Minimum-Action Method for Nonequilibrium Phase Transitions

Abstract

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 nongradient and a reaction-diffusion network based on the Schlögl 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.