Theory of Nonequilibrium Coexistence with Coupled Conserved and Nonconserved Order Parameters

Abstract

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 the phenomenological Active Model C+. We hope our work aids in the development of a comprehensive theory of high-dimensional nonequilibrium phase diagrams.

Figures (3)

• Figure 1: Three example systems where our theory would apply with $n_N=1$: active crystallization, chemotactic fluid-fluid separation, and active isotropic-nematic coexistence. A schematic of the order parameter profiles is shown, where the nonconserved order parameter represents the local crystallinity, fuel concentration, and nematic order. While chemoattractant fuel is likely to be subject to a constraint (i.e., mass conservation) common coarse-grained models treat chemotactic fuel as a wholly unconstrained variable Liebchen2018Zhao2023.
• Figure 2: Phase diagram of AMC with $n_N=1$, displaying the coexistence values of $\rho$/$\psi$ as a function of the ratio $(\chi + \alpha) / (\chi - \alpha)$. We set $\overline{\kappa}_{\rho \rho} = \overline{\kappa}_{\psi \psi} = \overline{\kappa}_{\rho \psi} = 0.01$, $\overline{\kappa}_{\psi \rho}=0.005$, and $\chi = 1/2$. The dotted green line indicates the limit where $\overline{\kappa}_{\rho \psi} (\chi - \alpha) = \overline{\kappa}_{\psi \rho} (\chi + \alpha)=2$ and the coexistence criteria are exact.
• Figure 3: Phase diagrams of AMC with $n_N=1$ as a function of $\rho$/$\psi$ and $\lambda_{\rho \rho \rho} / \overline{\kappa}_{\rho \rho}$. We set $\overline{\kappa}_{\psi \rho}=0$ and $\overline{\kappa}_{\rho \rho} = \overline{\kappa}_{\psi \psi} = 0.01$ in every case. We consider systems with (a) exact coexistence criteria by setting $\chi=-\alpha=1/4$ and $\overline{\kappa}_{\rho \psi} = \lambda_{\rho \psi \rho} = \lambda_{\rho \rho \psi} = \lambda_{\rho \psi \psi} = 0$, (b) approximate coexistence criteria by setting $\chi=1/2$, $\alpha=0$, and $\overline{\kappa}_{\rho \psi} = \lambda_{\rho \psi \rho} = \lambda_{\rho \rho \psi} = \lambda_{\rho \psi \psi} = 0$, and (c) no coexistence criteria within our theory by setting $\chi=1/2$, $\alpha=0$, and $\overline{\kappa}_{\rho \psi} = \lambda_{\rho \psi \rho} = \lambda_{\rho \rho \psi} = \lambda_{\rho \psi \psi} = 0.01$. The dotted green line in (c) indicates where $\lambda_{\rho \rho \rho} = \lambda_{\rho \psi \psi}$ and well-defined approximate coexistence criteria can be formulated. When this is the case, the interfacial Maxwell construction vector takes the form ${\boldsymbol{\mathcal{E}}^{\rm int} = \exp \left( 2 \lambda_{\rho \rho \rho} \rho / \overline{\kappa}_{\rho \rho} + 2 \lambda_{\rho \rho \psi} \psi / \overline{\kappa}_{\rho \psi} \right)0^T}$. When $\lambda_{\rho \rho \rho} \neq \lambda_{\rho \psi \psi}$, a solution to Eqs. \ref{['eq:G22']}-\ref{['eq:G21']} does not exist. In this case, we continue to use the weighted-area construction with $\mathbf{E}^{\rm int}$ to predict the phase diagram in (c) even though it is not a well-defined approximation here.