{"id":"128b040a-1ea5-4aed-959d-8c9122ab498f","arxiv_id":"2506.14548","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"In numerical simulations of Active Model B+, the rotational activity term changes coarsening from a t^(1/3) law to a slower t^(1/4) law during macroscale phase separation, and causes droplet size saturation in a hexagonal pattern during microscale phase separation.","lead":"This paper simulates a standard model of active particle phase separation and finds that adding a circulating-current term changes how droplets grow: larger droplets still win, but growth slows from the classical one-third to a one-quarter power law. It also finds a regime where droplets stop coarsening and settle into a regular hexagonal pattern.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The MPS claim of an asymptotic t^(1/4) growth law rests on effective-exponent data that have not plateaued by t=10^6, making the headline exponent an extrapolation from a non-monotonic transient rather than a measured asymptotic value.","rationale":"The reader's weakest_assumption concerns the use of 1D static-kink saturation values in 2D microPS morphologies. That concern is valid but does not threaten the two quantitative scaling laws that form the paper's central claim: the t^(1/4) MPS law and the Ls ~ (-ξ)^(-2/3) microPS law are empirical fits that do not rely on ψ1 and ψ2. The more load-bearing issue is the determination of the asymptotic exponent in MPS, where the effective exponent is still approaching 1/4 from below at the final simulation time and has not demonstrated a plateau. The reader's rationale does flag the limited time window and transient dip, so my concern is partially anticipated even though it was not named as the weakest assumption. The qualitative features—slower-than-LS growth in MPS and droplet-crystal saturation in microPS—are plausible and visually supported, so a rejection is not warranted. The CONDITIONAL verdict already captures the need for stronger evidence; a longer, larger-scale run would settle whether the exact t^(1/4) and t_c ~ ξ^(-2) claims survive. No code or data are provided, which makes the proposed computational check the appropriate path to verification.","tokens_in":13995,"tokens_out":11659,"duration_ms":124822,"concrete_test":"Repeat the λ=0, ξ=-2 and ξ=-1 cases on a 1024^2 lattice with Δx=0.5 (physical size 512) to t=10^7, computing L(t) and 1/z_eff in sliding log-time windows. The MPS claim is supported only if the last two decades show 1/z_eff plateauing at 0.250±0.01 with no systematic drift; also refit t_c from the same local-exponent curves and check whether t_c ~ ξ^(-2) survives for ξ=-1, -2, -3, -4. If the plateau is absent or the slope changes, the claimed asymptotic exponent and crossover scaling should be downgraded.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central MPS claim—that the asymptotic growth law is exactly L(t) ~ t^(1/4)—rests on effective-exponent data (Eq. (15), Figs. 2(b)–2(e)) that have not reached a plateau by the end of the run. For ξ<0, 1/z_eff dips below 1/4 for t>t_c and is still rising toward 1/4 at t=10^6; the claimed asymptotic value is therefore inferred from a non-monotonic transient rather than a measured plateau. The crossover time t_c is defined as the first crossing of the 1/4 line, which is arbitrary if the local exponent is non-monotonic, so the t_c ~ ξ^(-2) scaling in Fig. 3 inherits this arbitrariness and is fit to only a few ξ values. The proposed loop-transport mechanism gives t^(1/4) only through a heuristic τ ~ t/L argument, leaving the numerics as the only direct evidence for the exponent. If a longer run shows the late-time local exponent settling away from 0.25, or if finite-size effects appear by L~64 on the N=512 grid, the paper's headline MPS result would need revision.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript reports Euler-discretized numerical solutions of Active Model B+ (AMB+) for critical quenches in two dimensions. For parameters assigned to forward Ostwald ripening (macroscale phase separation, MPS), it claims a crossover from Lifshitz-Slyozov growth L(t) ~ t^(1/3) to an asymptotic slower law L(t) ~ t^(1/4), with crossover time tc ~ |xi|^(-2), and it attributes the slowdown to circulating interfacial current loops. For parameters assigned to reverse Ostwald ripening (microscale phase separation, muPS), it claims that the system saturates to a hexagonal crystal of monodisperse droplets with saturation length Ls satisfying Ls ~ (-xi)^(-2/3), and it derives a conservation-law relation between the hexagonal lattice spacing and Ls. The paper also reports dynamical scaling of the correlation function in both regimes, with a superuniversal early-time form and a parameter-dependent late-time form in MPS, and a crystalline oscillatory scaling function in muPS.","tokens_in":14194,"tokens_out":5805,"duration_ms":64075,"significance":"If the MPS exponent and the muPS saturation scaling are correct, the paper is a valuable extension of phase-separation kinetics to a minimal active field theory, showing that the rotational active current can change the growth law and produce a steady-state microphase-separated crystal. The numerical protocol is standard and clearly specified (N=512, dx=dy=0.5, dt=0.01, averages over 50 runs for the main data sets), and the use of effective exponents, correlation functions, and current-field visualization is transparent. The authors are also honest about limitations, explicitly noting the restricted parameter range for the Ls data and the lack of a predictive theory for Ls. However, the central quantitative claims are empirical and the asymptotic regime is not fully demonstrated: the effective exponent for the MPS case has not plateaued by the largest simulation time, and the Ls scaling is fitted over a narrow range. The significance is therefore conditional on the additional numerical evidence discussed below.","major_comments":[{"comment":"The claimed asymptotic growth law L(t) ~ t^(1/4) is not established by the presented data. For xi < 0, the inverse effective exponent 1/z_eff initially follows 1/3, then dips below 1/4, and only approaches 1/4 from below at the largest times; there is no plateau by t = 10^6. Since the asymptotic value is inferred from a non-monotonic transient, the paper should either extend the simulations to longer times (with a check for finite-size effects on the N = 512 lattice) or use an alternative analysis, for example windowed log-log slopes with an extrapolation procedure, that demonstrably converges to a plateau. Without this, the central MPS claim remains an interpretation of the final decade of data rather than a measured asymptotic exponent.","section":"Section III.A, Figs. 2(b)-2(e)"},{"comment":"The loop-transport mechanism is a post hoc heuristic: the assumption of ballistic transport along current loops, encoded in tau ~ t/L, is introduced after observing the 1/4 fit, and the step 'for diffusive transport L ~ tau^(1/3), yielding L ~ t^(1/4)' is not derived from the AMB+ equation. The manuscript should provide a direct numerical test of this mechanism, for instance by measuring the current-loop length and traversal time, or by checking whether the data collapse onto L ~ tau^(1/3) when t is renormalized as tau ~ t/L. As written, the mechanism does not independently support the fitted exponent.","section":"Section III.A, paragraph after Fig. 5"},{"comment":"Equation (20) uses the one-dimensional static-kink saturation values psi1 and psi2 from Eq. (12), but the manuscript itself notes in Section III.B and Fig. 9 that the droplet and background phases in the two-dimensional muPS state do not saturate to psi1 and psi2 because the droplets are small. This is exactly the regime in which Eq. (20) is used to connect the hexagonal lattice spacing to Ls. Please quantify the error by measuring the actual plateau values inside the droplets and in the background, and state whether Eq. (20) remains quantitatively accurate, or restrict the claim to cases where the phases are closer to saturation. Relatedly, the steady-state nature of the muPS morphology is only shown up to t = 10^4; longer runs are needed to rule out a slow coarsening process at later times.","section":"Section III.B, Eqs. (19)-(20) and Fig. 9"},{"comment":"The claimed scaling Ls ~ (-xi)^(-2/3) is fitted over a very narrow range of xi (roughly from -1 to -2 for each lambda), and the paper acknowledges that the accessible range is limited by finite-size effects and numerical stability. With only about a factor of 2 in xi, a power-law fit is not strongly discriminative; alternative functional forms, such as exponential or crossover forms, would also fit the data. To make this a load-bearing quantitative claim, the authors should broaden the parameter range (for example with adaptive mesh or larger systems) or provide a theoretical derivation of the exponent. The non-monotonic Ls vs. -lambda data in Fig. 10(c) are presented without a quantitative model, which further limits the predictive content of the muPS results.","section":"Section III.B, Fig. 10(b)"}],"minor_comments":[{"comment":"The x-axis label '124 xi' appears to be a typographical artifact; it should be '-xi' or '|xi|' to match the text.","section":"Fig. 3"},{"comment":"The rescaling that removes the coefficients of the Ginzburg-Landau free energy and defines the dimensionless lambda and xi is not shown explicitly; stating the rescaling factors would make the parameter values fully reproducible.","section":"Section II, Eq. (7)"},{"comment":"The sentence 'The total number of vortices is even due to the periodic domain, which enforces zero net vorticity' is imprecise: periodic boundary conditions enforce zero total vorticity, not an even number of vortices. Please rephrase to avoid a topological statement that is not justified.","section":"Section III.A, after Fig. 5(c)"},{"comment":"When citing Refs. [37,38] for the surface-diffusion growth law, the text should explicitly state that those works derive L ~ t^(1/4) in d = 2 for conserved order-parameter dynamics with surface diffusion, so that the analogy is concrete.","section":"Section III.A, paragraph after Fig. 5"},{"comment":"A data/code availability statement would strengthen the paper's reproducibility; the numerical scheme is standard, but the exact central-difference discretization of the xi-term (a mixed second-order derivative) is not fully specified, and providing the code or a detailed discretization formula would remove ambiguity.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The paper is a numerical extension of the authors' earlier AMB study [28], and the novelty lies in including the rotational current term xi in both the forward- and reverse-Ostwald regions. The central quantitative claims (the MPS 1/4 exponent and the muPS Ls scaling) are not yet conclusively demonstrated because the effective exponent has not plateaued and the Ls power law is fitted over a very narrow range. These are, however, testable with longer simulations and a more systematic analysis, so I would ask for major revision rather than rejection. I see no citation or novelty concerns beyond the need for the revised manuscript to distinguish clearly between measured asymptotic behavior and interpretation of transient data."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague,\n\nYou should know two things about this paper. First, it cleanly demonstrates that the rotational activity term in AMB+ produces the same 1/3 to 1/4 growth crossover seen for the lambda-term in AMB, and it gives the clearest visualization I've seen of the surface current loops that drive that slowdown. Second, the microPS steady-state droplet crystal is a real and interesting phenomenology, with a neat conservation-law relation between lattice spacing and droplet size. But don't take the headline exponent of 1/4 as measured. The effective exponent data dip below 1/4 and are still rising at t=10^6; the asymptotic value is extrapolated, not settled.\n\nWhat the paper does well: the simulations are carefully specified (512^2 lattice, Euler scheme, 50 runs for MPS, 15 for the lambda=2 cases), the correlation functions are scaled and compared to MB, and the current-field analysis is convincing. The proposed mechanism—mass moving along surface loops whose length scales with L, giving tau ~ t/L and hence L ~ t^{1/4}—is plausible and connects to earlier surface-diffusion work [37,38]. The microPS saturation length Ls ~ (-xi)^{-2/3} is fit over only a few points, but the authors are honest about the limited parameter range.\n\nThe soft spots: (1) The asymptotic exponent. The text says the late-time data 'tend towards' 1/4, which is accurate—they haven't reached it. The crossover time tc is defined as the first crossing of the 1/4 line, which is arbitrary when the effective exponent is non-monotonic. The tc ~ xi^{-2} scaling inherits that arbitrariness. (2) The mechanism is post hoc, though not circular—it explains the exponent after the fact without independent support. (3) In microPS, the paper acknowledges that the bulk domains don't saturate to psi1 and psi2 because the droplets are small, but Eq. (20) uses that relation anyway; that makes the computed lattice spacing approximate in exactly the regime where it matters. (4) No code or data is shipped, so the fits can't be independently checked.\n\nMy bottom line: the central qualitative picture—xi slows MPS growth and drives microPS crystallization—is well supported. The quantitative scalings (exponent 1/4, tc scaling, Ls scaling) are plausible but not ironclad. This deserves serious peer review, and I'd want the referee to push for longer runs or a softened claim about the asymptotic exponent.\n\nWould I bring it to reading group? Maybe. I'd cite it if I were working on active phase separation kinetics. Yes, it deserves a referee.","headline":"Solid numerical study of AMB+ coarsening, but the asymptotic t^{1/4} claim is supported by effective exponents that haven't plateaued.","tokens_in":14815,"tokens_out":2706,"would_cite":true,"duration_ms":27711,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Adding a rotational current to the minimal active Model B slows droplet growth from t^(1/3) to t^(1/4) in the macroscale regime, and in the reverse-Ostwald regime arrests coarsening in a hexagonal crystal of monodisperse droplets.","keywords":["active matter","Active Model B+","coarsening kinetics","phase separation","Ostwald ripening","microphase separation","surface diffusion","Lifshitz-Slyozov law"],"falsifier":"Extend the $\\lambda=0$, $\\xi=-2$ MPS simulation to a $2048^2$ lattice and $t=10^7$ and measure the running local exponent $1/z_{\\rm eff}$; if it rises back toward $1/3$, the apparent $t^{1/4}$ asymptotic regime is a transient. Similarly, run the $\\mu$PS case $\\lambda=-4$, $\\xi=-1$ on a $1024^2$ box and track $L(t)$ past $t=10^4$: if $L(t)$ keeps growing or the hexagonal ordering anneals away, the claimed steady-state droplet crystal is a finite-size artifact.","tokens_in":13707,"feed_emoji":"🌀","tokens_out":15616,"duration_ms":134327,"temperature":0.7,"pith_summary":"This paper investigates how the two additional activity terms in Active Model B+ (AMB+) alter the kinetics of phase separation in a conserved order parameter. The central result is that the rotational current term of strength $\\xi$ reshapes the mass-transport current into closed loops along domain walls, so that material travels around droplet surfaces rather than along straight bulk paths. In the forward-Ostwald region (e.g., $\\lambda=0$, $\\xi<0$), this changes the asymptotic growth law from the Lifshitz–Slyozov $L(t) \\sim t^{1/3}$ to $L(t) \\sim t^{1/4}$, with a crossover time scaling as $\\xi^{-2}$. In the reverse-Ostwald region ($\\lambda<0$, $\\xi<0$), coarsening stops: the system reaches a steady state of monodisperse droplets arranged on a hexagonal lattice, with saturation size $L_s(\\lambda,\\xi)$ that appears to scale as $(-\\xi)^{-2/3}$. These results matter because they show that a minimal nonequilibrium field theory can produce both slower coarsening and stable microphase-separated patterns without long-range interactions.","feed_headline":"Rotational currents cut coarsening to t^(1/4) and freeze droplets","feed_subtitle":"Two active terms turn ordinary coarsening into a t^(1/4) law and a frozen hexagonal droplet lattice.","key_machinery":"The central object is the dimensionless AMB+ equation $\\partial\\psi/\\partial t = -\\nabla\\cdot\\mathbf{J}$ with $\\mathbf{J} = -\\nabla(-\\psi + \\psi^3 - \\nabla^2\\psi + \\lambda|\\nabla\\psi|^2) + \\xi\\nabla^2\\psi\\nabla\\psi$, where $\\psi$ is the conserved order parameter. The $\\lambda$-term is a rotation-free nonequilibrium current, the $\\xi$-term a rotational current; neither can be written as the gradient of a free energy. The paper shows that this current forms closed loops on domain walls, and that the loop circumference scales with the droplet size $L$; mass is transported along these loops, effectively renormalizing time from $t$ to $\\tau \\sim t/L$ and converting a diffusive $L \\sim \\tau^{1/3}$ into $L \\sim t^{1/4}$. The second load-bearing ingredient is the static one-dimensional kink solution of the model, which gives the nonequilibrium chemical potential $\\mu_s = 4\\alpha/15$ with $\\alpha = \\lambda - \\xi/2$, and the coexisting phase values $\\psi_1 = 1 + \\mu_s/2$, $\\psi_2 = -1 + \\mu_s/2$; this asymmetry fixes which phase forms droplets in MPS and enters the conservation-law relation $a = \\sqrt{2\\pi(\\psi_1-\\psi_2)/(\\sqrt{3}|\\psi_2|)} L_s$ used to connect the hexagonal lattice spacing to the droplet size in $\\mu$PS.","core_discovery":"The paper establishes that AMB+—the scalar conserved-order-parameter theory with a zero-curl activity term of strength $\\lambda$ and a rotational activity term of strength $\\xi$—has two distinct kinetic regimes after a quench at critical composition. In the forward-Ostwald regime (for instance $\\lambda=0$, $\\xi<0$), the current becomes sharply peaked at the interfaces and organizes into alternating clockwise and anticlockwise loops separated by nodal points. Mass transfer from small to large droplets occurs only by moving along these surface loops, whose length grows with droplet size, so the effective transport is slowed; the domain size crosses over from the Model B Lifshitz–Slyozov law $L(t) \\sim t^{1/3}$ at early times to an asymptotic $L(t) \\sim t^{1/4}$, with crossover time $t_c \\sim \\xi^{-2}$. In the reverse-Ostwald regime ($\\lambda<0$, $\\xi<0$), the current loops between droplets cancel, so ripening is reversed and the system reaches a steady state: monodisperse droplets on a hexagonal lattice, with saturation size $L_s(\\lambda,\\xi)$ consistent with $L_s \\sim (-\\xi)^{-2/3}$ over the studied parameter range, and a lattice spacing set by the conservation law. The paper also shows that the correlation function obeys dynamical scaling in both regimes, that the early-time scaling function coincides with the Model B (superuniversal) function, and that at late times in MPS the scaling function becomes parameter-dependent because the two coexisting phases are asymmetric.","pith_inferences":["A testable extension not pursued in the paper would be to measure how the current-loop length grows with $L$ in MPS and to check whether $L(t)$ collapses when time is rescaled by $t_c(\\xi)\\sim \\xi^{-2}$, which would sharpen the surface-diffusion mechanism into a quantitative scaling prediction.","Because Eq. (20) is used in a regime where the bulk phases do not saturate to the kink values, measuring the actual droplet and background compositions in $\\mu$PS would quantify the error in the predicted lattice spacing and may explain the scatter in $L_s(-\\xi)$.","Restricting to critical composition means off-critical quenches are unexplored; by the model's $\\psi\\to-\\psi$ symmetry, off-critical initial conditions could select different droplet-size scaling or different steady-state patterns in $\\mu$PS.","If the $t^{1/4}$ law is exact rather than an effective exponent, it implies that mass transport is controlled by the interfaces, not the bulk; introducing a small but nonzero bulk mobility should eventually restore $t^{1/3}$ at extremely long times, a prediction that could be tested numerically."],"forward_implications":["In the forward-Ostwald region, the asymptotic growth law of AMB+ is $L(t) \\sim t^{1/4}$, not the $t^{1/3}$ of Model B, even at critical composition, because the morphology becomes droplet-like and transport is surface-diffusion mediated.","The crossover from $t^{1/3}$ to $t^{1/4}$ occurs at a time $t_c$ that decreases with increasing $|\\xi|$, consistent with $t_c \\sim \\xi^{-2}$, so stronger rotational activity reaches the slow-growth regime sooner.","In the reverse-Ostwald region, coarsening terminates in a steady-state hexagonal crystal of monodisperse droplets; the saturation size $L_s$ decreases with $|\\xi|$ ($L_s \\sim (-\\xi)^{-2/3}$ in the studied range) and the lattice spacing is tied to $L_s$ by the conservation law.","Both MPS and $\\mu$PS exhibit dynamical scaling of the correlation function: the early-time MPS scaling function is the superuniversal Model B form, while the late-time MPS function depends on $\\lambda$ and $\\xi$; the $\\mu$PS scaling function is universal and shows oscillations from the crystalline order.","Porod's law holds in both regimes, confirming that the interfaces are sharp even though the transport is nonequilibrium."],"supporting_citations":[{"why":"Introduces AMB+ and its (λ, ξ) phase diagram, which defines the forward- and reverse-Ostwald regions that organize the entire study.","marker":"[26]"},{"why":"Introduces the AMB model (the zero-curl λ-term) whose kinetics the present work extends by adding the ξ-term.","marker":"[25]"},{"why":"The authors' earlier numerical study of AMB kinetics, providing the baseline of a t^{1/3}→t^{1/4} crossover and crossover-time scaling that the present paper generalizes.","marker":"[28]"},{"why":"Establishes the L ~ t^{1/4} growth law for phase separation driven by surface diffusion, the mechanism invoked to explain the asymptotic MPS kinetics.","marker":"[37]"},{"why":"Provides numerical support for surface-diffusion-driven coarsening and the t^{1/4} law used to interpret the AMB+ morphology.","marker":"[38]"},{"why":"Standard reference for Model B coarsening, the Lifshitz-Slyozov law, dynamical scaling, and the Porod regime used as the passive baseline.","marker":"[27]"}],"fun_headline_variants":["Rotational currents slow coarsening to t^1/4 and freeze droplets","Two active terms yield t^1/4 coarsening and frozen droplet lattice","Active currents flip coarsening law and lock droplets into lattice","Rotational currents reshape kinetics: t^1/4 coarsening and frozen droplets","Active rotations: coarsening slows to t^1/4, droplets freeze into lattice"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The analysis assumes that the two coexisting phase compositions taken from a one-dimensional static interface also describe the saturated droplet and background phases in the two-dimensional patterns, although in the microphase-separated state the paper notes the small droplets do not reach those compositions.","fun_headline_variants_meta":{"raw":{"variants":["Rotational currents slow coarsening to t^1/4 and freeze droplets","Two active terms yield t^1/4 coarsening and frozen droplet lattice","Active currents flip coarsening law and lock droplets into lattice","Rotational currents reshape kinetics: t^1/4 coarsening and frozen droplets","Active rotations: coarsening slows to t^1/4, droplets freeze into lattice"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000959,"raw_usage":{"total_tokens":4134,"prompt_tokens":1040,"completion_tokens":3094,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":656,"completion_tokens_details":{"reasoning_tokens":2989}},"tokens_in":656,"tokens_out":3094,"duration_ms":20534,"temperature":1.0,"reasoning_tokens":2989,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:17:29.431646+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Extend the $\\lambda=0$, $\\xi=-2$ MPS simulation to a $2048^2$ lattice and $t=10^7$ and measure the running local exponent $1/z_{\\rm eff}$; if it rises back toward $1/3$, the apparent $t^{1/4}$ asymptotic regime is a transient. Similarly, run the $\\mu$PS case $\\lambda=-4$, $\\xi=-1$ on a $1024^2$ box and track $L(t)$ past $t=10^4$: if $L(t)$ keeps growing or the hexagonal ordering anneals away, the claimed steady-state droplet crystal is a finite-size artifact.","supporting_citations":[{"cited_title":"Wittkowski, A","cited_arxiv_id":null,"evidence_quote":"Introduces the AMB model (the zero-curl λ-term) whose kinetics the present work extends by adding the ξ-term."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"The authors' earlier numerical study of AMB kinetics, providing the baseline of a t^{1/3}→t^{1/4} crossover and crossover-time scaling that the present paper generalizes."},{"cited_title":"Zhao, Ann","cited_arxiv_id":null,"evidence_quote":"Establishes the L ~ t^{1/4} growth law for phase separation driven by surface diffusion, the mechanism invoked to explain the asymptotic MPS kinetics."}],"review_version":1}