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arxiv: 2605.10346 · v1 · submitted 2026-05-11 · ✦ hep-ph · cond-mat.stat-mech

Recognition: no theorem link

Non-equilibrium scaling across first-order transitions with relativistic scalar fields

Leon J. Sieke , Jessica Fuchs , Lorenz von Smekal
Authors on Pith no claims yet

Pith reviewed 2026-05-12 05:03 UTC · model grok-4.3

classification ✦ hep-ph cond-mat.stat-mech
keywords non-equilibrium scalingfirst-order phase transitionKibble-Zurek scalingfinite-time scalingrelativistic scalar fieldZ2 symmetryLangevin dynamicsout-of-equilibrium dynamics
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0 comments X

The pith

Rapid linear driving across first-order transitions in relativistic Z2 scalar fields produces finite-time scaling identical to mean-field theory, independent of temperature and dimension.

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

The paper examines the out-of-equilibrium dynamics of a relativistic scalar field theory with Z2 symmetry when linearly driven across a magnetic first-order phase transition. Classical-statistical lattice simulations with Langevin dynamics in two and three dimensions show that sufficiently fast driving produces scaling that matches mean-field behavior and does not depend on temperature or spatial dimension. Slower driving near the critical temperature leads to a crossover into critical Kibble-Zurek scaling, while much lower temperatures allow nucleation and growth to dominate and introduce corrections. The crossover near the point where the order parameter changes sign is captured by the leading algebraic correction to Kibble-Zurek scaling, and universal scaling functions for the order parameter are computed in the regime where they apply for temperatures up to Tc.

Core claim

If the driving timescale is sufficiently fast, the system exhibits finite-time scaling behavior independent of temperature and dimensionality, identical to that observed in mean-field simulations. In slow quenches near Tc this mean-field behavior crosses over to critical Kibble-Zurek scaling behavior, while for temperatures T ≪ Tc nucleation and growth dominate the transition dynamics, resulting in corrections to scaling. Near the transition point where the order parameter changes sign, the crossover between mean-field and critical out-of-equilibrium dynamics is found to be well described by the leading algebraic correction to Kibble-Zurek scaling. Universal non-equilibrium scaling behavior

What carries the argument

Finite-time scaling of the order parameter under linear driving, with the crossover to Kibble-Zurek scaling captured by its leading algebraic correction.

If this is right

  • Fast driving yields mean-field finite-time scaling for the order parameter regardless of temperature below Tc.
  • Slow driving near Tc replaces the mean-field scaling with critical Kibble-Zurek scaling.
  • At low temperatures, nucleation and growth produce corrections to the universal scaling.
  • The crossover between these regimes near the sign change of the order parameter follows the leading algebraic correction to Kibble-Zurek scaling.
  • Universal scaling functions for the order parameter can be extracted when driving is fast enough to suppress nucleation yet slow enough for correlations to develop.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The temperature independence suggests that the same finite-time scaling could appear in other models with first-order transitions under comparable linear driving protocols.
  • The results indicate that quench rate can be used to select between mean-field-dominated and nucleation-dominated regimes in systems described by similar field theories.

Load-bearing premise

Classical-statistical lattice simulations with Langevin dynamics accurately capture the out-of-equilibrium dynamics of the relativistic Z2-symmetric scalar field theory across the range of driving timescales and temperatures studied.

What would settle it

A simulation or observation in which the order parameter evolution during a fast linear quench depends on temperature or changes when dimensionality is varied would falsify the claimed temperature- and dimension-independent finite-time scaling.

Figures

Figures reproduced from arXiv: 2605.10346 by Jessica Fuchs, Leon J. Sieke, Lorenz von Smekal.

Figure 1
Figure 1. Figure 1: Qualitative phase diagram of the model in the [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Snapshots of field configurations during the phase transition across the critical point ( [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Quench rate dependence of the mean-field hysteresis exponents [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: Scaled mean-field magnetization data for multiple slow quench rates [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Mean-field scaling function fM extracted from the rescaled magnetization data in fast quenches for different values of the damping coefficient γ. The left panel shows the rescaled magnetization assuming the inertial term is irrelevant, i.e [PITH_FULL_IMAGE:figures/full_fig_p011_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Magnetization M as a function of external field J for different quench rates R close to the critical temperature (τ = −0.001, left) and away from the critical point (τ = −0.1, right). Statistical uncertainties, drawn as shaded regions, are not visible at this scale. 11 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Coercive field Jc ≡ J(M = 0) as a function of quench rate R and reduced temperature τ in d = 2 (left) and d = 3 (right) spatial dimensions. Open yellow circles correspond to data at the critical temperature τ = 0 obtained in [91]. The expected critical scaling Jc ∼ R βδ/νrc is indicated by the solid black line, with the critical exponents being βδ/νrc ≈ 0.464 for the d = 2 Ising universality class and βδ/ν… view at source ↗
Figure 8
Figure 8. Figure 8: Scaled coercive field in d = 2 (left) and d = 3 (right) spatial dimensions. Data points corresponding to fast quenches (R > 10−3 for d = 2 and R > 2.5 × 10−5 for d = 3) are drawn semi-transparent. The solid black line indicates the expected finite-time scaling behavior with Ising critical exponents. The dashed black line indicates finite-time scaling with a mean-field exponent of 3/4. The upper panels show… view at source ↗
Figure 9
Figure 9. Figure 9: Remanent magnetization Mr ≡ M(J = 0) as a function of quench rate R and reduced temperature τ in d = 2 (left) and d = 3 (right) spatial dimensions. The expected critical scaling Mr ∼ R β/νrc is indicated by the solid black line, with the exponent being β/νrc ≈ 0.031 for the d = 2 Ising universality class and β/νrc ≈ 0.115 for d = 3. The values of the critical exponents used, as well as their references are… view at source ↗
Figure 10
Figure 10. Figure 10: Scaled remanent magnetization in d = 2 (left) and d = 3 (right) spatial dimensions. Data points corresponding to fast quenches (R > 10−3 for d = 2 and R > 2.5 × 10−5 for d = 3) are drawn semi-transparent. The solid black lines show the critical finite-time scaling behavior, while the black dashed lines indicate the asymptotic mean-field scaling behavior of the underdamped system. 5.3. Scaling functions fo… view at source ↗
Figure 11
Figure 11. Figure 11: Left: Magnetization M as a function of the external field J for different quench rate and temperature combinations corresponding to fixed values of the scaling variable xR = −τ/R 1/νrc ; xR = 0.1 (blue), xR = 1 (yellow) and xR = 10 (green). Different quench rates are represented by different line styles: R = 10−5 , R = 10−4 , R = 2 × 10−4 , R = 5 × 10−4 . The reduced temperature τ was chosen accordingly t… view at source ↗
Figure 12
Figure 12. Figure 12: Two-parameter scaling collapse of the magnetization in [PITH_FULL_IMAGE:figures/full_fig_p017_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: Scaling collapse of the magnetization to the underdamped mean-field scaling function (dashed black line) for fast quenches ( [PITH_FULL_IMAGE:figures/full_fig_p018_13.png] view at source ↗
read the original abstract

We investigate the out-of-equilibrium dynamics of a relativistic $Z_2$-symmetric scalar field theory with Langevin dynamics in two and three spatial dimensions under linear driving across magnetic first-order phase transitions, close to and far below the critical temperature $T_c$. Using classical-statistical lattice simulations, we find that if the driving timescale is sufficiently fast, the system exhibits finite-time scaling behavior independent of temperature and dimensionality, identical to that observed in mean-field simulations. In slow quenches near $T_c$ this mean-field behavior crosses over to critical Kibble-Zurek scaling behavior, while for temperatures $T \ll T_c$ nucleation and growth dominate the transition dynamics, resulting in corrections to scaling. Near the transition point where the order parameter changes sign, the crossover between mean-field and critical out-of-equilibrium dynamics is found to be well described by the leading algebraic correction to Kibble-Zurek scaling. We find that universal non-equilibrium scaling behavior can be observed for $T \lesssim T_c$, provided the driving is fast enough to avoid nucleation but slow enough for correlations to form, and compute the associated universal scaling functions for the order parameter.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript reports classical-statistical lattice simulations of a relativistic Z_2-symmetric scalar field with Langevin dynamics, driven linearly across first-order phase transitions in two and three spatial dimensions. It claims that fast driving produces finite-time scaling independent of temperature and dimensionality that matches mean-field simulations; slow quenches near T_c cross over to Kibble-Zurek scaling; far below T_c nucleation and growth dominate and produce corrections; and near the order-parameter sign-change point the mean-field to critical crossover is captured by the leading algebraic correction to Kibble-Zurek scaling. Universal scaling functions for the order parameter are extracted for T ≲ T_c when driving is fast enough to suppress nucleation but slow enough for correlations to develop.

Significance. If the numerical results hold, the work supplies concrete evidence for the coexistence of distinct non-equilibrium scaling regimes (mean-field finite-time, Kibble-Zurek, and nucleation-corrected) in driven first-order transitions of relativistic scalars. The direct extraction of universal scaling functions from simulations constitutes a strength, offering falsifiable predictions that can be tested in other models or dimensions. The findings are relevant to out-of-equilibrium dynamics in cosmology and heavy-ion physics.

major comments (2)
  1. [Numerical methods] Numerical methods section: the manuscript does not report lattice volumes, lattice spacings, number of independent realizations, or the precise procedures used for error estimation and data exclusion. These parameters are load-bearing for the central claims, as the reported crossovers and scaling functions rest entirely on the statistical reliability of the Langevin trajectories.
  2. [Results] Results on fast-quench regime: the claim that the finite-time scaling is identical to mean-field and independent of T and d requires explicit quantitative comparison (e.g., collapse quality or fitted exponents) between the lattice data and the mean-field reference simulations; without these, the independence statement remains qualitative.
minor comments (2)
  1. [Abstract] The abstract and introduction would benefit from a brief statement of the precise definition of the driving timescale and the observable used to locate the order-parameter sign-change point.
  2. [Figures] Figure captions should explicitly state the range of driving rates and temperatures shown, to allow readers to map the plotted data onto the three regimes discussed in the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major point below and have revised the manuscript to incorporate the requested information and strengthen the quantitative support for our claims.

read point-by-point responses

  1. Referee: [Numerical methods] Numerical methods section: the manuscript does not report lattice volumes, lattice spacings, number of independent realizations, or the precise procedures used for error estimation and data exclusion. These parameters are load-bearing for the central claims, as the reported crossovers and scaling functions rest entirely on the statistical reliability of the Langevin trajectories.

    Authors: We agree that these details are necessary for assessing the statistical reliability of the results. In the revised manuscript we have expanded the Numerical Methods section to report the lattice volumes and spacings employed in both two and three dimensions, the number of independent realizations for each quench rate and temperature, and the precise error estimation procedure (including jackknife resampling and criteria for data exclusion based on trajectory convergence). revision: yes

  2. Referee: [Results] Results on fast-quench regime: the claim that the finite-time scaling is identical to mean-field and independent of T and d requires explicit quantitative comparison (e.g., collapse quality or fitted exponents) between the lattice data and the mean-field reference simulations; without these, the independence statement remains qualitative.

    Authors: We acknowledge that the original presentation of the fast-quench regime relied primarily on visual data collapse. In the revised manuscript we have added quantitative comparisons: we report the goodness-of-fit measures for the scaling collapse of the order parameter between the relativistic lattice data and the mean-field reference runs, together with the fitted effective exponents and their uncertainties, demonstrating consistency across the temperatures and dimensions considered. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper's central results are extracted from direct classical-statistical lattice simulations of Langevin dynamics for the relativistic Z2 scalar field. Scaling regimes (fast-quench finite-time scaling, crossover to Kibble-Zurek near Tc, nucleation corrections far below Tc) and associated universal functions are reported as numerical outputs, not as analytical derivations or predictions that presuppose the observed behavior. No self-definitional equations, fitted parameters renamed as predictions, or load-bearing self-citation chains appear in the methodology or claims.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the standard assumptions of the Z2-symmetric relativistic scalar field model and the validity of the classical-statistical approximation for its Langevin dynamics on a lattice; no new entities are postulated and no parameters are fitted to produce the scaling results.

axioms (2)
  • domain assumption The scalar field obeys relativistic dynamics with Z2 symmetry.
    Core model assumption defining the theory under study.
  • domain assumption Langevin dynamics governs the out-of-equilibrium evolution on the lattice.
    Standard framework for classical-statistical simulations of scalar fields.

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Reference graph

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