{"id":"6fd30827-52ab-4fc6-a395-cdd07f640ba7","arxiv_id":"2411.19643","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":4,"one_line_summary":"At the Curie-Weiss temperature-driven transition, heat current noise diverges with system size, while at the field-driven transition it diverges only near, not at, the transition point.","lead":"A Curie-Weiss magnet connected to two heat baths shows different heat current fluctuation behavior at its two kinds of phase transitions: noise diverges at the temperature-driven transition, but stays finite exactly at the field-driven transition while peaking exponentially nearby. The work tests whether divergent current noise is a universal feature of nonequilibrium phase transitions and finds the picture is more nuanced.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"No significant objection identified.","rationale":"The reader's weakest assumption identifies the additivity of path-integral and two-state contributions as the main limitation. I agree that this is a genuine limitation of the methodological claim, but it is not load-bearing for the central physical results. The exact spectral calculations directly produce the finite-size scaling shown in Figs. 4 and 12; the path-integral and two-state models provide interpretation and asymptotic support, but they are not the sole evidence. The qualitative conclusions, namely power-law divergence at the continuous transition and exponentially growing but non-divergent-at-h=0 behavior at the field-driven transition, would survive even if the additivity approximation were imperfect. The fitted exponents carry no error bars, which argues for a conditional rather than an unqualified acceptance, but does not overturn the paper's core findings. Therefore I leave the reader's conditional verdict unchanged.","tokens_in":19953,"tokens_out":11983,"duration_ms":116746,"concrete_test":"Decompose the exact variance in Eq. (32) into the slowest-eigenvalue (switching) contribution and the sum over all remaining modes, then compare those two pieces against the two-state variance of Eq. (48) and the path-integral variance for an untested parameter set, e.g., Teff = 0.7 Tc, Delta beta = 0.3 beta_eff, at N = 200, 300, and 400. If the slow-mode term deviates from Eq. (48) by more than a few percent of the peak height, the additivity assumption should be treated as an uncontrolled approximation rather than a validated decomposition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"I do not find a load-bearing flaw in the central claims. The power-law divergence at the temperature-driven transition and the exponentially growing noise peaks near the field-driven transition are established by exact finite-size spectral calculations (Secs. IV A and IV C, Figs. 4 and 12), independent of the approximate decomposition methods. The closest thing to a soft spot is the additive combination of path-integral and two-state variances in Sec. IV C (Fig. 10), which is verified for only two system sizes and one parameter set. If that additivity failed for other parameters, the attribution of the exponential peak specifically to stochastic switching would lose some support, but the exact numerical scaling behavior would remain intact. Similarly, the fitted scaling exponents in Figs. 6, 8, and 12 lack uncertainty estimates and cover modest N ranges, but the qualitative divergence patterns are clear from the plotted data and from the large-deviation structure.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper analyzes the finite-size scaling of heat current fluctuations in the Curie-Weiss model coupled to two thermal baths at different temperatures. Using exact finite-size spectral cumulant calculations, a path-integral approach, and an effective two-state model, the authors find that at the temperature-driven continuous transition the equilibrium contribution to the current variance vanishes with system size while the nonequilibrium contribution diverges as a power law, leading to nonmonotonic scaling for small temperature bias. At the magnetic-field-driven transition below the critical temperature, the variance evaluated exactly at h=0 does not diverge, but noise maxima at finite fields grow exponentially with system size while their positions shift toward h=0. The paper also proposes that combining path-integral and two-state descriptions characterizes current fluctuations in bistable macroscopic systems.","tokens_in":20095,"tokens_out":16194,"duration_ms":141016,"significance":"If the results hold, they substantially refine the emerging classification of current fluctuations at phase transitions: continuous transitions are not always associated with power-law divergence at the transition point, and first-order-like transitions can produce exponentially growing noise peaks away from the transition point rather than at it. The main scaling statements are established by exact diagonalization of the finite-size generator, with analytic support from the equilibrium fluctuation-response relation and the Delta-beta expansion in Eq. (49). The availability of Wolfram Mathematica notebooks and data at a DOI is a clear strength. The methodological combination of path-integral and two-state models is useful, although its validation is narrower than the exact scaling results.","major_comments":[],"minor_comments":[{"comment":"The additive decomposition of the noise into path-integral and two-state contributions is verified only for N=100 and 200 at Teff=0.8Tc and Delta beta=0.5 beta_eff. Since the abstract's methodological claim is stated generally, please either test at least one additional parameter set or explicitly state that the quantitative agreement is demonstrated only for the shown parameters.","section":"Sec. IV C, Fig. 10"},{"comment":"The fitted scaling exponents 0.65 and -0.46 are reported without uncertainty estimates or fit ranges; please report standard errors and the range of N used in each fit, and do the same for the exponents in Figs. 8 and 12.","section":"Sec. IV A, Fig. 6"},{"comment":"The exact data for the exponential peak growth are restricted to N<=450 because of the small spectral gap in the bistable regime, so the exponential scaling at larger N rests mainly on the two-state model; a brief comment on numerical reliability, such as the size of the spectral gap or condition estimates, would strengthen the claim.","section":"Sec. IV C, Fig. 12"},{"comment":"The caption says 'T = 0.8Teff', which appears to be a typo; it should read Teff = 0.8Tc.","section":"Fig. 2 caption"},{"comment":"The phrase 'The green solid line in represents' is incomplete; it should read 'in (a) represents'.","section":"Fig. 13 caption"},{"comment":"The finite-difference evaluation of the path-integral variance is mentioned only in a footnote; because this is the numerical tool behind several asymptotic curves, a sentence in the main text describing the choice of epsilon and the consistency checks would improve reproducibility.","section":"Sec. III D, Eq. (45)"}],"recommendation":"minor_revision","confidential_remarks":null},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The punchline: this paper shows that the common picture of diverging current fluctuations at nonequilibrium phase transitions is not universal in the way people assumed. In the Curie-Weiss model driven by two heat baths, the heat current noise at the temperature-driven continuous transition does diverge with system size as a power law, but at the field-driven transition the noise at the transition point does not diverge. What actually diverges is the peak of the noise at small nonzero field, which grows exponentially and moves toward the transition as N increases. That is a clear, non-obvious answer to the question the paper poses.\n\nThe paper does its job well. The central scalings are pinned down by exact finite-size master equation calculations, so the results do not rest on the approximations. The path integral approach and the two-state model are each checked against the exact numerics, and the agreement is convincing where they should apply. The Delta-beta expansion nicely explains the nonmonotonic scaling at small bias as a competition between a vanishing equilibrium contribution and a divergent nonequilibrium one. The authors also flag the order-of-limits subtlety explicitly and connect it to the familiar magnetization case. Data and notebooks are deposited, so the numerical claims are reproducible.\n\nThe soft spots are proportionate. The biggest one is the additivity of path-integral and two-state variances in Sec. IV C: it is a plausible decomposition, but it is not derived, and it is verified for only two system sizes at one parameter set. If it fails elsewhere, the attribution of the exponential peak to stochastic switching would weaken, though the exact numerical scaling at h=0 would remain intact. The fitted scaling exponents (0.65, -0.46, and the critical-isotherm exponents) come with no uncertainty estimates and modest N ranges; they are indicative, not precise. The peak-position shift is said to be close to hyperbolic but fitted as a power law; that section could be tightened. These are revision-level issues, not fatal ones.\n\nThis paper is for people working on current statistics at nonequilibrium phase transitions, particularly finite-size scaling. It deserves a serious referee: the question is central, the method mix is sound, and the claims are honestly circumscribed. I would send it to review. The main requests would be error bars on the exponents and a broader check of the additivity assumption.","headline":"A careful finite-size study showing heat current noise diverges at the temperature-driven transition but only near—not at—the field-driven transition, with exact numerics carrying the argument.","tokens_in":20628,"tokens_out":2762,"would_cite":true,"duration_ms":24182,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"At the temperature-driven continuous phase transition of the Curie-Weiss model, heat current noise diverges as a power law with system size, while at the field-driven transition the noise instead peaks slightly away from the transition…","keywords":["heat current fluctuations","Curie-Weiss model","nonequilibrium phase transitions","finite-size scaling","large deviation theory","full counting statistics","two-state model","critical phenomena"],"falsifier":"Evaluate the exact heat-current variance by the spectral method for a parameter set outside the one used to benchmark the two-state addition, for example Teff=0.7Tc with $\\Delta$ $\\beta$=0.3 beta_eff, across N=100 to 450, and compare with the sum of the path-integral and two-state predictions; a mismatch that grows with N would falsify the decomposition. Alternatively, at h=0, Teff=0.8Tc, checking whether Var(q1) continues to saturate rather than diverging as N grows would settle the no-divergence-at-the-transition claim.","tokens_in":19735,"feed_emoji":"🔥","tokens_out":7267,"duration_ms":58683,"temperature":0.7,"pith_summary":"The paper asks whether current fluctuations always blow up at a nonequilibrium phase transition, and answers that the behavior depends on which control parameter drives the transition. In the Curie-Weiss model coupled to two heat baths, the heat current noise at the temperature-driven continuous transition splits into an equilibrium part that vanishes with system size and a nonequilibrium part that diverges as a power law; for small temperature bias this produces a nonmonotonic dependence on system size. At the magnetic-field-driven transition below the critical temperature, the noise evaluated exactly at h=0 does not diverge, but peaks at small nonzero fields whose height grows exponentially with system size while their position moves toward h=0, an effect traced to stochastic switching between two current values. The paper also demonstrates a two-part method for large systems: a path-integral calculation of fluctuations around a fixed point plus an effective two-state model for switching. If correct, it means the thermodynamic limit and the order of limits matter for defining divergence of current noise near a transition.","feed_headline":"Heat noise diverges at one transition, not the other","feed_subtitle":"In a Curie-Weiss magnet, field-driven noise peaks still grow exponentially just off the transition point.","key_machinery":"The load-bearing object is the normalized heat current variance Var(q1), computed exactly for finite N from the full counting statistics of the mesoscopic master equation and in the thermodynamic limit from the nonequilibrium quasipotential V(m) via a path-integral large-deviation scheme. At equilibrium the variance is fixed by the fluctuation-response relation Var(q1)=2 d<Q1>/dDelta $\\beta$, which gives the thermodynamic-limit noise directly. In the bistable regime the additional mechanism is an effective two-state Markov model: two fixed-point current values <q>+ and <q>- with transition rates r+ and r- suppressed exponentially by the quasipotential barrier, yielding the switching-noise formula Var(q) approximately 2(<q>+ - <q>-)^2 p+ p-/(r+ + r-). Combining these two pieces reproduces the exact finite-size noise in the bistable region.","core_discovery":"The central discovery is that heat current fluctuations witness the Curie-Weiss phase transition in a way that is not universal across transition types. At the continuous temperature-driven transition (Teff=Tc, h=0), the normalized heat current variance behaves as Var(q1)=Vareq(q1)+$\\Delta$ $beta^{2}$[<$q1^{4}$>eq/12+G3(q1)/3]+O($\\Delta$ $beta^{4}$): the equilibrium variance vanishes as a power law, while the nonequilibrium contribution diverges as a power law, driven mainly by divergent equilibrium kurtosis; for small $\\Delta$ $\\beta$ the two competing power laws produce nonmonotonic scaling with N. At the field-driven transition (h=0, Teff<Tc), the variance at h=0 saturates to a finite value, but near the transition it develops peaks whose maximum grows exponentially with N and whose position shifts to h=0 as a power law; the paper attributes the peaks to stochastic switching between the stable and metastable magnetization states, which carry different heat currents, and confirms this with an effective two-state model appended to the path-integral result. Consequently, fluctuations diverge infinitely close to the field-driven transition even though they do not diverge at the transition point.","pith_inferences":["If the divergent nonequilibrium noise at Teff=Tc is indeed controlled by the equilibrium fourth cumulant, then higher-order current cumulants may serve as early-warning indicators of criticality in other all-to-all or mean-field-type systems before the variance itself diverges.","The infinitely close divergence at the field-driven transition suggests that the order of the thermodynamic and long-time limits, or of h approaching 0 and N tending to infinity, controls observable noise; a finite-size device could show large noise peaks in a field region where the infinite-size steady state shows none.","The paper's two-state decomposition, if it holds, implies that the exponential noise peaks are a kinetic effect dependent on measurement time relative to switching rates; a finite-time experiment would see the peak only when the observation window is long enough for switches to occur, which is testable by time-resolved current measurements.","Following a suggestion the authors leave open, comparing the heat-current-noise scaling with the heat-capacity behavior of the same nonequilibrium Curie-Weiss model could reveal whether the divergence pattern is inherited from equilibrium-like energy fluctuations."],"forward_implications":["Heat current noise is a genuine critical observable: at equilibrium it develops a kink at the transition even though the average heat current is zero, so noise measurements can locate a phase transition without any net current.","At the temperature-driven transition, the equilibrium and nonequilibrium noise contributions scale differently with N, so the total noise can first drop and then rise; simulations probing only small systems could misread the scaling.","At the field-driven transition, the divergence is shifted off the transition point; locating critical current fluctuations requires scanning a neighborhood of the transition, not just the transition point itself.","The exponential growth of noise peaks even though the heat current is continuous at h=0 breaks the simple rule that continuous transitions give power-law divergence and discontinuous transitions give exponential divergence.","The two-part method, path integral plus two-state model, offers a practical way to compute current fluctuations in large systems where direct numerics is limited by the closing spectral gap."],"supporting_citations":[{"why":"Supplies the mesoscopic master-equation description, rate construction, local detailed balance, and path-integral framework used throughout the paper.","marker":"[47]"},{"why":"Provides the large-deviation quasipotential formalism that defines the magnetization rate function V(m) and the thermodynamic-limit stationary state.","marker":"[56]"},{"why":"Gives the full-counting-statistics symmetry underlying the equilibrium fluctuation-response relation used to obtain thermodynamic-limit noise.","marker":"[74]"},{"why":"Supplies the spectral recursion used to compute exact finite-size current cumulants from the rate matrix.","marker":"[68]"},{"why":"Provides the effective two-state switching model and its variance formula used to describe the bistable regime.","marker":"[33]"},{"why":"Gives an earlier example of power-law divergent current fluctuations at a continuous phase transition, which motivates the universality question addressed here.","marker":"[29]"},{"why":"Provides a comparison model at a critical line, showing where the Curie-Weiss critical-isotherm behavior differs from previous results.","marker":"[30]"},{"why":"Supplies the nonequilibrium Curie-Weiss phase diagram and specific-heat behavior to which the paper connects its fluctuation scaling at the end.","marker":"[48]"}],"fun_headline_variants":["Heat noise diverges only at temperature-driven Curie-Weiss transition","Near field-driven transition, heat noise peaks grow exponentially with size","Curie-Weiss heat noise: divergent at one transition, exponential peaks near the other","Heat current noise does not diverge at field-driven transition, but peaks nearby","Heat noise divergence distinguishes Curie-Weiss transitions"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that, in the bistable region, the total heat-current noise is the sum of the local fixed-point noise and the switching noise, an additivity that is assumed and checked only for Teff=0.8Tc, $\\Delta$ $\\beta$=0.5 beta_eff, and two system sizes; if this sum fails elsewhere, the attribution of the exponential peaks to switching loses its basis.","fun_headline_variants_meta":{"raw":{"variants":["Heat noise diverges only at temperature-driven Curie-Weiss transition","Near field-driven transition, heat noise peaks grow exponentially with size","Curie-Weiss heat noise: divergent at one transition, exponential peaks near the other","Heat current noise does not diverge at field-driven transition, but peaks nearby","Heat noise divergence distinguishes Curie-Weiss transitions"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000713,"raw_usage":{"total_tokens":3259,"prompt_tokens":1048,"completion_tokens":2211,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":664,"completion_tokens_details":{"reasoning_tokens":2119}},"tokens_in":664,"tokens_out":2211,"duration_ms":14822,"temperature":1.0,"reasoning_tokens":2119,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-12T05:58:55.453923+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Evaluate the exact heat-current variance by the spectral method for a parameter set outside the one used to benchmark the two-state addition, for example Teff=0.7Tc with $\\Delta$ $\\beta$=0.3 beta_eff, across N=100 to 450, and compare with the sum of the path-integral and two-state predictions; a mismatch that grows with N would falsify the decomposition. Alternatively, at h=0, Teff=0.8Tc, checking whether Var(q1) continues to saturate rather than diverging as N grows would settle the no-divergence-at-the-transition claim.","supporting_citations":[{"cited_title":"Meibohm and M","cited_arxiv_id":null,"evidence_quote":"Provides the large-deviation quasipotential formalism that defines the magnetization rate function V(m) and the thermodynamic-limit stationary state."},{"cited_title":"Walldorf, F","cited_arxiv_id":null,"evidence_quote":"Supplies the spectral recursion used to compute exact finite-size current cumulants from the rate matrix."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the effective two-state switching model and its variance formula used to describe the bistable regime."},{"cited_title":"Nguyen, U","cited_arxiv_id":null,"evidence_quote":"Provides a comparison model at a critical line, showing where the Curie-Weiss critical-isotherm behavior differs from previous results."}],"review_version":1}