Theory of Nonequilibrium Multicomponent Coexistence

TL;DR

This work develops a mechanical theory for nonequilibrium multicomponent phase coexistence that casts coexistence as equality of bulk state functions via species pseudopotentials and a generalized Gibbs-Duhem relation, extending equilibrium concepts to nonequilibrium settings. It introduces a transformation tensor linking the per-species effective body forces to pseudopotentials and a Maxwell-construction framework with a global quantity, enabling exact coexistence criteria when a full-rank solution exists and practical approximations otherwise. The theory is validated on the phenomenological Multicomponent Active Model B+ (MAMB+) and related NRCH models, where phase diagrams determined from the bulk criteria quantitatively agree with full spatial-density simulations, including nonlinear Maxwell constructions. The approach provides a unified framework bridging equilibrium thermodynamics and nonequilibrium phase behavior, while outlining practical paths to determine the governing state functions and highlighting limitations and directions for extending to more complex, high-dimensional systems.

Abstract

Multicomponent phase separation is a routine occurrence in both living and synthetic systems. Thermodynamics provides a straightforward path to determine the phase boundaries that characterize these transitions for systems at equilibrium. The prevalence of phase separation in complex systems outside the confines of equilibrium motivates the need for a genuinely nonequilibrium theory of multicomponent phase coexistence. Here, we develop a mechanical theory for coexistence that casts coexistence criteria into the familiar form of equality of state functions. Our theory generalizes traditional equilibrium notions such as the species chemical potential and thermodynamic pressure to systems out of equilibrium. Crucially, while these notions may not be identifiable for all nonequilibrium systems, we numerically verify their existence for a variety of systems by introducing the phenomenological Multicomponent Active Model B+. Our work establishes an initial framework for understanding multicomponent coexistence that we hope can serve as the basis for a comprehensive theory for high-dimensional nonequilibrium phase transitions.

Figures (3)

• Figure 1: A schematic of multiphase coexistence (six phases shown) and possible nonequilibrium constituents comprising the phases (including biomolecules in a driven environment Doan2024, multi-temperature particle mixtures Grosberg2015Weber2016Han2017, active-passive particle mixtures Angelani2011Stenhammar2015Wittkowski2017Omar2019Batton2024Manson2024Kreienkamp2024, and nonreciprocally interacting systems You2020Fruchart2021Chiu2023Dinelli2023). While curved interfaces are displayed, our theory assumes that the sizes of the phases are such that the radii of curvature far exceed any length scale associated with the constituents and thus the interfaces can be approximated as planar.
• Figure 2: A visual procedure to obtain the coexistence criteria for multicomponent nonequilibrium systems. First, determine the stationary species mechanical balance [Eq. \ref{['eq:species_linear_momentum_1']}]. Next, transform the effective body forces to species pseudopotentials with the transformation tensor $\boldsymbol{\mathcal{T}}$ [Eq. \ref{['eq:T_intro']}]. Only proceed if solutions for $\boldsymbol{\mathcal{T}}$ and $\mathbf{u}$ can be found. If so, identify the Maxwell construction vector, $\boldsymbol{\mathcal{E}}$, used in the generalized Gibbs-Duhem relation [Eq. \ref{['eq:generalized_GD']}]. If solutions for $\boldsymbol{\mathcal{E}}$ and $\mathcal{G}$ can be found, we have exact coexistence criteria [Eq. \ref{['eq:final_coex_criteria']}]. If $\boldsymbol{\mathcal{E}}^{\rm bulk} \neq \boldsymbol{\mathcal{E}}^{\rm int}$ but solutions for $\boldsymbol{\mathcal{E}}^{\rm bulk}$ and/or $\boldsymbol{\mathcal{E}}^{\rm int}$ can be found, we can approximate the coexistence criteria.
• Figure 3: Theoretical and numerical phase diagrams of the considered MAMB models. NRCH models with (a) $\overline{\kappa}_{AB} (\chi - \alpha) = \overline{\kappa}_{BA} (\chi + \alpha)$, where $\boldsymbol{\mathcal{E}}^{\rm bulk}=\boldsymbol{\mathcal{E}}^{\rm int}$ and (b) $\overline{\kappa}_{AB} (\chi - \alpha) \neq \overline{\kappa}_{BA} (\chi + \alpha)$, where $\boldsymbol{\mathcal{E}}^{\rm bulk} \neq \boldsymbol{\mathcal{E}}^{\rm int}$. For both (a) and (b) we set $a=-1$, $\chi = -0.5$, $\alpha = -0.2$, and $\overline{\kappa}_{AA} = \overline{\kappa}_{BB} = 1/(6\pi)$, with $\overline{\kappa}_{AB} = \overline{\kappa}_{BA} = 0$ for (a) and $\overline{\kappa}_{AB} = \overline{\kappa}_{BA} = 0.01/(6\pi)$ for (b). (c) A MAMB system which results in an $\boldsymbol{\mathcal{E}}$ with nonlinear elements, achieved by introducing $\lambda_{AAA}=0.01$, $\chi = - \alpha = -1$ with $a= -1.5$, $\overline{\kappa}_{AA} = \overline{\kappa}_{BB} = 1/8\pi$, and $\overline{\kappa}_{AB} = \overline{\kappa}_{BA} = \lambda_{AAB} = \lambda_{ABB} = \lambda_{ABA} = 0$.