Active Model B+: Nonequilibrium Phase Ordering

• Active Model B+ is a continuum field theory for phase ordering in active matter that extends passive Model B by introducing nonequilibrium gradient nonlinearities and a non-potential interfacial current.
• It enables the simulation of phenomena such as reverse Ostwald ripening, arrested coarsening, and microphase separation by breaking time-reversal symmetry and creating circulating vortex-like currents at interfaces.
• Analytical, numerical, and functional-renormalization-group studies establish AMB+ as a minimal RG-complete framework that bridges passive phase separation dynamics with emergent active matter behaviors.

Searching arXiv for recent and related work on Active Model B+. Active Model B+ (AMB+) is a continuum field theory for phase ordering in active matter with a conserved scalar order parameter, used in particular as a coarse-grained description of motility-induced phase separation (MIPS). It generalizes passive Model B (the Cahn–Hilliard dynamics) by adding nonequilibrium gradient nonlinearities and, crucially, a non-potential current that breaks detailed balance and time-reversal symmetry. In consequence, AMB+ retains the conserved transport structure of passive phase separation while permitting circulating interfacial currents, reverse Ostwald ripening, arrested coarsening, and microphase separation. Across recent numerical, analytical, and functional-renormalization-group studies, it has emerged as a minimal RG-complete active extension of Model B and as a reference model for distinguishing macroscale phase separation from intrinsically nonequilibrium microstructured steady states (Yadav et al., 17 Jun 2025, Bhowmick et al., 8 Apr 2026, Fejős et al., 16 Aug 2025).

1. Constitutive structure and relation to Model B

AMB+ evolves a conserved scalar field through a continuity equation,

$\partial_t \phi = -\nabla\cdot \mathbf{J},$

or, in an alternative notation used for critical-composition coarsening, $\partial_t \psi = -\nabla\cdot \mathbf{J}$. In passive Model B the current is purely diffusive,

$$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$

with Landau–Ginzburg free energy

$$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$

so that $\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$ in the notation of the critical-scaling study, or $\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$ in the coarsening study (Yadav et al., 17 Jun 2025, Bhowmick et al., 8 Apr 2026).

The AMB current keeps the current potential, but adds active terms inside $-\nabla[\cdots]$. AMB+ augments this further by a genuinely non-potential term. In the minimal AMB+ formulation,

$$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$

The coefficients $\lambda$, $\nu$, and $\partial_t \psi = -\nabla\cdot \mathbf{J}$0 parameterize activity. The $\partial_t \psi = -\nabla\cdot \mathbf{J}$1-term cannot be written as $\partial_t \psi = -\nabla\cdot \mathbf{J}$2; it explicitly breaks time-reversal symmetry and encodes circulating interfacial currents. In the reduced parametrization used for deterministic coarsening below $\partial_t \psi = -\nabla\cdot \mathbf{J}$3,

$\partial_t \psi = -\nabla\cdot \mathbf{J}$4

where $\partial_t \psi = -\nabla\cdot \mathbf{J}$5 is the strength of a rotation-free current and $\partial_t \psi = -\nabla\cdot \mathbf{J}$6 the strength of a rotational current (Yadav et al., 17 Jun 2025, Bhowmick et al., 8 Apr 2026).

A central physical distinction from passive Model B is that in $\partial_t \psi = -\nabla\cdot \mathbf{J}$7,

$\partial_t \psi = -\nabla\cdot \mathbf{J}$8

so AMB+ supports vortex-like interfacial transport localized at interfaces. This separates it both from Model B, which is relaxational, and from AMB, where active corrections remain potential at this order. In the RG analysis, the active couplings are marginal in $\partial_t \psi = -\nabla\cdot \mathbf{J}$9, which underlies logarithmically slow flow of the effective couplings and the associated corrections to scaling (Yadav et al., 17 Jun 2025, Fejős et al., 16 Aug 2025).

2. Interfaces, coexistence, and nonequilibrium binodals

A distinctive feature of AMB+ is that interface structure and coexistence are already modified in one dimension. In the reduced $$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$0 description, the active contributions collapse into the combination

$$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$1

For a static kink profile $$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$2, the interface equation becomes

$$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$3

with a constant static chemical potential $$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$4. For small $$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$5,

$$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$6

and the saturation values satisfy

$$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$7

Hence $$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$8 enriches the $$\mathbf{J}_{\rm MB} = -M \nabla \mu_{\rm eq}, \qquad \mu_{\rm eq}=\frac{\delta \mathcal{F}}{\delta \phi},$$9 phase, whereas $$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$0 enriches the $$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$1 phase. This asymmetry is present even at critical composition $$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$2, and it underlies the observed crossover from bicontinuous to droplet-in-matrix morphologies during coarsening (Yadav et al., 17 Jun 2025).

In the more general $$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$3 AMB+ formulation, coexistence across a flat interface is obtained by a generalized equal-area construction. Writing the effective one-dimensional stationary chemical potential as

$$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$4

with

$$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$5

one introduces a pseudodensity $$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$6 satisfying

$$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$7

and a pseudopotential $$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$8 through $$\mathcal{F}[\phi]=\int d^dr\left[\frac{r}{2}\phi^2+\frac{u}{4}\phi^4+\frac{\kappa}{2}|\nabla\phi|^2\right],$$9. The coexistence conditions are then

$\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$0

together with equality of the pseudopressure,

$\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$1

This construction recovers the binodal densities in excellent agreement with direct numerics. Within the parameter scans reported, activity chiefly changes the binodal width and breaks the $\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$2 symmetry, while leaving the apex near $\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$3 (Bhowmick et al., 8 Apr 2026).

A common misconception is that conserved scalar models at zero global mean must remain morphologically symmetric. AMB+ shows that this is not generally true: conserved mean does not preclude phase-amplitude asymmetry once nonequilibrium interfacial terms shift the coexistence values.

3. Linear instability, critical dynamics, and finite-size scaling

At linear order around a homogeneous state, the active current terms do not contribute because they are quadratic in gradients. The linearized dispersion relation is therefore the same as in passive Model B,

$\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$4

or equivalently $\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$5 in density notation. The homogeneous phase is unstable for $\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$6, or $\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$7 in the density formulation. Thus activity does not alter the spinodal condition at this truncation; its effects enter through nonlinear transport, coexistence asymmetry, and interfacial currents (Bhowmick et al., 8 Apr 2026, Vrugt et al., 2022).

At criticality $\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$8, deterministic two-dimensional simulations of both AMB and AMB+ show identical mean-field critical scaling. Using

$\mu_{\rm eq}=r\phi+u\phi^3-\kappa\nabla^2\phi$9

the order parameter decays as

$\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$0

with dynamic exponent $\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$1. Finite-size scaling takes the form

$\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$2

and in the supercritical steady state $\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$3, where $\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$4 and $\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$5. The same analysis gives $\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$6, $\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$7, $\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$8, and $\mu_{\rm eq}=a(T-T_c)\psi+b\psi^3-K\nabla^2\psi$9 in the mean-field description reported there (Bhowmick et al., 8 Apr 2026).

Finite size shifts the apparent critical point because the smallest periodic mode $-\nabla[\cdots]$0 is stabilized unless $-\nabla[\cdots]$1. This yields

$-\nabla[\cdots]$2

so $-\nabla[\cdots]$3. The consequence is an extended algebraic regime $-\nabla[\cdots]$4 at $-\nabla[\cdots]$5 that eventually crosses over at long times in finite systems (Bhowmick et al., 8 Apr 2026).

These results delimit one robust aspect of AMB+: the nonequilibrium current does not alter the reported mean-field critical exponents in $-\nabla[\cdots]$6, even though it strongly modifies nonlinear ordering kinetics away from criticality.

4. Macroscale phase separation, microphase separation, and arrest

The most extensively documented nonequilibrium effects of AMB+ occur in phase-ordering kinetics after a quench. In the deterministic two-dimensional coarsening study at critical composition, AMB+ exhibits either macroscale phase separation (MPS), in which the domain size $-\nabla[\cdots]$7 diverges, or microscale phase separation ($-\nabla[\cdots]$8PS), in which coarsening arrests at a finite steady length $-\nabla[\cdots]$9 (Yadav et al., 17 Jun 2025).

For MPS, the characteristic length is defined from the equal-time correlation function

$$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$0

via the first zero of its spherical average $$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$1. In the parameter sets studied, such as $$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$2 with $$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$3 and $$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$4 with $$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$5, the system lies in the forward Ostwald region. The kinetics shows an early diffusive regime

$$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$6

with bicontinuous morphology and a scaling function matching Model B, followed asymptotically by

$$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$7

when rotational interfacial currents dominate transport. The effective exponent $$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$8 crosses smoothly from approximately $$\frac{\mathbf{J}}{M} = -\nabla\!\left[\mu_{\rm eq}+\lambda |\nabla\phi|^2+\frac{\nu}{2}\nabla^2(\phi^2)\right] +\zeta (\nabla^2\phi)\nabla\phi.$$9 to approximately $\lambda$0, and the crossover time obeys

$\lambda$1

for $\lambda$2 in the range studied. At the same time the morphology crosses over from bicontinuous to droplet-like, consistent with $\lambda$3 (Yadav et al., 17 Jun 2025).

For $\lambda$4PS, the parameter sets $\lambda$5 with $\lambda$6 lie in the reverse Ostwald region. Here $\lambda$7 initially grows and then saturates, the saturation time decreases with increasing $\lambda$8 at fixed $\lambda$9, and increases with $\nu$0 at fixed $\nu$1. Over the accessible range,

$\nu$2

while $\nu$3 versus $\nu$4 is nonmonotonic and diverges as $\nu$5. The arrested state in $\nu$6 is a hexagonal droplet lattice. The proposed mechanism is that reverse Ostwald ripening first drives the droplet population toward a monodisperse size, after which alternating clockwise and anticlockwise interfacial-current loops cancel mass transfer between neighbors and pin the structure at $\nu$7 (Yadav et al., 17 Jun 2025).

The geometry of the arrested state is constrained by conservation. For a hexagonal tiling with droplet radius $\nu$8, cell side $\nu$9, and coexistence values $\partial_t \psi = -\nabla\cdot \mathbf{J}$00, the reported relation is

$\partial_t \psi = -\nabla\cdot \mathbf{J}$01

hence

$\partial_t \psi = -\nabla\cdot \mathbf{J}$02

This expresses the lattice spacing in terms of the arrested droplet size and the nonequilibrium phase asymmetry (Yadav et al., 17 Jun 2025).

A second deterministic study of AMB+ emphasizes a complementary supercritical-coarsening scenario. Using the structure-factor length

$\partial_t \psi = -\nabla\cdot \mathbf{J}$03

it finds

$\partial_t \psi = -\nabla\cdot \mathbf{J}$04

for supercritical quenches in $\partial_t \psi = -\nabla\cdot \mathbf{J}$05, consistent with functional-RG expectations for marginal couplings, while for sufficiently large positive $\partial_t \psi = -\nabla\cdot \mathbf{J}$06 coarsening saturates and a long-lived microphase-separated state appears. In that parametrization, the crossover is controlled by

$\partial_t \psi = -\nabla\cdot \mathbf{J}$07

and for $\partial_t \psi = -\nabla\cdot \mathbf{J}$08, $\partial_t \psi = -\nabla\cdot \mathbf{J}$09, one has $\partial_t \psi = -\nabla\cdot \mathbf{J}$10 when $\partial_t \psi = -\nabla\cdot \mathbf{J}$11 (Bhowmick et al., 8 Apr 2026).

Taken together, these studies do not support a single universal late-time growth scenario for all AMB+ regimes. A plausible implication is that the observed asymptotics depend sensitively on parametrization, phase-space sector, and on whether the dominant mechanism is marginally renormalized diffusive transport or explicitly rotational interfacial-current transport.

5. Dynamic scaling, structure, and renormalization-group organization

When a single length scale governs morphology, the standard scaling form

$\partial_t \psi = -\nabla\cdot \mathbf{J}$12

is expected. In AMB+, good collapse is reported both in early and late stages. At early times in the bicontinuous regime, the scaling function is superuniversal and matches Model B irrespective of $\partial_t \psi = -\nabla\cdot \mathbf{J}$13. At late times in MPS, however, the scaling function depends on $\partial_t \psi = -\nabla\cdot \mathbf{J}$14 and $\partial_t \psi = -\nabla\cdot \mathbf{J}$15 because the asymmetry between the coexisting values makes the morphology effectively off-critical. In the saturated $\partial_t \psi = -\nabla\cdot \mathbf{J}$16PS regime, the scaled correlation function exhibits oscillations reflecting the crystalline, hexagonal arrangement and is approximately superuniversal across the studied parameter sets at fixed late time (Yadav et al., 17 Jun 2025).

The sharp-interface character of the domains yields Porod behavior. For small $\partial_t \psi = -\nabla\cdot \mathbf{J}$17,

$\partial_t \psi = -\nabla\cdot \mathbf{J}$18

and the structure factor satisfies

$\partial_t \psi = -\nabla\cdot \mathbf{J}$19

with $\partial_t \psi = -\nabla\cdot \mathbf{J}$20 in the reported simulations. The observed linear decay of $\partial_t \psi = -\nabla\cdot \mathbf{J}$21 at small $\partial_t \psi = -\nabla\cdot \mathbf{J}$22 is consistent with this sharp-interface regime (Yadav et al., 17 Jun 2025).

The field-theoretic RG treatment places these kinetic observations in a broader scaling framework. Using an MSRJD effective action and the Wetterich flow, the FRG study derives $\partial_t \psi = -\nabla\cdot \mathbf{J}$23-functions for all couplings in generic $\partial_t \psi = -\nabla\cdot \mathbf{J}$24, identifies invariant subspaces corresponding to AMB, equilibrium models, Model B, cKPZ+, and a new $\partial_t \psi = -\nabla\cdot \mathbf{J}$25 class with fixed $\partial_t \psi = -\nabla\cdot \mathbf{J}$26, and confirms a bicritical fixed point conjectured to control the transition between bulk phase separation and microphase separation (Fejős et al., 16 Aug 2025). It also reports regulator-independent contributions at special dimensions, notably triangle contributions in $\partial_t \psi = -\nabla\cdot \mathbf{J}$27 and bubble contributions in $\partial_t \psi = -\nabla\cdot \mathbf{J}$28, consistent with perturbative $\partial_t \psi = -\nabla\cdot \mathbf{J}$29-expansion expectations.

The global FRG flows differ qualitatively from perturbative momentum-shell RG. In particular, around the bicritical AMB+ fixed point $\partial_t \psi = -\nabla\cdot \mathbf{J}$30, FRG finds no trajectories reaching the Wilson–Fisher fixed point in $\partial_t \psi = -\nabla\cdot \mathbf{J}$31; flows either run to strong coupling or encounter the singular locus $\partial_t \psi = -\nabla\cdot \mathbf{J}$32. On that basis, the study argues that the microphase-separation transition is not second order. At the same time, the FRG uncovers an additional fixed point in the equilibrium subspace that prevents the runaways seen in perturbative RG for large $\partial_t \psi = -\nabla\cdot \mathbf{J}$33 (Fejős et al., 16 Aug 2025).

This RG picture clarifies two points. First, AMB+ contains a bona fide nonequilibrium sector not reducible to passive Model B by parameter renormalization. Second, the distinction between macro- and microphase separation is organized by flow topology in coupling space, not merely by a shift of coexistence densities.

6. Extensions, numerical practice, and open problems

The recent numerical literature on AMB+ is predominantly deterministic and two-dimensional. One coarsening study uses a square periodic $\partial_t \psi = -\nabla\cdot \mathbf{J}$34 grid with $\partial_t \psi = -\nabla\cdot \mathbf{J}$35, Euler time stepping, central spatial differences, $\partial_t \psi = -\nabla\cdot \mathbf{J}$36, and $\partial_t \psi = -\nabla\cdot \mathbf{J}$37, starting from random fluctuations $\partial_t \psi = -\nabla\cdot \mathbf{J}$38 at critical composition and averaging typically over 50 runs. Another uses a pseudo-spectral Euler scheme with periodic boundary conditions, $\partial_t \psi = -\nabla\cdot \mathbf{J}$39, $\partial_t \psi = -\nabla\cdot \mathbf{J}$40, and system sizes up to $\partial_t \psi = -\nabla\cdot \mathbf{J}$41 depending on the observable (Yadav et al., 17 Jun 2025, Bhowmick et al., 8 Apr 2026). These implementations provide the current quantitative basis for the reported growth laws, scaling collapses, and arrested lengths.

AMB+ also serves as the overdamped limit of a broader inertial theory, Active Model I+ (AMI+). In that extension, the hydrodynamic variables are density and velocity, the density-dependent swim speed

$\partial_t \psi = -\nabla\cdot \mathbf{J}$42

generates an effective viscosity

$\partial_t \psi = -\nabla\cdot \mathbf{J}$43

and the thermodynamic and mechanical velocities no longer coincide. In the overdamped limit one recovers AMB+ with $\partial_t \psi = -\nabla\cdot \mathbf{J}$44, activity-renormalized bulk free energy, and non-potential current coefficients mapped from the inertial theory. Under a special choice of parameters, the inertial model reduces to Madelung hydrodynamics and hence to the Schrödinger equation, which the authors use to discuss an active analogue of tunneling and analogies with fuzzy dark matter (Vrugt et al., 2022).

Several open problems remain explicit in the current literature. The role of conserved noise is unresolved in both the coarsening and critical-scaling studies, which set $\partial_t \psi = -\nabla\cdot \mathbf{J}$45. Three-dimensional morphologies and kinetics have not been systematically explored in the reported AMB+ coarsening work. Finite-size effects become severe when $\partial_t \psi = -\nabla\cdot \mathbf{J}$46 is large, while large activity can induce numerical instabilities in fixed-grid schemes. Analytical predictions for the amplitudes of logarithmic corrections, for the full parameter dependence of $\partial_t \psi = -\nabla\cdot \mathbf{J}$47, and for late-time scaling functions remain incomplete. The FRG study further indicates that stronger truncations may be needed to determine whether the singular locus $\partial_t \psi = -\nabla\cdot \mathbf{J}$48 is a physical obstruction or a truncation artifact (Yadav et al., 17 Jun 2025, Bhowmick et al., 8 Apr 2026, Fejős et al., 16 Aug 2025).

Within this developing landscape, the central lesson is stable: the non-potential interfacial current is the defining ingredient of AMB+. It preserves conserved scalar dynamics while enabling interfacial circulation, reverse Ostwald ripening, arrested coarsening, and nonequilibrium coexistence structures that are inaccessible to passive Model B and only partially accessible to AMB.