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arxiv: 1906.09803 · v1 · pith:VZXK7WTFnew · submitted 2019-06-24 · ❄️ cond-mat.stat-mech

Multinucleation in the first-order phase transition of the 2d Potts model

Federico Corberi , Leticia F Cugliandolo , Marco Esposito , Marco Picco This is my paper

Pith reviewed 2026-05-25 17:15 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords Potts modelfirst-order phase transitionnucleationcoarseningmetastabilityquench dynamics2d lattice models
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The pith

In the 2d q-Potts model for q>4, shallow quenches near the first-order transition cause the metastable state to decay through simultaneous formation of multiple nuclei rather than a single one.

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

The paper studies the relaxation of the two-dimensional q-state Potts model after a quench from a disordered high-temperature state to a temperature slightly below the first-order transition. Numerical simulations show that close to the transition the metastable phase ends through multi-nucleation, in which many ordered domains appear at once. The subsequent evolution is a coarsening process in which domains belonging to the q equivalent ground states compete and grow until one dominates.

Core claim

After a shallow subcritical quench in the 2d q-Potts model for q>4, the system escapes metastability via multi-nucleation when the final temperature is close to critical; the ensuing relaxation proceeds through coarsening with competition between the equivalent ground states.

What carries the argument

The multi-nucleation process, in which multiple nuclei of the ordered phases form simultaneously to destroy the metastable state.

If this is right

  • Relaxation times near the transition become controlled by the density of nuclei rather than by the nucleation barrier of a single droplet.
  • Domain growth after multi-nucleation exhibits competition among q equivalent ordered phases instead of simple two-phase coarsening.
  • The multi-nucleation regime is limited to a window of final temperatures close to the transition; deeper quenches recover conventional single-nucleation behavior.
  • The mechanism is expected to hold for other first-order transitions in two dimensions when the quench is shallow.

Where Pith is reading between the lines

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

  • Similar multi-nucleation should appear in other lattice models with first-order transitions when quenched close to their transition temperature.
  • The crossover from multi- to single-nucleation could be mapped by systematically varying quench depth and system size.
  • The late-stage coarsening dynamics may produce distinct scaling exponents compared with equilibrium domain growth because of the initial multi-nuclei configuration.

Load-bearing premise

The large-scale lattice simulations faithfully reproduce the continuous-time kinetics and that finite-size effects do not change the observed multi-nucleation route.

What would settle it

Direct observation, in systems large enough to eliminate boundary artifacts, that only a single nucleus forms and grows when the quench depth is small.

Figures

Figures reproduced from arXiv: 1906.09803 by Federico Corberi, Leticia F Cugliandolo, Marco Esposito, Marco Picco.

Figure 1
Figure 1. Figure 1: Magnetization density, m = M/N of the Ising model, (a) as a function of magnetic field at T < Tc (first-order phase transition), (b) as a function of temperature at zero magnetic field (second-order phase transition). it is useful to consider the familiar Ising model, described by the Hamiltonian HI = −J X hiji σiσj − H X i σi , (1) where J is a coupling constant, the first sum is restricted to nearest-nei… view at source ↗
Figure 2
Figure 2. Figure 2: Pictorial representation of a system during coarsening at [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: φE vs t for q = 9 (Tc = 0.71235), L = 1000 and different values of the temperature Tf that are given in the key (Left Panel). In the right panel, φE vs t for q = 100 (Tc = 0.41703), L = 700 and different values of Tf also in the key. In the inset, the same but for q = 5 (Tc = 0.85153). For q < 50, the value of φ ∗ E is well approximated by the logarithm of q − 4. For greater values of q one has saturation … view at source ↗
Figure 4
Figure 4. Figure 4: Snapshots of the lattice at different times for [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Value of φE on the plateau as a function of the logarithm of q − 4. φE ∝ t −1/2 . The behavior of the plateau energy φ ∗ E (q) is consistent with the logarithm of q − 4. During coarsening, phases are eliminated. In the light of our results, it can be interesting to investigate the spontaneous symmetry breaking mechanism whereby this elimination takes place, and compare it with the coarsening process that c… view at source ↗
read the original abstract

Using large-scale numerical simulations we studied the kinetics of the 2d q-Potts model for q > 4 after a shallow subcritical quench from a high-temperature homogeneous configuration. This protocol drives the system across a first-order phase transition. The initial state is metastable after the quench and, for final temperatures close to the critical one, the system escapes from it via a multi-nucleation process. The ensuing relaxation towards equilibrium proceeds through coarsening with competition between the equivalent ground states. This process has been analyzed for different choices of the parameters such as the number of states and the final quench temperature.

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

1 major / 1 minor

Summary. The paper uses large-scale numerical simulations to study the kinetics of the 2d q-Potts model (q>4) after a shallow subcritical quench from a high-temperature state. It finds that near the critical temperature, the system escapes the metastable state via multi-nucleation, followed by coarsening with competition between the q ground states. The behavior is analyzed for different q and quench temperatures.

Significance. If substantiated by the simulations, the results illustrate the multi-nucleation pathway in first-order transitions of the Potts model for shallow quenches, consistent with expectations for low barrier heights, and highlight the subsequent domain coarsening dynamics in a system with multiple degenerate states.

major comments (1)
  1. [Abstract] The abstract and manuscript description do not specify the lattice sizes, Monte Carlo algorithm, number of samples, or criteria for identifying nucleation events. These details are load-bearing for validating the multi-nucleation claim and ensuring the protocol captures the continuous-time kinetics without significant finite-size artifacts.
minor comments (1)
  1. The title uses 'Multinucleation' while the abstract uses 'multi-nucleation'; consistency in terminology would improve clarity.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading and constructive comment on our manuscript. We address the point below and will incorporate the requested details in the revised version.

read point-by-point responses

  1. Referee: [Abstract] The abstract and manuscript description do not specify the lattice sizes, Monte Carlo algorithm, number of samples, or criteria for identifying nucleation events. These details are load-bearing for validating the multi-nucleation claim and ensuring the protocol captures the continuous-time kinetics without significant finite-size artifacts.

    Authors: We agree that these details should be stated more explicitly. The full manuscript already contains a dedicated Numerical Methods section describing the single-spin-flip Metropolis algorithm, lattice sizes (L=128 to 512 with periodic boundaries), number of independent samples (typically 100–200 per (q,T) pair), and the nucleation criterion (first appearance of a cluster whose linear size exceeds the critical radius estimated from the measured interface tension). However, we acknowledge that the abstract and opening paragraphs do not highlight them. In the revised manuscript we will add a concise sentence to the abstract and expand the introductory description to include these parameters, together with a brief statement on finite-size checks performed to confirm that the observed multi-nucleation is not an artifact. revision: yes

Circularity Check

0 steps flagged

No significant circularity: simulation outputs only

full rationale

The paper reports results exclusively from direct Monte Carlo simulations of the 2d q-state Potts model after quenches. No derivations, ansatzes, fitted parameters renamed as predictions, or self-citations are used to establish the central claim of multi-nucleation followed by coarsening. The observed escape from metastability and subsequent domain competition are direct outputs of the numerical protocol rather than quantities constructed from prior results or equations within the paper. This is a standard, self-contained numerical study with no load-bearing theoretical steps that could reduce to inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The study rests on the standard definition and known first-order character of the 2d Potts model for q>4 together with the assumption that Monte Carlo dynamics faithfully represent the physical kinetics; no new entities are postulated and no parameters are fitted to produce the reported mechanism.

axioms (2)
  • domain assumption The 2d q-Potts model undergoes a first-order phase transition for q>4
    Invoked in the abstract to justify the quench protocol across the transition.
  • domain assumption Monte Carlo simulation on finite lattices reproduces the metastable escape and coarsening dynamics of the infinite system
    Implicit in the claim that large-scale simulations reveal the multi-nucleation process.

pith-pipeline@v0.9.0 · 5637 in / 1491 out tokens · 35011 ms · 2026-05-25T17:15:39.744708+00:00 · methodology

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Reference graph

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