  Understanding nonequilibrium systems and the consequences of irreversibility
for the system's behavior as compared to the equilibrium case, is a fundamental
question in statistical physics. Here, we investigate two types of
nonequilbrium phase transitions, a second-order and an infinite-order phase
transition, in a prototypical q-state vector Potts model which is driven out of
equilibrium by coupling the spins to heat baths at two different temperatures.
We discuss the behavior of the quantities that are typically considered in the
vicinity of (equilibrium) phase transitions, like the specific heat, and
moreover investigate the behavior of the entropy production (EP), which
directly quantifies the irreversibility of the process. For the second-order
phase transition, we show that the universality class remains the same as in
equilibrium. Further, the derivative of the EP rate with respect to the
temperature diverges with a power-law at the critical point, but displays a
non-universal critical exponent, which depends on the temperature difference,
i.e., the strength of the driving. For the infinite-order transition, the
derivative of the EP exhibits a maximum in the disordered phase, similar to the
specific heat. However, in contrast to the specific heat, whose maximum is
independent of the strength of the driving, the maximum of the derivative of
the EP grows with increasing temperature difference. We also consider entropy
fluctuations and find that their skewness increases with the driving strength,
in both cases, in the vicinity of the second-order transition, as well as
around the infinite-order transition.
