{"id":"c7b0d406-82f9-43bc-bb10-779134c1b4d7","arxiv_id":"2502.03372","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In a run-and-tumble coagulation model, the competition between motility and reaction rates switches wave fronts from traveling waves to diffusive fronts.","lead":"This paper studies a one-dimensional model of self-propelled particles that can die and duplicate, and shows that slow tumbling makes density fronts travel as waves while fast tumbling makes them diffuse. It also maps the long-time fluctuations onto a known growth equation, linking active motion to pattern formation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The diffusive closure used for the KPZ mapping is regular only for small β, but the advertised long-lived-current regime requires β > α+µ; the two halves of the paper address opposite parameter regimes.","rationale":"The reader's weakest-assumption analysis already flags Eq. (36), the diffusive closure. My stress-test sharpens this into a parameter-regime contradiction: Eq. (18) locates the mean-field current peak at β > α+µ, while the regularization of D[ρ] assumes β small. This means the two headline mechanisms — traveling waves from density-current coupling, and Fisher-Kolmogorov/KPZ diffusive behavior — are demonstrated in disjoint regions of parameter space. The front-propagation crossover from the exact linear stability analysis is not affected by this closure, so the paper should not be rejected; but the pattern-formation/KPZ claims need either a derivation valid at finite β or a clear restriction showing they do not address the long-lived-current regime. The other issues (the sign in the stability condition, the missing absolute value in kc, and the apparent mismatch between Eq. (24) and the dispersion relation in Eq. (28)) are typographical and repairable; they do not alter the corrected dispersion relation. The verdict therefore remains conditional, with the added requirement being to demonstrate that the closure is valid where the phenomenon lives, or to restrict the claims accordingly.","tokens_in":11090,"tokens_out":12709,"duration_ms":128644,"concrete_test":"With v=1, µ=1, compare the full two-field system (4)-(5) against the closed equation (37) for two parameter sets: β=5, α=1 (β > α+µ, so Eq. (18) predicts a current peak and ρ_c = 0.4, close to ρ∞ = 0.2) and β=0.1, α=1 (β < α+µ, so D stays positive). Use identical Gaussian initial data, the same grid, and the same CFL condition. Record ρ(x,t), J(x,t), and the minimum over space-time of α+µ-βρ. If the closed equation departs from the full system or develops a negative/divergent effective diffusion precisely in the β>α+µ case, while the full solution shows a current peak, then the diffusive-limit and KPZ conclusions are not applicable in the regime where the central effect is claimed.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's long-time and pattern-formation claims, including the KPZ mapping, rest on the adiabatic closure J[ρ] = -D[ρ]ρ' with D[ρ] = v²/(α+µ-βρ) (Eqs. 35-36). The paper acknowledges that D[ρ] diverges at ρ_c = (α+µ)/β and changes sign, calling this 'a much serious problem,' and then restricts to small β to keep D positive. But the mean-field current-peak time, Eq. (18), gives t>0 exactly when β > α+µ. That is the regime in which the density-current coupling generates the long-lived currents the paper advertises. There, the singular density ρ_c lies only α/β above the stationary density µ/β, so modest density fluctuations can cross the singular point and the closure is uncontrolled. In the small-β regime the closure is regular, but the mean-field current decays monotonically; the advertised mechanism is absent unless spatial gradients restore it. Hence the diffusive/KPZ claims and the traveling-wave phenomenon are established in opposite parameter regimes unless an additional argument connects them. The linear-stability crossover does not depend on this closure, but the abstract's Fisher-Kolmogorov-to-KPZ pattern-formation claim does.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript presents a one-dimensional continuum model of run-and-tumble particles with coagulation and decoagulation reactions, summarized in Eqs. (4)-(5). It solves the mean-field logistic density and the current peak time, performs a linear stability analysis of the mixed stationary state, derives a diffusive closure for the current, and uses that closure to discuss Fisher-Kolmogorov front propagation and a formal mapping to the Kardar-Parisi-Zhang equation. Numerical solutions of the PDEs are used to illustrate the qualitative regimes.","tokens_in":11334,"tokens_out":20469,"duration_ms":196955,"significance":"The paper is clearly written and has the virtue of exact mean-field and linear-stability results with no parameter fitting; in particular, the predicted crossover wavenumber kc=|mu-alpha|/(2v) and the mean-field current peak time are explicit and falsifiable. The identification of the singular diffusive closure is honest and important. However, several algebraic errors in the linear-stability and diffusive-limit derivations, and the questionable KPZ mapping at the absorbing-state critical point, currently undermine the pattern-formation claims and require correction before the results can be fully endorsed.","major_comments":[{"comment":"Eq. (24) is inconsistent with Eq. (28). Linearizing Eq. (5) around rho_infinity = mu/beta gives delta J_dot = -v^2 delta rho' - alpha delta J, because beta rho_infinity = mu; the printed coefficient (alpha+mu) would lead to different eigenvalues. Eq. (28), which is the correct result for the physical linearization, should be derived from the corrected equation. In addition, the text states that stability requires Re(sigma) > 0, whereas the eigenvalues in Eq. (28) have Re(sigma) < 0 for all k; the condition is Re(sigma) < 0. Finally, Eq. (30) should read kc = |mu-alpha|/(2v), not (mu-alpha)/(2v), since the threshold follows from (mu-alpha)^2 - (2vk)^2 < 0.","section":"Linear Stability Analysis, Eqs. (24), (28), (30)"},{"comment":"The small-beta expansion of D[rho] is algebraically incorrect. For rho = rho_infinity + delta rho with rho_infinity = mu/beta, the denominator in Eq. (36) is alpha+mu-beta rho = alpha - beta delta rho. Hence D(rho_infinity) = v^2/alpha and the Taylor coefficient is v^2 beta/alpha^2. Equations (39) and (42) instead use tilde D = v^2/(alpha+mu) and tilde beta = beta/(beta+alpha). The resulting diffusion constant and the coefficient of (delta rho')^2 in Eq. (42) are therefore not the small-beta limit of Eq. (36). In particular, the claimed dependence of the KPZ amplitude on mu/alpha is an artifact of this expansion error; the corrected coefficient is v^2 beta/alpha^2.","section":"Diffusive limit and pattern formation, Eqs. (35)-(42)"},{"comment":"The long-lived current effect advertised in the abstract is the mean-field peak in J(t), which by Eq. (18) exists only for beta > alpha+mu. In this regime the density rho_c = (alpha+mu)/beta lies only alpha/beta above the stationary density mu/beta, so the denominator in D[rho] is close to zero and the adiabatic closure J = -D[rho] rho' is uncontrolled. The diffusive and KPZ results are derived in the small-beta regime, where D is regular but the mean-field current decays monotonically. The manuscript acknowledges the singularity but does not supply an argument connecting the two regimes; as written, the traveling-wave/diffusive-wave crossover does not require the closure, while the FK/KPZ pattern-formation claims are confined to the opposite parameter regime. This limitation should be stated explicitly, and the phrase ranging from Fisher-Kolmogorov to Kardar-Parisi-Zhang in the abstract should be qualified.","section":"Diffusive limit and pattern formation, Eqs. (18), (35)-(36)"},{"comment":"The step from Eq. (42) to the KPZ statement is not justified. Equation (42) is an expansion around the mixed stationary state with mu > 0; its linear term is -mu delta rho. Setting mu = 0 does not describe the absorbing-state critical point in the same expansion, because at mu = 0 the stationary state is rho = 0 and the reaction term is -beta rho^2, which is not negligible at criticality. The Langevin equation normally associated with this absorbing-state transition belongs to the directed-percolation universality class, as the final paragraph itself states. The relation between the claimed KPZ universality for pattern formation at criticality and the stated DP universality class of the transition must be clarified; as it stands, the formal equivalence in Eq. (43) is not established.","section":"Diffusive limit and pattern formation, Eq. (43) and Conclusions"}],"minor_comments":[{"comment":"Eq. (32) has J[rho] = DA rho'; the correct closure is J[rho] = -DA rho', and the Fisher-Kolmogorov equation (33) uses the latter sign. Please fix the sign in Eq. (32).","section":"Large-alpha limit, Eq. (32)"},{"comment":"The statement 'with t = 0 if beta < alpha+mu' should be 'if beta <= alpha+mu', since at beta = alpha+mu the argument of the logarithm is unity.","section":"Mean-field approximation, Eq. (18)"},{"comment":"The text before Fig. 6 says the two tumbling rates are alpha = 0.5 and alpha = 2, while the caption says alpha = 0.1 and alpha = 2; please reconcile the values.","section":"Fig. 6 and surrounding text"},{"comment":"The explicit Euler scheme with central differences is dispersive for the hyperbolic terms, and no convergence test or resolution study is reported; a brief convergence statement would strengthen the numerical evidence.","section":"Appendix A"},{"comment":"The phrase 'birth/date processes' appears to be a typo for 'birth/death processes'.","section":"Introduction"}],"recommendation":"major_revision","confidential_remarks":"The core mean-field and linear-stability results are likely correct once the Eq. (24) typo and the kc absolute value are fixed, and the paper's candid discussion of the singular diffusive closure is a strength. The main risk is the KPZ claim, which is not derived from the model's actual absorbing-state critical dynamics; I would recommend that the revision be evaluated by a referee familiar with directed percolation and stochastic PDEs."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear [Colleague],\n\nThe short version: this paper has a clean, correct core result—a linear-stability crossover between damped traveling waves and purely diffusive relaxation in a run-and-tumble coagulation model—and a shaky secondary claim about KPZ universality that rests on a closure that diverges in exactly the parameter regime the primary effect lives in. Worth a serious look, but not without revision.\n\nWhat's actually new: the model itself, one-dimensional run-and-tumble particles with coagulation/decoagulation, is not in the cited literature. The mean-field analysis is done properly: ρ follows the logistic equation, J develops a non-monotonic peak when β > α+µ, and the stationary density is unchanged by activity. The linear stability calculation around ρ∞ = µ/β gives a simple dispersion relation, and the numerical integration of the continuum equations does show the predicted crossover from waves at low tumbling rate to diffusive relaxation at high tumbling rate. That part holds up.\n\nThe soft spots are real but mostly mechanical. The text says the stability condition is Re σ > 0; it should be Re σ < 0. The critical wavenumber is written as kc = (µ−α)/2v without an absolute value, which makes the α=15 example internally inconsistent—with the signed formula, k0=2π is larger than kc, yet they call it diffusive. That's fixable by writing |µ−α|/2v. The paper's own diffusive-limit section acknowledges the divergence of D[ρ] at ρc = (α+µ)/β, and the stress-test concern is on target: the small-β limit that regularizes the closure is precisely the regime where the mean-field current does not peak, so the advertised long-lived currents and the KPZ mapping are established in opposite parameter regimes. The linear stability result doesn't depend on the closure, so the central claim survives; the KPZ claim does not, as stated.\n\nThe citation pattern is clean: self-citations are used for context and regularization, not fitted to data. No circularity.\n\nWho is this for? Anyone working on active matter, population dynamics, or reaction-diffusion systems with non-conserved particles. It deserves a serious referee, but I'd send it back for a substantive revision: fix the sign and absolute value, and either restrict the claims about pattern formation to the small-β regime with an explicit caveat about the divergence, or provide an alternative argument connecting the two regimes.\n\nRecommendation: engage with it, but require the revision before accepting.","headline":"A clean linear-stability crossover in a new active coagulation model, with a KPZ claim that only works in the parameter regime where the main effect is absent.","tokens_in":11841,"tokens_out":2627,"would_cite":true,"duration_ms":22374,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["82C22","82C31","82C27"],"pacs":[],"model":"deepseek-v4-flash","headline":"In an active coagulation model, density perturbations switch from damped traveling waves to diffusive relaxation at a critical wavenumber $k_c=|\\mu-\\alpha|/(2v)$, and the diffusive limit is well defined only for small reaction rates…","keywords":["run-and-tumble particles","coagulation model","absorbing state phase transition","wave front propagation","long-lived currents","diffusive limit","KPZ universality","linear stability analysis"],"falsifier":"Track the relaxation of a cosinusoidal density perturbation with wavenumber $k$ in a particle simulation or exact numerical solution of the full density-current equations; the claim predicts damped oscillations with a nonzero imaginary part in the dispersion relation whenever $k>|\\mu-\\alpha|/(2v)$. Observing purely monotonic decay at all wavenumbers for $\\alpha<\\mu$ would falsify the traveling-wave regime, and observing no sign change in $D[\\rho]$ at $\\beta\\rho=\\alpha+\\mu$ would falsify the predicted breakdown of the diffusive limit.","tokens_in":1893,"feed_emoji":"🌊","tokens_out":4508,"duration_ms":88738,"temperature":0.7,"pith_summary":"This paper studies a minimal population model: self-propelled run-and-tumble particles that coagulate (annihilate) at rate $\\beta$ and are born at rate $\\mu$. It argues that active motion does not move the absorbing-state transition point, but it changes how the system relaxes: density and current stay coupled through the coagulation rate, long-lived currents form, and perturbations of wavenumber $k$ relax as damped traveling waves when $k>|\\mu-\\alpha|/(2v)$ and as purely diffusive modes when $k$ is smaller. The paper further claims that in the diffusive limit the effective diffusion coefficient is $D[\\rho]=v^2/(\\alpha+\\mu-\\beta\\rho)$, which can diverge or become negative, so the diffusive description is only safe at small reaction rates; in that regime a noisy version of the dynamics maps to the KPZ equation. If correct, the result identifies the tumbling rate relative to the birth rate as the control parameter that decides whether a spreading population invades as a wave or as a diffusive blur.","feed_headline":"Tumbling rate tunes spreading fronts from waves to diffusion","feed_subtitle":"In a minimal coagulation model, the ratio of tumbling to birth rate controls how a population relaxes toward its mixed state.","key_machinery":"The central object is the coupled density-current system $\\dot{\\rho}=-J'-\\mu\\rho+\\beta\\rho^2$ and $\\dot{J}=-v^2\\rho'-J(\\alpha+\\mu-\\beta\\rho)$, where $\\rho=R+L$ is the particle density and $J=v(R-L)$ is the current of right- and left-moving run-and-tumble particles. The mechanism that carries the argument is the coupling term $-J(\\alpha+\\mu-\\beta\\rho)$ in the current equation, which makes the current relaxation rate depend on the local density and thereby generates long-lived currents and the wavenumber-dependent traveling-to-diffusive crossover. Linear stability analysis of this system supplies the critical wavenumber $k_c$, while the adiabatic closure $\\dot{J}=0$ supplies the effective diffusion coefficient whose divergence marks the breakdown of the diffusive limit.","core_discovery":"The central discovery is that for the active coagulation model the stationary mixed state $\\rho_\\infty=\\mu/\\beta$ is unchanged by activity, but the relaxation toward it is controlled by the ratio of the tumbling rate $\\alpha$ to the reaction scales. Linearizing the coupled density-current equations around the mixed state gives a critical wavenumber $k_c=|\\mu-\\alpha|/(2v)$: perturbations with longer wavelengths ($k<k_c$) relax diffusively, while those with shorter wavelengths ($k>k_c$) first propagate as damped waves. The same density-current coupling produces a peak in the mean-field current $J(t)$, a signature of long-lived currents, and makes the diffusive closure $J=-D[\\rho]\\rho'$ pathological whenever $\\beta\\rho$ approaches $\\alpha+\\mu$, because $D[\\rho]=v^2/(\\alpha+\\mu-\\beta\\rho)$ diverges and changes sign. In the small-$\\beta$ regime the theory is well defined, and the fluctuating equation for the perturbation reduces to a KPZ-like equation; the paper therefore concludes that pattern formation at criticality should belong to the KPZ universality class.","pith_inferences":["The same density-current mechanism should appear whenever active motion is coupled to any reaction with a density-dependent rate; replacing coagulation by SIS or SIR reactions and checking for the same $k_c$ crossover would be a direct test of the generality of this picture.","The divergence of $D[\\rho]$ at $\\rho_c=(\\alpha+\\mu)/\\beta$ suggests that the diffusive limit is not merely a technical approximation but marks a regime where the current must be kept as a slow field, potentially leading to a two-field KPZ-like description instead of a single scalar equation.","The KPZ mapping at $\\mu=0$ implies that active noise could alter the roughening exponents of an expanding front, giving a measurable signature of activity even though the phase boundary is unchanged.","Because the model is one-dimensional, a natural extension is to two dimensions, where the current is a vector and the traveling-to-diffusive crossover may interact with vorticity or banding instabilities; this is beyond what the paper addresses."],"forward_implications":["The absorbing-state transition point and the stationary mixed-state density remain at their mean-field values even when activity is strong, so directed-percolation universality is preserved despite the changed dynamics.","The tumbling rate $\\alpha$ tunes between traveling-wave relaxation (small $\\alpha$) and purely diffusive relaxation (large $\\alpha$), meaning the relaxation of density perturbations directly exposes $k_c$ and hence the active speed $v$.","In the large-$\\alpha$ Brownian limit the model reduces to the Fisher-Kolmogorov equation, so standard reaction-diffusion wave-front results apply in that regime.","The diffusion coefficient becoming negative or divergent signals that the current cannot always be eliminated; a well-defined diffusive description exists only for small reaction rates.","Near the absorbing-state critical point ($\\mu\\to 0$), the noisy dynamics is formally equivalent to KPZ, predicting KPZ scaling for large-scale pattern formation."],"supporting_citations":[{"why":"Supplies the reaction scheme and the absorbing-state phase-transition framework in which the coagulation model is defined.","marker":"[3]"},{"why":"Provides the Fisher-Kolmogorov equation and wave-front propagation background used for the large-alpha and diffusive limits.","marker":"[9]"},{"why":"Used together with [25] for the continuum pattern-formation description and surface-term regularization in the negative-diffusion regime.","marker":"[17]"},{"why":"Cited for inserting $\\nabla^4\\rho$ surface terms that regularize a negative diffusion coefficient during pattern formation.","marker":"[25]"},{"why":"Gives the standard run-and-tumble hydrodynamic equations that the model extends with coagulation and decoagulation reactions.","marker":"[27]"},{"why":"Supplies the telegrapher equation, the no-reaction baseline that interpolates between ballistic waves and diffusion.","marker":"[28]"},{"why":"Defines the KPZ equation used for the universality-class claim in the diffusive limit at criticality.","marker":"[29]"}],"fun_headline_variants":["Tumble speed sets how fronts spread in the active coagulation model","Crossover from traveling to diffusive waves in active coagulation","Active dynamics doesn't shift transition, but alters relaxation waves","Motility controls relaxation and wave type in absorbing-state model","From Fisher-Kolmogorov to KPZ via active coagulation dynamics"],"cache_read_input_tokens":13952,"weakest_assumption_plain":"The conclusions about the diffusive limit and the KPZ mapping rest on the assumption that the reaction rates $\\beta$ and $\\mu$ are small enough that the effective diffusion coefficient $D[\\rho]$ stays positive and finite; the paper itself flags that outside this regime $D[\\rho]$ diverges and then changes sign.","fun_headline_variants_meta":{"raw":{"variants":["Tumble speed sets how fronts spread in the active coagulation model","Crossover from traveling to diffusive waves in active coagulation","Active dynamics doesn't shift transition, but alters relaxation waves","Motility controls relaxation and wave type in absorbing-state model","From Fisher-Kolmogorov to KPZ via active coagulation dynamics"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000252,"raw_usage":{"total_tokens":1556,"prompt_tokens":935,"completion_tokens":621,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":551,"completion_tokens_details":{"reasoning_tokens":537}},"tokens_in":551,"tokens_out":621,"duration_ms":6800,"temperature":1.0,"reasoning_tokens":537,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T04:57:52.051537+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Track the relaxation of a cosinusoidal density perturbation with wavenumber $k$ in a particle simulation or exact numerical solution of the full density-current equations; the claim predicts damped oscillations with a nonzero imaginary part in the dispersion relation whenever $k>|\\mu-\\alpha|/(2v)$. Observing purely monotonic decay at all wavenumbers for $\\alpha<\\mu$ would falsify the traveling-wave regime, and observing no sign change in $D[\\rho]$ at $\\beta\\rho=\\alpha+\\mu$ would falsify the predicted breakdown of the diffusive limit.","supporting_citations":[{"cited_title":"Hinrichsen, Advances in physics , 2000, 49, 815–958","cited_arxiv_id":null,"evidence_quote":"Supplies the reaction scheme and the absorbing-state phase-transition framework in which the coagulation model is defined."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the Fisher-Kolmogorov equation and wave-front propagation background used for the large-alpha and diffusive limits."},{"cited_title":"Paoluzzi, M","cited_arxiv_id":null,"evidence_quote":"Used together with [25] for the continuum pattern-formation description and surface-term regularization in the negative-diffusion regime."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Cited for inserting $\\nabla^4\\rho$ surface terms that regularize a negative diffusion coefficient during pattern formation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the standard run-and-tumble hydrodynamic equations that the model extends with coagulation and decoagulation reactions."},{"cited_title":"Kac, The Rocky Mountain Journal of Mathematics , 1974, 4, 497–509","cited_arxiv_id":null,"evidence_quote":"Supplies the telegrapher equation, the no-reaction baseline that interpolates between ballistic waves and diffusion."},{"cited_title":"Kardar, G","cited_arxiv_id":null,"evidence_quote":"Defines the KPZ equation used for the universality-class claim in the diffusive limit at criticality."}],"review_version":1}