{"id":"1fa0f053-6dd2-49b6-b562-19e399959331","arxiv_id":"2607.05194","paper_version":1,"verdict":"ACCEPT","confidence":"UNKNOWN","novelty_score":7.0,"correctness_risk":"unknown","formal_verification":"none","parameter_count":4,"one_line_summary":"Extends classical nucleation theory to nonequilibrium non-conserved scalar fields by showing the time-reversed-relaxation ansatz fails and deriving corrected quasipotentials via two independent routes, validated numerically.","lead":"The paper extends classical nucleation theory to nonequilibrium systems with non-conserved order parameters (like active matter and population dynamics). It shows that the standard assumption—that the nucleation path is the time-reverse of relaxation—fails for these systems, and provides a corrected analytical framework that predicts different nucleation barriers.","discovery_kind":"unclear","skeptic_critique":{"model":"glm-5.2","headline":"Numerical validation assumes radial symmetry, same as the analytical theory, so the key independent check is not fully independent of the central assumption it tests.","rationale":"The reader correctly identified the perturbative 1/R expansion and radial symmetry as the weakest assumptions. I focus specifically on the radial symmetry aspect because it creates a circularity in the validation logic: the analytical theory assumes radial symmetry, and the numerical check also assumes it, so the agreement between them cannot independently confirm that assumption. The perturbative expansion, by contrast, is directly tested by Fig. 6 (showing ε_A ~ O(v_0)) and is the same well-established framework as CNT. The capillary wave analysis provides analytical evidence for radial stability but only for small perturbations — it cannot exclude qualitatively different non-radial instantons. Despite this concern, I recommend UNCHANGED because: (1) radial symmetry of the instanton is standard for isotropic nucleation problems and is expected on symmetry grounds; (2) the positive capillary wave tension provides partial support; (3) the two analytical routes agree; and (4) the radially-constrained numerics show excellent agreement with NNT. The concern is real but does not rise to the level of changing the verdict from ACCEPT. It would, however, be the natural next validation step for the authors or independent groups to perform.","tokens_in":27292,"tokens_out":6252,"duration_ms":89826,"concrete_test":"Run the gMAM action minimization in full 2D (or 3D) without imposing radial symmetry, for the AMA parameters used in Fig. 5 (e.g., h=0.2, λ=λ*(0.2)+δλ). If the converged instanton remains radially symmetric (to within discretization error) and the action matches the radially-constrained result to within ~5%, the radial symmetry assumption and the NNT barrier prediction are independently confirmed. If the action is lower or the instanton develops significant non-radial structure, the single-reaction-coordinate reduction underlying NNT would need revision.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim — that NNT (Eqs. 14-16) gives the correct quasipotential barrier while the TRR ansatz (Eqs. 22-24) does not — is supported by two analytical derivations and numerical gMAM minimization. However, both analytical routes share the same key structural assumption (ψ' ∈ Ker(L†) and the 1/R perturbative expansion), so their agreement tests internal consistency but not independence from that shared assumption. The genuinely independent check is the gMAM computation in Sec. V, but this computation explicitly imposes radial symmetry: 'We invoke rotational symmetry to impose that the instanton density field remains a function of the radial coordinate only.' This is the same radial symmetry assumed in the analytical theory. The capillary wave analysis (Secs. IIIF, IVB) shows positive interfacial tension, which addresses stability to small transverse perturbations around the spherical shape, but it does not rule out qualitatively different non-radial instanton configurations that could have lower action. If such configurations existed, the true nucleation barrier would differ from both NNT and the radially-constrained gMAM result, and the agreement in Fig. 5 would be an artifact of the shared constraint. This concern is moderate: radial symmetry of the instanton is standard for isotropic nucleation problems, and the positive capillary wave tension provides partial analytical support. But the strongest possible validation — an unconstrained 2D action minimization — has not been performed.","agreement_with_reader":"partial"},"referee_report":{"model":"glm-5.2","summary":"This paper presents Nonequilibrium Nucleation Theory (NNT), extending Classical Nucleation Theory to non-conserved scalar field theories that break detailed balance. The central result is that the quasipotential barrier for nucleation, given by Eqs. (14)-(16), differs from the prediction obtained via the time-reversed relaxation (TRR) ansatz, Eqs. (22)-(24). The authors provide two analytical derivations: a stochastic route (Sec. IID) projecting the field dynamics onto the droplet radius using a test function ψ' chosen in Ker(L†), and an action route (Sec. IIF) via self-consistent minimization of the Freidlin-Wentzell action. Both yield identical results. The theory is applied to Active Model A (Sec. III) and a population dynamics model (Sec. IV), with closed-form expressions for barriers and mobilities. Capillary wave stability is analyzed to justify the spherical droplet assumption. Numerical validation via geometric minimum action method (gMAM) and a Ritz method is presented in Sec. V. The framework is also shown to recover known results for the conserved case (AMB+, Sec. VI), where the TRR ansatz is valid to the required order.","tokens_in":27527,"tokens_out":2894,"duration_ms":67816,"significance":"The paper addresses a genuinely difficult problem: analytical computation of nucleation barriers in nonequilibrium systems where no free energy exists and the instanton is not the time-reversal of relaxation. The key technical insight — that choosing ψ' ∈ Ker(L†) eliminates the need to explicitly compute the deviation between instanton and relaxation profiles — is elegant and powerful. The two-route derivation (stochastic and action-based) provides a strong internal consistency check. The closed-form results for AMA and population dynamics, the perturbative near-equilibrium expansions, and the recovery of AMB+ results from the same framework all add value. The numerical validation via gMAM provides quantitative support. The distinction between NNT and TRR-ansatz barriers (factors of ~2 difference in examples) is physically significant and falsifiable.","major_comments":[{"comment":"Sec. V: The numerical validation via gMAM explicitly imposes radial symmetry ('We invoke rotational symmetry to impose that the instanton density field remains a function of the radial coordinate only'). This is the same radial symmetry assumed in the analytical theory. Consequently, the agreement between NNT predictions and numerical results in Fig. 5, while encouraging, is not a fully independent test of the central claim: both theory and numerics share the radial-symmetry constraint. The capillary wave analysis (Secs. IIIF, IVB) shows positive interfacial tension for the examples studied, which addresses stability to small transverse perturbations, but does not rule out qualitatively different non-radial instanton configurations with lower action. The authors should discuss this limitation explicitly and state whether an unconstrained (full 2D) action minimization is feasible as a未来的校","section":null},{"comment":"Sec. IIF, Eqs. (26)-(30): The self-consistency argument showing ε_A ~ O(v_0) and that neglected terms are subleading is verified numerically only for specific AMA parameters (Fig. 6, with h=0.2, λ=-1.0333). The argument relies on the scaling ˙R ~ O(1/R, v_0) and ¨R ~ O(˙R/R²) for instanton paths. While this is standard for CNT-like regimes, the paper would benefit from a brief discussion of the parameter range over which this self-consistency is expected to hold, and any indications of where it might break down (e.g., very small R_c or strong activity).","section":null}],"minor_comments":[{"comment":"Eq. (15): The expression for U(R) contains the ratio ∫ψ'φ'_0 / ∫ψ'²D(φ_0), which is the inverse of the mobility prefactor in Eq. (14). This connection could be stated more explicitly for the reader's benefit.","section":null},{"comment":"Sec. IIIA, Eq. (38): The choice ψ' = φ'_0 exp(-2λ*φ_0/K) is derived perturbatively near the coexistence line (λ*, h*). The paper should briefly state how far from this line the expression remains accurate, given that the numerical validation in Sec. V uses λ = -1.033 (which may not be in the perturbative regime).","section":null},{"comment":"Fig. 2(f): The barrier heights U(R_c) and U_φ(R_c) are plotted but the axes labels and parameter values (δλ range) could be stated more clearly in the caption.","section":null},{"comment":"Sec. IV, Eq. (65): The modulus signs in D^(i)(ϕ) = |ϕ(1-ϕ)| + D_0 are noted as redundant given the dynamics, but the paper states they ensure positivity. A brief clarification that D_0 > 0 is required for well-posedness (not just for nucleation) would help.","section":null},{"comment":"Sec. V, Fig. 4 caption: The parameter D_0 = 10^{-2} is stated in the caption but D_0 = 10^{-3} is used in Fig. 3. This inconsistency should be clarified or corrected.","section":null},{"comment":"Reference [60] (the companion article) is cited frequently but listed as arXiv:2606.18911. Since both papers appear to be submitted simultaneously, the authors should ensure cross-references are consistent upon publication.","section":null}],"recommendation":"minor_revision","confidential_remarks":"The reader's concern about the radial symmetry assumption in the numerical validation is legitimate but, in my assessment, does not rise to the level of a major revision. Radial symmetry of the instanton is standard for isotropic nucleation problems, and the capillary wave analysis provides partial analytical support. The two analytical routes, while sharing core assumptions, represent genuinely different mathematical approaches (direct Langevin projection vs. action minimization), and their agreement is a meaningful consistency check. The paper is a substantial and careful contribution. The main improvement needed is a more transparent discussion of the limitations of the numerical validation, not a reworking of the theory."},"author_rebuttal":{"model":"glm-5.2","summary":"We thank the referee for a careful and constructive report. Both major comments identify legitimate limitations that we will address in the revised manuscript. The first concerns the circularity inherent in imposing radial symmetry in both the analytical theory and the numerical validation; we agree this is an important caveat and will discuss it explicitly, including the feasibility of unconstrained 2D action minimization. The second concerns the parameter range over which the self-consistency argument in Sec. IIF holds; we will add a discussion of the expected regime of validity and potential breakdown mechanisms. Neither comment requires changes to our central results, but both warrant transparent discussion that is currently missing from the manuscript.","responses":[{"response":"The referee is correct that our numerical validation does not constitute a fully independent test of the radial symmetry assumption, since both the analytical theory and the gMAM computation impose it. We agree this limitation should be stated explicitly. In the revised manuscript, we will add a discussion in Sec. V making the following points. First, the capillary wave analysis (Secs. IIIF, IVB) demonstrates positive interfacial tension in the examples studied, which addresses stability to small transverse perturbations but, as the referee notes, does not rule out non-radial instanton configurations with lower action. Second, in equilibrium nucleation theory, the spherical droplet assumption is justified not only by capillary wave stability but also by the fact that the critical nucleus is a saddle point of the free energy functional, and non-spherical saddles are known to have higher action in the large-Rc limit. Our NNT framework inherits the large-Rc perturbative structure from CNT, so the same physical reasoning applies, but we have not proven this rigorously for the nonequilibrium case. Third, regarding feasibility of unconstrained 2D action minimization: this is in principle possible using gMAM on a 2D spatial grid without imposing radial symmetry, and we will state this. However, the computational cost is substantially higher (the number of degrees of freedom scales as Nr^2 * N_ell rather than Nr * N_ell), and the radial discretization used in our current implementation would need to be replaced by a full 2D mesh. We will note that such a computation would provide a stronger test and is a natural direction for future work, while clarifying that the agreement we do observe is consistent with — though not a proof of — the validity of the radial assumption. We will","revision_made":"yes","referee_comment":"Sec. V: The numerical validation via gMAM explicitly imposes radial symmetry, so the agreement between NNT predictions and numerical results is not a fully independent test since both theory and numerics share the radial-symmetry constraint. The capillary wave analysis shows positive interfacial tension but does not rule out qualitatively different non-radial instanton configurations with lower action. The authors should discuss this limitation and state whether unconstrained full 2D action minimization is feasible."},{"response":"The referee correctly points out that our self-consistency argument is verified numerically only for specific AMA parameters and that the manuscript lacks a discussion of the parameter regime where the scaling assumptions hold. We will add such a discussion in Sec. IIF. The self-consistency argument relies on two key assumptions: (1) that the instanton path satisfies R_dot ~ O(1/R, v_0) and R_ddot ~ O(R_dot/R^2), which is standard for CNT-like regimes where the critical radius is large and the instanton traverses the quasipotential barrier slowly, and (2) that epsilon_A remains small, specifically O(v_0). Both assumptions are expected to hold when v_0 is small (equivalently, when Rc is large), which is the regime where the perturbative expansion in 1/R is controlled. The numerical verification in Fig. 6 confirms this for AMA with h=0.2 and lambda near lambda*(h), where v_0 is indeed small. We expect the same scaling to hold throughout the near-equilibrium perturbative regime (small h, small lambda) and more generally whenever |delta_lambda| is small enough that Rc >> interfacial width. Potential breakdown mechanisms include: (a) very small Rc, where the 1/R expansion is not controlled; (b) strong activity, where v_0 is not small and the instanton may deviate significantly from the relaxation path; and (c) cases where the capillary wave tension approaches zero or becomes negative, invalidating the spherical droplet assumption entirely. We will add a paragraph to Sec. IF stating these points and noting that a systematic numerical study of the breakdown regime is beyond the scope of the present work.","revision_made":"yes","referee_comment":"Sec. IIF, Eqs. (26)-(30): The self-consistency argument showing epsilon_A ~ O(v_0) and that neglected terms are subleading is verified numerically only for specific AMA parameters (Fig. 6, with h=0.2, lambda=-1.0333). The argument relies on the scaling R_dot ~ O(1/R, v_0) and R_ddot ~ O(R_dot/R^2) for instanton paths. The paper would benefit from a brief discussion of the parameter range over which this self-consistency is expected to hold, and where it might break down."}],"tokens_in":27183,"tokens_out":1126,"duration_ms":63896,"standing_objections":[]},"desk_editor":{"model":"glm-5.2","letter":"Bottom line: this paper extends classical nucleation theory to non-equilibrium non-conserved scalar fields, and the key result is that the time-reversed relaxation (TRR) ansatz gives wrong barrier heights — differing by factors of ~2 in the examples studied. The correct result requires projecting onto the reaction coordinate using ψ' ∈ Ker(L†), which sidesteps explicitly computing the instanton-relaxation deviation. This is a genuine advance over the conserved case the same authors handled in [29], because the non-conserved case is harder: the TRR ansatz actually fails, and the paper's technique for dealing with that failure is new and well-motivated.","headline":"Solid paper extending nucleation theory to non-equilibrium non-conserved fields; main soft spot is shared radial symmetry assumption between theory and numerics.","tokens_in":28114,"tokens_out":205,"would_cite":true,"duration_ms":18486,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"glm-5.2","headline":"Nucleation far from equilibrium breaks time-reversal symmetry","keywords":[],"falsifier":"If numerical action minimization in regimes beyond the perturbative limit (small critical radius) shows that the NNT quasipotential barrier disagrees with the true minimum action, or if capillary waves are found to be unstable in a nonconserved nonequilibrium model, the theory's domain of validity would be narrower than claimed.","tokens_in":27434,"feed_emoji":"🔬","tokens_out":966,"duration_ms":51244,"temperature":0.7,"pith_summary":"Classical Nucleation Theory (CNT) describes how a new phase forms from a metastable one by tracking a single variable: the radius of a growing droplet. This paper extends that framework to nonequilibrium systems with a non-conserved order parameter, such as active matter and population dynamics models. The central result is that in these systems, the most probable path along which a droplet nucleates is NOT the time-reversal of the path along which it relaxes back. This distinction matters because it changes the effective mobility and the quasipotential barrier height. The authors derive the correct theory by carefully projecting the full field dynamics onto the droplet radius using a specific reaction coordinate, and they show that the resulting barrier heights differ from those obtained via the time-reversed relaxation ansatz by factors of roughly 2 in the examples studied. The theory is validated by numerical action minimization.","feed_headline":"Nonequilibrium nucleation breaks time-reversal symmetry","feed_subtitle":"Extending classical nucleation theory to active matter and population dynamics reveals that the standard time-reversal assumption fails, hal","key_machinery":"The key technical object is the reaction coordinate: the droplet radius is defined via a weighted integral of the field deviation, where the weighting function's derivative is chosen to lie in the kernel of the adjoint linear operator. This choice ensures that deviations of the instanton density profile from the relaxational profile contribute only at subleading order, making the projection self-consistent without requiring explicit computation of the instanton-relaxation difference.","core_discovery":"The authors show that for nonconserved nonequilibrium scalar field theories, the quasipotential barrier for nucleation requires projecting the field dynamics onto the droplet radius using a reaction coordinate defined by a function whose derivative lies in the kernel of the adjoint linear operator. This projection yields a modified interfacial mobility and quasipotential that differ from the results obtained by assuming the instanton equals the time-reversed relaxation path. The difference arises because, unlike in conserved-order-parameter systems, the density profile along the nucleation path deviates from the relaxational profile at leading order. The theory is validated numerically for a","pith_inferences":["If the perturbative expansion in inverse critical radius breaks down, for instance in systems where the critical droplet is small or where capillary waves are unstable, the single-reaction-coordinate reduction may fail, and the theory would need modification.","The distinction between conserved and nonconserved cases suggests a general principle: conservation laws can protect the time-reversal ansatz, while their absence exposes the full nonequilibrium structure of the instanton path.","The framework could be tested experimentally in active colloidal systems or microbial populations by measuring nucleation rates near coexistence and comparing with predictions from the time-reversed relaxation ansatz."],"forward_implications":["Nucleation rates in active matter systems with nonconserved dynamics could be systematically over- or under-estimated if one assumes the time-reversal symmetry of the instanton, with errors of order a factor of 2 in the barrier height.","The framework can be applied to reaction-diffusion systems, ecological invasion models, and synthetic biological systems where detailed balance is broken, providing analytical predictions where previously only numerical action minimization was available.","The method extends naturally to systems with multiple coupled order parameters or mixed conserved/nonconserved dynamics, broadening the class of nonequilibrium nucleation problems amenable to analytical treatment.","The derivation of capillary wave stability within the same framework confirms that the spherical-droplet assumption underlying the theory is self-consistent, at least for the models studied."],"fun_headline_variants":["Nucleation theory extended to nonequilibrium scalar fields","Time-reversal symmetry breaking reshapes nucleation barriers","Reaction coordinate projection recovers nonequilibrium nucleation","Nonequilibrium droplet profiles deviate from relaxation paths","Capillary wave stability constrains nonequilibrium nucleation"],"cache_read_input_tokens":0,"weakest_assumption_plain":"The derivation assumes a perturbative regime where the critical radius is large, equivalently the flat-interface velocity is small, and that the instanton remains radially symmetric. If the critical droplet is not large or if shape fluctuations become unstable, the single-reaction-coordinate reduction underlying the theory would not hold.","fun_headline_variants_meta":{"raw":{"variants":["Nucleation theory extended to nonequilibrium scalar fields","Time-reversal symmetry breaking reshapes nucleation barriers","Reaction coordinate projection recovers nonequilibrium nucleation","Nonequilibrium droplet profiles deviate from relaxation paths","Capillary wave stability constrains nonequilibrium nucleation"]},"model":"glm-5.2","effort":"high","cost_usd":0.0,"raw_usage":{"total_tokens":813,"prompt_tokens":750,"completion_tokens":63,"prompt_tokens_details":null},"tokens_in":750,"tokens_out":63,"duration_ms":18053,"temperature":1.0,"reasoning_tokens":null,"cache_read_input_tokens":0,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-07T23:58:18.512765+00:00","model_set":{"reader":"glm-5.2"},"falsifier":"If numerical action minimization in regimes beyond the perturbative limit (small critical radius) shows that the NNT quasipotential barrier disagrees with the true minimum action, or if capillary waves are found to be unstable in a nonconserved nonequilibrium model, the theory's domain of validity would be narrower than claimed.","supporting_citations":[],"review_version":1}