  First-order nonequilibrium phase transitions observed in active matter, fluid
dynamics, biology, climate science, and other systems with irreversible
dynamics are challenging to analyze because they cannot be inferred from a
simple free energy minimization principle. Rather the mechanism of these
transitions depends crucially on the system's dynamics, which requires us to
analyze them in trajectory space rather than in phase space. Here we consider
situations where the path of these transitions can be characterized as the
minimizer of an action, whose minimum value can be used in a nonequilibrium
generalization of the Arrhenius law to calculate the system's phase diagram. We
also develop efficient numerical tools for the minimization of this action.
These tools are general enough to be transportable to many situations of
interest, in particular when the fluctuations present in the microscopic system
are non-Gaussian and its dynamics is not governed by the standard Langevin
equation. As an illustration, first-order phase transitions in two
spatially-extended nonequilibrium systems are analyzed: a modified
Ginzburg-Landau equation with a chemical potential which is non-gradient, and a
reaction-diffusion network based on the Schl\"ogl model. The phase diagrams of
both systems are calculated as a function of their control parameters, and the
paths of the transitions, including their critical nuclei, are identified.
These results clearly demonstrate the nonequilibrium nature of the transitions,
with differing forward and backward paths.
