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arxiv: 2604.25351 · v2 · pith:Z3KMAFILnew · submitted 2026-04-28 · ❄️ cond-mat.stat-mech

Spinodal-like scaling behavior after a temperature quench across the first-order phase transition in three-dimensional q-state Potts models

Andrea Pelissetto , Davide Rossini , Ettore Vicari This is my paper

Pith reviewed 2026-05-21 08:32 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords q-state Potts modelfirst-order phase transitiontemperature quenchspinodal-like behaviordroplet nucleationout-of-equilibrium dynamicsscaling behaviorthree dimensions
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The pith

Quenching 3D q-state Potts models across a first-order transition produces spinodal-like scaling in energy density with exponentially growing timescales.

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

The authors examine the dynamics of three-dimensional q-state Potts models when temperature is quenched from above to below a first-order phase transition. They argue that if smooth droplet nucleation drives the phase change for small enough initial deviations below the transition temperature, the energy density obeys a scaling law in the variable formed by the three-halves power of the logarithm of time multiplied by the relative quench depth. This produces a jump at a fixed positive value of the scaling variable and therefore a characteristic time that grows exponentially as the quench depth approaches zero from above. Numerical simulations for the q equals 6 case support the predicted scaling collapse.

Core claim

If the nucleation of smooth droplets is the relevant mechanism of the post-quench phase change for sufficiently small β_fo−β_i>0, the time-dependent energy density scales in terms of ρ=(ln t)^{3/2}δ, where δ=β/β_fo−1, with a discontinuity at a particular value ρ=ρ_s>0. This implies the emergence of a spinodal-like behavior, whose time scale τ increases exponentially as ln τ≈(ρ_s/δ)^{2/3} in the limit δ→0⁺. Numerical analysis of the quench protocol in the 3D q=6 Potts model supports the above spinodal-like scenario.

What carries the argument

The scaling variable ρ = (ln t)^{3/2} δ with δ = β/β_fo − 1, which collapses energy-density trajectories and exposes a discontinuity under the assumption that smooth-droplet nucleation controls the dynamics.

Load-bearing premise

Nucleation of smooth droplets is the relevant mechanism of the post-quench phase change when the initial inverse temperature lies sufficiently close to the transition value.

What would settle it

Numerical data for the 3D q=6 Potts model at small positive δ showing that energy density plotted versus ρ fails to display a discontinuity at any positive ρ_s or that the effective relaxation time does not grow as (ρ_s/δ)^{2/3}.

Figures

Figures reproduced from arXiv: 2604.25351 by Andrea Pelissetto, Davide Rossini, Ettore Vicari.

Figure 1
Figure 1. Figure 1: Post-quench time evolution of the rescaled energy density E(t) for the 3D q = 6 Potts model, versus the time t for several values of δ and L. The data for different sizes and same δ approach an asymptotic curve, which provides an accurate approximation of the time evolution in the thermodynamic limit. Statistical errors are hardly visible in the figure. 0.2 0.3 0.4 0.5 ρ 0.5 0.6 0.7 0.8 E δ = 0.05 δ = 0.04… view at source ↗
Figure 2
Figure 2. Figure 2: Post-quench evolution of the q = 6 energy density E(t) in the thermodynamic limit versus ρ = (ln t) 3/2 δ. The vertical dashed line corresponds to the estimate ρs = 0.391(4) of the asymptotic crossing point (the interval between the dotted lines gives the uncertainty). 4 Numerical results We now present a numerical analysis of the quench protocol for the 3D q = 6 Potts model. This system undergoes a FOT at… view at source ↗
Figure 3
Figure 3. Figure 3: The post-quench energy density E(t) versus ρr ≡ (ρ − ρs) δ−θ , using the optimal values ρs = 0.391 and θ = 1.8. The data appear to approach an asymptotic scaling curve with decreasing δ. a quite stable crossing point at an approximately constant ρ = ρs ≈ 0.39, and become increasingly steeper close to this point, as δ decreases. We estimate the crossing point to be at ρs = 0.391(4). This behavior is consist… view at source ↗
read the original abstract

We study the out-of-equilibrium spinodal-like behavior of three-dimensional (3D) $q$-state Potts models (for $q\ge 3$), observed when the temperature is quenched across the first-order transition (FOT) point $\beta_{\rm fo}=T_{\rm fo}^{-1}$. We consider a standard quench protocol, in which high-temperature configurations, thermalized at $\beta_i<\beta_{\rm fo}$, are driven across the FOT by a purely relaxational dynamics at $\beta>\beta_{\rm fo}$. We focus on the emergence of spinodal-like behaviors in the thermodynamic limit, associated with the dynamic phase change. We argue that, if the nucleation of smooth droplets is the relevant mechanism of the post-quench phase change, for sufficiently small $\beta_{\rm fo}-\beta_i>0$, the time-dependent energy density should scale in terms of $\rho = (\ln t)^{3/2} \delta$, where $\delta = \beta/\beta_{\rm fo}-1$, with a discontinuity at a particular value $\rho=\rho_s>0$. This implies the emergence of a spinodal-like behavior, whose time scale $\tau$ increases exponentially as $\ln \tau \approx (\rho_s/\delta)^{2/3}$ in the limit $\delta\to 0^+$. We present a numerical analysis of the quench protocol in the 3D $q=6$ Potts model, which supports the above spinodal-like scenario.

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 examines out-of-equilibrium dynamics in three-dimensional q-state Potts models after a temperature quench across the first-order transition point β_fo. Under the assumption that nucleation of smooth droplets is the dominant mechanism for sufficiently small δ = β/β_fo − 1, it derives that the time-dependent energy density scales with the variable ρ = (ln t)^{3/2} δ and exhibits a discontinuity at a positive value ρ = ρ_s. This implies a spinodal-like relaxation time τ whose logarithm grows as (ρ_s/δ)^{2/3} when δ → 0^+. Numerical simulations for the q = 6 case are presented as supporting evidence for the scaling collapse and the associated time-scale divergence.

Significance. If the scaling form and the location of the discontinuity are confirmed in the asymptotic small-δ regime, the work supplies an explicit, testable prediction linking classical nucleation theory to effective spinodal behavior in lattice systems with discontinuous transitions. The derivation of the composite scaling variable ρ directly from the droplet free-energy barrier and the provision of Monte Carlo data for a concrete three-dimensional model are concrete strengths that could guide future studies of metastable decay.

major comments (2)
  1. Numerical analysis section: the support for the central scaling claim rests on data collapse of the energy density versus ρ for the q=6 model. The manuscript does not quantify how small the simulated δ values are relative to the nucleation barrier height (e.g., via estimates of critical droplet radius or barrier ΔF), leaving open whether the observed discontinuity at ρ_s is resolved in the true asymptotic regime or affected by finite-size rounding and crossover to other relaxation channels.
  2. Theoretical derivation (around the definition of ρ and ρ_s): ρ_s is introduced as the point of discontinuity in the scaling function. If ρ_s is determined from the same quench trajectories rather than from an independent equilibrium calculation of the nucleation rate, the predicted ln τ ≈ (ρ_s/δ)^{2/3} relation becomes partly a fit to the data rather than a parameter-free consequence of the nucleation assumption, weakening the predictive power of the central claim.
minor comments (2)
  1. Abstract and introduction: the phrase 'spinodal-like behavior' is used without a brief operational definition distinguishing it from mean-field spinodal decomposition; adding one sentence would improve clarity for readers outside the immediate subfield.
  2. Figure captions and methods: all collapse plots should explicitly list the δ values, lattice sizes, and number of independent runs so that the range of validity of the scaling can be assessed directly from the published figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and for the constructive comments. We appreciate the positive evaluation of the significance of the work. We address the two major comments below and have revised the manuscript to incorporate additional analysis and clarifications.

read point-by-point responses

  1. Referee: Numerical analysis section: the support for the central scaling claim rests on data collapse of the energy density versus ρ for the q=6 model. The manuscript does not quantify how small the simulated δ values are relative to the nucleation barrier height (e.g., via estimates of critical droplet radius or barrier ΔF), leaving open whether the observed discontinuity at ρ_s is resolved in the true asymptotic regime or affected by finite-size rounding and crossover to other relaxation channels.

    Authors: We agree that explicit estimates of the nucleation barrier and critical droplet size are needed to confirm that the simulated δ values place the system in the asymptotic nucleation regime. In the revised manuscript we have added a new subsection in the numerical analysis that computes the barrier height ΔF and critical radius r_c for each simulated δ using classical nucleation theory, employing the known values of the latent heat and interface tension for the 3D q=6 Potts model. These estimates show that for the smallest δ, ΔF exceeds 30 k_B T while r_c remains much smaller than the linear system size L=64, indicating that finite-size rounding is negligible and that the observed discontinuity occurs in the nucleation-dominated regime. revision: yes

  2. Referee: Theoretical derivation (around the definition of ρ and ρ_s): ρ_s is introduced as the point of discontinuity in the scaling function. If ρ_s is determined from the same quench trajectories rather than from an independent equilibrium calculation of the nucleation rate, the predicted ln τ ≈ (ρ_s/δ)^{2/3} relation becomes partly a fit to the data rather than a parameter-free consequence of the nucleation assumption, weakening the predictive power of the central claim.

    Authors: The functional form of the scaling variable ρ = (ln t)^{3/2} δ and the prediction of a discontinuity at some ρ_s > 0 follow directly from the theoretical assumption of nucleation with a barrier that diverges as 1/δ^2; these are parameter-free consequences of the derivation. The numerical value of ρ_s is extracted from the location of the jump in the collapsed data, which serves as a test of the predicted scaling rather than an input. The resulting expression for τ is then a direct consequence. We acknowledge that an independent equilibrium calculation of the nucleation rate would allow a fully parameter-free prediction of ρ_s. In the revised manuscript we have expanded the discussion section to clarify this distinction and to outline how future equilibrium Monte Carlo simulations could determine ρ_s independently. revision: partial

Circularity Check

1 steps flagged

Time-scale prediction reduces to fitted discontinuity location ρ_s from the same numerical quench data

specific steps
  1. fitted input called prediction [Abstract]
    "We argue that, if the nucleation of smooth droplets is the relevant mechanism of the post-quench phase change, for sufficiently small β_fo−β_i>0, the time-dependent energy density should scale in terms of ρ = (ln t)^{3/2} δ, where δ = β/β_fo−1, with a discontinuity at a particular value ρ=ρ_s>0. This implies the emergence of a spinodal-like behavior, whose time scale τ increases exponentially as ln τ ≈ (ρ_s/δ)^{2/3} in the limit δ→0⁺."

    The scaling function and its discontinuity location ρ_s are not derived from first principles; ρ_s is identified numerically from the quench simulations that are also used to test the collapse. Consequently the concrete prediction for the divergence of τ is a reparametrization of the same numerical input rather than an independent consequence.

full rationale

The paper derives the scaling form ρ=(ln t)^{3/2}δ and the implied ln τ≈(ρ_s/δ)^{2/3} conditionally on the smooth-droplet nucleation assumption. ρ_s is introduced as the (unspecified) location of the discontinuity in the scaling function. The numerical analysis for the 3D q=6 model is used both to support the scaling collapse and to locate the discontinuity, so that the specific exponential growth law for τ is effectively read off from the fitted/observed ρ_s rather than predicted independently. This matches the 'fitted input called prediction' pattern at moderate strength; the central claim remains conditional on the nucleation assumption and is not self-definitional or reduced by self-citation.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The claim rests on the nucleation mechanism as a domain assumption and the introduction of the scaling variable ρ together with the parameter ρ_s. No free parameters are explicitly fitted in the abstract description, but ρ_s functions as a threshold value whose origin is not detailed here.

free parameters (1)
  • ρ_s
    The particular positive value of the scaling variable at which the energy density exhibits a discontinuity; its specific numerical value is not derived from first principles in the abstract.
axioms (1)
  • domain assumption Nucleation of smooth droplets is the relevant mechanism of the post-quench phase change for sufficiently small β_fo−β_i>0
    Invoked explicitly to justify the scaling form ρ = (ln t)^{3/2} δ and the resulting time-scale prediction.

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Reference graph

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