Theory of nonequilibrium coexistence with coupled conserved and nonconserved order parameters

Phase separation routinely occurs in both living and synthetic systems. These phases are often complex and distinguished by features including crystallinity, nematic order, and a host of other nonconserved order parameters. For systems at equilibrium, the phase boundaries that characterize these transitions can be straightforwardly determined through the framework of thermodynamics. The prevalence of phase separation in active and driven systems motivates the need for a genuinely nonequilibrium theory for the coexistence of complex phases. Here, we develop a dynamical theory of coexistence when both conserved and nonconserved order parameters are present, casting coexistence criteria into the familiar form of equality of state functions. Our theory generalizes thermodynamic notions such as the chemical potential and Gibbs-Duhem relation to systems out of equilibrium. While these notions may not exist for all nonequilibrium systems, we numerically verify their existence for a variety of systems by introducing phenomenological active field theories. We hope our work aids in the development of a comprehensive theory of high-dimensional nonequilibrium phase diagrams.