  We study the equilibrium simplex of Nica-Pimsner algebras arising from
product systems of finite rank on the free abelian semigroup. First we show
that every equilibrium state has a convex decomposition into parts parametrized
by ideals on the unit hypercube. Secondly we associate every gauge-invariant
part to a sub-simplex of tracial states of the diagonal algebra. We show how
this parametrization lifts to the full equilibrium simplices of non-infinite
type.
  The finite rank entails an entropy theory for identifying the two critical
inverse temperatures: (a) the least upper bound for existence of non
finite-type equilibrium states, and (b) the least positive inverse temperature
below which there are no equilibrium states at all. We show that the first one
can be at most the strong entropy of the product system whereas the second is
the infimum of the tracial entropies (modulo negative values). Thus phase
transitions can happen only in-between these two critical points and possibly
at zero temperature.
