Critical Motility-Induced Phase Separation in Three Dimensions is Consistent with Ising Universality

TL;DR

This work demonstrates that the critical point of three-dimensional motility-induced phase separation (MIPS) in active Brownian particles falls into the $3D$ Ising universality class with a conserved order parameter, supported by large-scale simulations and finite-size scaling. The dynamic critical exponent is found to be $z \approx 3.964$, consistent with Model B dynamics, while a coarse-grained fluctuating-hydrodynamics analysis maps the system to AMB+ and places it in the attractive basin of the Wilson–Fisher fixed point. The combination of numerical evidence and RG analysis suggests that 3D MIPS shares the universality class of passive fluids, though the proximity to a separatrix implies that modest changes in microscopic parameters could, in principle, access a strong-coupling regime and potential novel behavior. Overall, the study reinforces Ising-like criticality in 3D active matter and highlights a framework for assessing universality by linking particle-based simulations with continuum RG descriptions.

Abstract

Identifying the universality class of critical active phase transitions has been the subject of recent interest and controversy. Resolving these controversies will require robust numerical investigations to determine whether active critical exponents point to novel universality classes or are consistent with established ones. Here, we conduct large-scale computer simulations and a finite-size scaling analysis of the motility-induced phase separation (MIPS) of active Brownian hard spheres in three dimensions (3D), finding that the static and dynamic critical exponents all closely match those of the 3D Ising universality class with a conserved scalar order parameter. This finding is corroborated by a fluctuating hydrodynamic description of the critical dynamics of the order parameter field which flows to the Wilson-Fisher fixed point in three dimensions. Our work suggests that 3D MIPS and likely the entire phase diagram of active Brownian hard spheres is similar to that of molecular passive fluids despite the absence of Boltzmann statistics.

Figures (3)

• Figure 1: Finite-size scaling analysis for determining static critical exponents. System size and activity dependence of (a) $\mathcal{B}$, (b) $\chi$, and (c) $m$. (d) Collapse of data in (a) as a function of the scaling variable $\lambda \overline{L}^{1/\nu}$ with $\nu=0.630$ [Inset: Slope of $\mathcal{B}$ at $\lambda=0$ ($\ell_0=\ell_{\rm c}$) as a function of system size, where the dashed line is the best fit of $1/\nu=1.302(0.076)$ and the solid line is $1/\nu=1.587$]. (e) Collapse of data in panel (b) with the exponents $\gamma=1.237$ and $\nu=0.630$ [Inset: $\chi$ at $\lambda=0$ as a function of system size, where the dashed line is the best fit of $\gamma/\nu=2.003(0.083)$ and the solid line is $\gamma/\nu=1.963$]. (f) Collapse of data in panel (c) with the exponents $\beta=0.326$ and $\nu=0.630$ [Inset: $m$ at $\lambda=0$ as a function of system size, where the dashed line is the best fit of $\beta/\nu=0.556(0.068)$ and the solid line is $\beta/\nu=0.518$]. We define $\overline{L}=L/L_0$ where $L_0/\sigma \approx 9.2$ is the size of the smallest sub-box ($N=1.6 \times 10^4$).
• Figure 2: The correlation length $\xi(t)$ at criticality ($\lambda=0$, purple circles) and inside phase-separation regime ($\lambda\approx 0.59$, green triangles), as determined from the inverse of the first moment of the static structure factor. Each point represents the average of ten consecutive data points, computed at intervals of $\sigma/U_0$. The purple line shows $t^{1/z}$, where $z=4-\eta\approx 3.964$ for model B in 3D. The green line shows $t^{1/3}$ which is the classical scaling for Ostwald ripening and/or coalescence. The red dashed line denotes the upper limit for $\xi$, which is the system size $L$.
• Figure 3: Numerical integration of the full RG flow in Ref. Caballero18 obtained for different initial conditions, all of them with initial parameters $\overline{\nu}=2.6$ and $\overline{\lambda}=3.25$ in $d=3$. The flow is represented in the $(\overline{\zeta}, \overline{u})$ plane. A separatrix (green dashed line) between two different regimes appears: when $\overline{\zeta}$ is small (red lines with arrow), the flow converges back to the Wilson-Fisher fixed point (we checked that $\overline{\nu}$ and $\overline{\lambda}$ are flowing to 0 in this case). On the right of the separatrix (blue lines with arrow), the flow diverges to infinity (strong coupling). The nontrivial $F_4$ fixed point resides on the separatrix, which has codimension one and thus represents a weak-to-strong coupling phase transition. The critical MIPS in our fluctuating hydrodynamics model lies on the orange dotted line (at $\overline{u} \gg 30$) which is to the left of the separatrix. The Gaussian fixed point lies at the origin.