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arxiv: 2604.07247 · v1 · submitted 2026-04-08 · ❄️ cond-mat.stat-mech · cond-mat.soft

Critical scaling and supercritical coarsening in Active Model B+

Abir Bhowmick , P. K. Mohanty This is my paper

Pith reviewed 2026-05-10 17:38 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft
keywords active model Bcritical dynamicsphase orderingcoarseningnonequilibrium systemsmicrophase separationlogarithmic corrections
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0 comments X

The pith

Despite nonequilibrium currents, Active Model B and AMB+ show the same mean-field critical scaling as equilibrium systems, with supercritical domain growth modified by logarithmic corrections.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines critical dynamics and phase-ordering kinetics in Active Model B and its minimal extension Active Model B+ through deterministic two-dimensional simulations. At the critical point r_c=0 both models exhibit identical mean-field behavior, with the order parameter decaying as m(t) ~ t^{-1/4} and dynamical exponent z=4, even though active currents are present. For quenches above criticality the characteristic domain size grows as L(t) ~ t^{1/3}(1 + c / ln t), a form that includes logarithmic corrections to the classic Lifshitz-Slyozov law and matches functional renormalization group expectations for marginal activity in two dimensions. In AMB+ the additional active current suppresses these corrections and eventually arrests coarsening by stabilizing a long-lived microphase-separated state.

Core claim

At criticality r_c=0, both AMB and AMB+ display identical mean-field scaling despite nonequilibrium currents, with order-parameter decay m(t)∼t^{-1/4} and dynamical exponent z=4. A generalized equal-area construction yields the binodal densities and phase diagram of AMB+. For supercritical quenches, domain size grows as L(t)∼t^{1/3}(1+c/ln t), revealing logarithmic corrections; these corrections are prominent in AMB but suppressed in AMB+ where growth is arrested in a microphase-separated state.

What carries the argument

Deterministic time-evolution simulations in two dimensions that track order-parameter decay and domain-size evolution, together with a generalized equal-area construction that locates the coexistence densities for AMB+.

Load-bearing premise

Deterministic two-dimensional simulations without stochastic noise or finite-size corrections can extract the universal scaling exponents and asymptotic growth laws.

What would settle it

If stochastic simulations that include noise produce a different order-parameter decay exponent or eliminate the logarithmic correction to domain growth, the reported mean-field scaling and coarsening laws would not hold.

Figures

Figures reproduced from arXiv: 2604.07247 by Abir Bhowmick, P. K. Mohanty.

Figure 2
Figure 2. Figure 2: FIG. 2. (a) Log-log plot of order parameter [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Scaling collapse in AMB ( [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. (a) Binodal diagram in the [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Active Model B ( [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Active Model B [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
read the original abstract

We study critical dynamics and phase-ordering kinetics in Active Model B (AMB) and its minimal extension, Active Model B$+$ (AMB$+$), using deterministic simulations in two dimensions. At criticality $r_c=0$, both models display identical mean-field scaling despite nonequilibrium currents, with order-parameter decay with time as $m(t)\sim t^{-\alpha}$, with $\alpha=\frac14$, and dynamical exponent being $z=4$. A generalized equal-area construction yields the binodal densities and phase diagram of AMB$+$. For supercritical quenches, domain size grows as $L(t)\sim t^{1/3}(1+c/\ln t)$, revealing logarithmic corrections to the classic $t^{1/3}$ growth-law; moreover it is consistent with the functional renormalization group predictions for marginal activity in $d=2$. While the logarithmic corrections are quite prominent in AMB, in AMB$+$ they are suppressed as the active current acts against the formation of macro-clusters; the growth is eventually arrested when a long-lived microphase-separated state appears.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript examines critical dynamics and phase-ordering kinetics in Active Model B (AMB) and its extension Active Model B+ (AMB+) using deterministic two-dimensional simulations. At criticality (r_c=0), both models exhibit identical mean-field scaling with order-parameter decay m(t)∼t^{-1/4} and dynamical exponent z=4 despite nonequilibrium currents. A generalized equal-area construction determines the binodal densities and phase diagram for AMB+. For supercritical quenches, domain growth follows L(t)∼t^{1/3}(1+c/ln t) with logarithmic corrections consistent with functional renormalization group predictions for marginal activity in d=2; these corrections are prominent in AMB but suppressed in AMB+, eventually arresting growth in a long-lived microphase-separated state.

Significance. If the reported scalings hold under proper stochastic conditions, the work would provide direct numerical evidence that mean-field critical exponents persist in active systems and that activity modulates coarsening via logarithmic corrections, offering support for FRG predictions in two dimensions. The generalized equal-area rule and the contrast between AMB and AMB+ growth arrest represent concrete advances in characterizing nonequilibrium phase separation.

major comments (2)
  1. [Abstract and simulation methodology] Abstract and simulation methodology: the claims of exact exponents α=1/4, z=4 at r_c=0 and the functional form L(t)∼t^{1/3}(1+c/ln t) rest on deterministic integration of the AMB/AMB+ PDEs. The underlying models are stochastic field theories; additive noise is required to access the fluctuation-driven asymptotic regime and to realize marginal-operator renormalization in d=2. Deterministic runs risk capturing only transient or initial-condition-dependent behavior rather than universal scaling, and no finite-size scaling analysis or error bars are indicated to control 2D corrections or numerical dissipation.
  2. [Supercritical coarsening] Supercritical coarsening section: the generalized equal-area construction for binodals and the claim that activity suppresses logarithmic corrections (leading to microphase arrest in AMB+) are untested against stochastic noise. Without noise, it is unclear whether the observed arrest and the specific log-correction coefficient c are robust or artifacts of the deterministic dynamics.
minor comments (1)
  1. [Abstract] The abstract reports precise functional forms and exponents without referencing system sizes, integration times, or validation against the passive Model B limit; adding these details would strengthen reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. Our manuscript explicitly employs deterministic simulations of the AMB and AMB+ equations, as stated in the abstract and methods. We address the concerns regarding stochastic effects and robustness below, while maintaining the focus on the deterministic dynamics that reveal mean-field scaling and activity-modulated coarsening.

read point-by-point responses

  1. Referee: Abstract and simulation methodology: the claims of exact exponents α=1/4, z=4 at r_c=0 and the functional form L(t)∼t^{1/3}(1+c/ln t) rest on deterministic integration of the AMB/AMB+ PDEs. The underlying models are stochastic field theories; additive noise is required to access the fluctuation-driven asymptotic regime and to realize marginal-operator renormalization in d=2. Deterministic runs risk capturing only transient or initial-condition-dependent behavior rather than universal scaling, and no finite-size scaling analysis or error bars are indicated to control 2D corrections or numerical dissipation.

    Authors: We agree that AMB and AMB+ are stochastic field theories and that additive noise is required for the full fluctuation-driven asymptotic regime. Our study, however, deliberately uses deterministic integration to isolate the mean-field critical dynamics and deterministic coarsening kinetics. In this limit we observe the mean-field exponents α=1/4 and z=4 at r_c=0, demonstrating that these scalings persist even without noise despite the presence of nonequilibrium currents. The logarithmic corrections to L(t)∼t^{1/3} are likewise obtained deterministically and remain consistent with FRG predictions for marginal activity in d=2. We acknowledge the absence of finite-size scaling analysis and error bars in the present version. In the revised manuscript we will add finite-size scaling checks across multiple system sizes together with error estimates on the fitted exponents and the coefficient c, thereby controlling for 2D corrections and numerical dissipation while preserving the deterministic character of the reported results. revision: partial

  2. Referee: Supercritical coarsening section: the generalized equal-area construction for binodals and the claim that activity suppresses logarithmic corrections (leading to microphase arrest in AMB+) are untested against stochastic noise. Without noise, it is unclear whether the observed arrest and the specific log-correction coefficient c are robust or artifacts of the deterministic dynamics.

    Authors: The generalized equal-area construction follows directly from the steady-state deterministic equations and yields the binodal densities for AMB+. In the deterministic supercritical regime we find that active currents in AMB+ suppress the logarithmic corrections relative to AMB and ultimately arrest coarsening into a long-lived microphase-separated state. While we recognize that stochastic noise could modify the long-time behavior and the precise value of c, the deterministic contrast between AMB and AMB+ illustrates how activity modulates coarsening even in the absence of fluctuations. In the revision we will add an explicit discussion noting that these findings pertain to the deterministic limit and that full stochastic simulations would be required to assess robustness in the complete model. This addition clarifies the scope without changing the reported deterministic results. revision: partial

Circularity Check

0 steps flagged

No significant circularity; scaling results obtained via direct simulation

full rationale

The paper's central results for critical scaling (m(t)∼t^{-1/4}, z=4) and supercritical coarsening (L(t)∼t^{1/3}(1+c/ln t)) are extracted from deterministic numerical integration of the AMB/AMB+ PDEs. These quantities are not defined in terms of themselves, nor are they obtained by fitting a parameter to a subset of data and then relabeling the fit as a prediction. The generalized equal-area construction and consistency with external FRG predictions are presented as independent checks rather than load-bearing self-referential steps. No self-citation chain, ansatz smuggling, or renaming of known results reduces the derivation to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The paper extends the established Active Model B framework with a minimal addition; no new free parameters, axioms beyond standard model assumptions, or invented entities are introduced in the abstract.

pith-pipeline@v0.9.0 · 5492 in / 1247 out tokens · 100244 ms · 2026-05-10T17:38:11.329742+00:00 · methodology

discussion (0)

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