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Critical heat current fluctuations in Curie-Weiss model in and out of equilibrium

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Pith's one-line read 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…

desk verdict 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. read the letter →

arxiv 2411.19643 v3 pith:FUZCJETN submitted 2024-11-29 cond-mat.stat-mech

Krzysztof Ptaszynski , Massimiliano Esposito This is my paper · ORCID
classification cond-mat.stat-mech
keywords heatcurrentfluctuationsCurie-Weissmodelnonequilibriumphasetransitionsfinite-sizescalinglargedeviationtheoryfullcountingstatisticstwo-statecriticalphenomena
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

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.

What carries the argument

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.

What would settle it

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.

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Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

  • 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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • 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.
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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Referee Report

0 major / 6 minor

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.

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.

minor comments (6)
  1. [Sec. IV C, Fig. 10] 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.
  2. [Sec. IV A, Fig. 6] 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.
  3. [Sec. IV C, Fig. 12] 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.
  4. [Fig. 2 caption] The caption says 'T = 0.8Teff', which appears to be a typo; it should read Teff = 0.8Tc.
  5. [Fig. 13 caption] The phrase 'The green solid line in represents' is incomplete; it should read 'in (a) represents'.
  6. [Sec. III D, Eq. (45)] 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.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity; central claims rest on exact finite-size numerics and model-derived large-deviation formulas.

full rationale

The derivation chain is self-contained for the model. The central results, namely the power-law divergence of heat-current noise at the temperature-driven transition and the exponentially growing off-transition noise maxima at the field-driven transition, are obtained from exact finite-size spectral calculations (Sec. III A; Figs. 3, 4, 7, 9, 11, 12) and from large-deviation expressions derived within the paper from the master equation, such as the quasipotential in Eq. (14), the average current in Eq. (16), the equilibrium fluctuation-response result in Eq. (37), and the two-state switching formula in Eq. (48). The path-integral and two-state methods are taken from prior work, including some by the authors, but they are benchmarked against the exact numerical data rather than fitted to the central claims. In particular, the exponential peak growth and the nondivergence at h=0 in the field-driven case follow directly from Eq. (48) together with the symmetry-induced cancellation of the prefactor at h=0, and the two-state predictions in Fig. 12 are compared with exact dots rather than used as the sole evidence. The only soft point is the additive combination of path-integral and two-state variances in Sec. IV C, which is verified for limited parameters and is an approximation robustness concern, not a circularity: even if that additivity failed, the exact finite-size scaling behavior would remain established. No equation reduces by construction to a fitted parameter, and no load-bearing claim rests solely on a self-citation chain.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

No new physical entities (particles, forces, dimensions) are introduced. The effective two-state model is a reduction of the original dynamics, not a new postulated entity. The free parameters are limited to scaling exponents fitted to finite-size data; the central qualitative claims do not depend on their exact values.

free parameters (4)
  • fitted scaling exponent for noise peak magnitude at temperature-driven transition = ~0.65
    Estimated by power-law fit in Fig. 6(a); not derived analytically. The reported value is fitted to exact finite-size data.
  • fitted scaling exponent for noise peak displacement from critical temperature = ~-0.46
    Estimated by power-law fit in Fig. 6(b); the exponent value is a fit, not an analytic prediction.
  • fitted scaling exponents for critical-isotherm noise peaks = not reported in text
    Fig. 8 shows fitted power-law lines for peak magnitude and position at the critical isotherm, but the numerical exponents are not stated.
  • fitted scaling exponent for peak position shift in field-driven transition = close to -1 (hyperbolic)
    Reported as a power law close to hyperbolic in Fig. 12(b); the exponent is obtained by fitting exact results.
assumptions (5)
  • domain assumption The master equation with Arrhenius rates (Eq. 8) satisfying local detailed balance (Eq. 6) is the correct mesoscopic description of the Curie-Weiss model coupled to thermal baths.
    Sec. II A; the model definition and rate choice are assumed, not derived. All results depend on this specific dynamics.
  • standard math For symmetric coupling Gamma1 = Gamma2, the quasipotential coincides with the equilibrium free energy at inverse temperature beta_eff (Eqs. 20-21).
    Sec. II C; exact algebraic result from the rate ratio, not an approximation.
  • domain assumption The path integral (Freidlin-Wentzell) Hamiltonian fixed point gives the scaled cumulant generating function in the thermodynamic limit (Eqs. 40-42).
    Sec. III D; standard large deviation result taken from Refs. [45,47,75,76], assumed valid for this model.
  • domain assumption The two-state model transition rates in Eq. (46) and switching variance in Eq. (48) from Ref. [33] apply to this model after reparameterization.
    Sec. III E; adapted from Ref. [33] and tested against exact numerics in Fig. 10.
  • standard math The Laplace (saddle-point) method is used to evaluate integrals in the large-N limit.
    Used in Sec. II B and elsewhere for converting sums to integrals and evaluating them asymptotically.

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Cite this review

Pith. "Pith review of Critical heat current fluctuations in Curie-Weiss model in and out of equilibrium." pith.science (2026). https://pith.science/paper/FUZCJETN

@misc{pith2026241119643,
  author       = {Pith},
  title        = {Pith review of: Critical heat current fluctuations in Curie-Weiss model in and out of equilibrium},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FUZCJETN}},
  note         = {Machine review of arXiv:2411.19643}
}
read the original abstract

In some models of nonequilibrium phase transitions, fluctuations of the analyzed currents have been observed to diverge with system size. To assess whether this behavior is universal across phase transitions, we examined heat current fluctuations in the Curie-Weiss model, a paradigmatic model of the paramagnetic-ferromagnetic phase transition, coupled to two thermal baths. This model exhibits phase transitions driven by both the temperature and the magnetic field. We find that at the temperature-driven phase transition, the heat current noise consists of two contributions: the equilibrium part, which vanishes with system size, and the nonequilibrium part, which diverges with system size. For small temperature differences, this leads to nonmonotonic scaling of fluctuations with system size. In contrast, at the magnetic-field-driven phase transition, heat current fluctuations do not diverge when observed precisely at the phase transition point. Instead, out of equilibrium, the noise is enhanced at the magnetic field values away but close to the phase transition point, due to stochastic switching between two current values. The maximum value of noise increases exponentially with system size, while the position of this maximum shifts towards the phase transition point. Finally, on the methodological side, the paper demonstrates that current fluctuations in large systems can be effectively characterized by combining a path integral approach for macroscopic fluctuations together with an effective two-state model describing subextensive transitions between the two macroscopic states involved in the phase transition.

Figures

Figures reproduced from arXiv: 2411.19643 by the authors.

Figure 1
Figure 1. FIG. 1. The normalized stationary magnetization [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. The normalized stationary magnetization [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. The effective temperature dependence of the heat [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (9 more)
Figure 4
Figure 4. Figure 4: FIG. 4. The finite-size scaling (on the log-log scale) of the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6. The finite-size scaling of the magnitude of the peak [PITH_FULL_IMAGE:figures/full_fig_p009_6.png]
Figure 5
Figure 5. Figure 5: FIG. 5. The finite size scaling (on a log-log scale) of the third [PITH_FULL_IMAGE:figures/full_fig_p009_5.png]
Figure 8
Figure 8. Figure 8: FIG. 8. The finite-size scaling of the magnitude of the peak [PITH_FULL_IMAGE:figures/full_fig_p010_8.png]
Figure 10
Figure 10. Figure 10: FIG. 10. The magnetic field dependence of the heat current [PITH_FULL_IMAGE:figures/full_fig_p011_10.png]
Figure 9
Figure 9. Figure 9: FIG. 9. The magnetic field dependence of the heat current [PITH_FULL_IMAGE:figures/full_fig_p011_9.png]
Figure 11
Figure 11. Figure 11: FIG. 11. The finite-size scaling of the heat current variance [PITH_FULL_IMAGE:figures/full_fig_p011_11.png]
Figure 12
Figure 12. Figure 12: FIG. 12. The finite-size scaling of the magnitude of the peak [PITH_FULL_IMAGE:figures/full_fig_p012_12.png]
Figure 14
Figure 14. Figure 14: FIG. 14. The same plot as in Fig [PITH_FULL_IMAGE:figures/full_fig_p013_14.png]

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