are not covered by the known classifications of these phenomena.
