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arxiv: 2509.17408 · v2 · submitted 2025-09-22 · ❄️ cond-mat.stat-mech · nlin.PS

Coarsening dynamics for spiral and disordered waves in active Potts models

Hiroshi Noguchi This is my paper

Pith reviewed 2026-05-18 15:14 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech nlin.PS
keywords active Potts modelcoarsening dynamicsLifshitz-Allen-Cahn lawspiral wavesdisordered wavesdomain growthMonte Carlo simulationcyclic symmetry
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The pith

Active Potts models with cyclic flips show domain coarsening that follows the Lifshitz-Allen-Cahn law with a temporary speedup before locking into stable waves.

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

The paper studies how domains grow in q-state active Potts models on lattices when states flip in a cycle, producing either spiral or disordered waves of finite length. Starting from random mixtures, the correlation length and average cluster size increase proportionally to the square root of time during the middle stage of evolution, matching the classic Lifshitz-Allen-Cahn coarsening behavior. Just before the domains reach the wavelength set by activity and temperature, the growth rate rises temporarily, producing a higher effective exponent; this boost is stronger for disordered waves than for spirals and becomes larger as q increases from 3 to 8. The same square-root growth appears even in factorized symmetry sectors at q=6. The findings remain unchanged when the lattice type or Monte Carlo update rule is switched.

Core claim

In q-state active Potts models under cyclic symmetry, the correlation length and mean cluster size grow as t to the power 1/2 in the intermediate regime, in accordance with the Lifshitz-Allen-Cahn law, before saturating at the characteristic wavelengths of the emerging waves; a transient elevation of the coarsening exponent occurs prior to saturation, and this elevation is larger for disordered waves than for spiral waves and increases with q.

What carries the argument

Active cyclic state flipping in the q-state Potts model, which generates finite-length moving waves whose wavelength is controlled by activity strength and temperature.

If this is right

  • Correlation length and cluster size increase as the square root of time until they reach the wave wavelength set by activity.
  • The temporary rise in growth rate is larger when the final state is disordered waves rather than spirals.
  • Increasing the number of states q amplifies the transient boost in the coarsening exponent.
  • Domains restricted to two- or three-state factorized modes at q=6 still obey the same Lifshitz-Allen-Cahn growth.
  • The coarsening trajectory is insensitive to whether a square or hexagonal lattice is used and to whether Metropolis or Glauber updates are chosen.

Where Pith is reading between the lines

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

  • Activity strength may serve as a control parameter that tunes how quickly patterns reach their final scale in other lattice-based active systems.
  • The observed saturation suggests a dynamical balance between domain merging and wave propagation that could be examined in continuum active-matter equations.
  • Similar transient accelerations might appear in experimental realizations of cyclic Potts-like systems, such as certain chemical reaction-diffusion media or multicellular aggregates.
  • The greater transient effect in disordered waves implies that symmetry breaking in the final state influences the approach to saturation.

Load-bearing premise

The Monte Carlo dynamics at the chosen activity and temperature produce stable waves whose wavelength is fixed by the model parameters rather than by the finite size of the simulation box or lattice details.

What would settle it

Running the same model on lattices at least ten times larger in linear size and checking whether the transient exponent increase and the subsequent saturation both disappear or shift to much later times.

Figures

Figures reproduced from arXiv: 2509.17408 by Hiroshi Noguchi.

Figure 2
Figure 2. Figure 2: FIG. 2. Coarsening exponents (a) [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 1
Figure 1. Figure 1: FIG. 1. Coarsening dynamics of the three-state standard [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Coarsening dynamics of the six-state standard Potts [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Coarsening exponents (a) [PITH_FULL_IMAGE:figures/full_fig_p005_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Coarsening dynamics of the six-state attractive [PITH_FULL_IMAGE:figures/full_fig_p006_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Snapshots ( [PITH_FULL_IMAGE:figures/full_fig_p007_9.png] view at source ↗
read the original abstract

This study examines the domain-growth dynamics of $q$-state active Potts models ($q=3$--$8$) under the cyclically symmetric conditions using Monte Carlo simulations on square and hexagonal lattices. By imposing active cyclic flipping of states, finite-length waves emerge in the long-term limit. This study focuses on coarsening dynamics from an initially random mixture of states to these moving-domain states. The correlation length and mean cluster size grow, following the Lifshitz--Allen--Cahn (LAC) law ($\propto t^{1/2}$) in the intermediate time range, and in the late range, saturation is observed at the characteristic wavelengths. It is found that the growth rate is raised prior to saturation, leading to a transient increase in the coarsening exponent. The coarsening dynamics to disordered waves exhibit greater transient increases than those to spiral waves. Moreover, the transient increase is greater at higher $q$. In factorized symmetry modes at $q=6$, domains composed of two or three states similarly follow the LAC law. Finally, this study confirms that the choice of lattice type (square or hexagonal) and update scheme (Metropolis or Glauber) does not alter the dynamic behavior.

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 examines coarsening dynamics in q-state active Potts models (q=3--8) with cyclic symmetry using Monte Carlo simulations on square and hexagonal lattices. From random initial states, finite-length waves emerge, and the correlation length and mean cluster size are shown to grow as t^{1/2} following the Lifshitz-Allen-Cahn law in the intermediate time regime. Saturation occurs at characteristic wavelengths in the late stage, preceded by a transient increase in the effective coarsening exponent, which is larger for disordered waves than for spiral waves and increases with q. The dynamics are consistent across lattice types and update schemes (Metropolis and Glauber). In factorized symmetry modes at q=6, similar behavior is observed for domains of two or three states.

Significance. If the reported scaling and transients hold as intrinsic features, this work contributes to the study of non-equilibrium coarsening in active systems by demonstrating how wave formation influences domain growth beyond standard LAC behavior. The consistency across different lattices and dynamics strengthens the reliability of the numerical findings. The distinction between spiral and disordered waves, and the q-dependence, offers new insights into symmetry-breaking and activity-driven dynamics in Potts models.

major comments (1)
  1. [Results] Results section on correlation length and cluster size evolution: The central claim that saturation of the correlation length and mean cluster size occurs at wavelengths fixed by activity strength and temperature (rather than finite lattice size) is not supported by explicit system-size scaling checks. No comparison of saturation times, saturated values, or transient exponent peaks is reported across at least two different linear sizes L, leaving open the possibility that the observed transient increase in the coarsening exponent is contaminated by the approach to the box size.
minor comments (1)
  1. [Abstract] Abstract: The statement that growth follows the Lifshitz-Allen-Cahn law provides no quantitative measures such as fitted exponents with uncertainties, time-range definitions for the intermediate regime, or system-size details.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We are grateful to the referee for their thorough review and constructive feedback on our manuscript. We address the major comment in detail below and have made revisions to strengthen the presentation of our results.

read point-by-point responses

  1. Referee: [Results] Results section on correlation length and cluster size evolution: The central claim that saturation of the correlation length and mean cluster size occurs at wavelengths fixed by activity strength and temperature (rather than finite lattice size) is not supported by explicit system-size scaling checks. No comparison of saturation times, saturated values, or transient exponent peaks is reported across at least two different linear sizes L, leaving open the possibility that the observed transient increase in the coarsening exponent is contaminated by the approach to the box size.

    Authors: We thank the referee for highlighting this important point. Upon reflection, we acknowledge that the manuscript does not present explicit finite-size scaling analyses comparing results for different linear system sizes L. The saturation lengths reported are tied to the activity and temperature parameters through the observed wave wavelengths, which are independent of L in our simulations. Nevertheless, to fully address this concern and rule out any contamination from the finite box size, we will conduct additional simulations using at least two different system sizes (e.g., L=256 and L=512) and include comparisons of the saturation times, saturated correlation lengths and cluster sizes, as well as the transient peaks in the effective exponent. These new results will be added to the revised manuscript, confirming that the observed behavior is intrinsic. revision: yes

Circularity Check

0 steps flagged

No circularity: direct numerical outputs benchmarked externally

full rationale

The paper reports Monte Carlo simulation results on coarsening in active Potts models. Growth of correlation length and cluster size is stated to follow the Lifshitz-Allen-Cahn law in an intermediate regime and to saturate at characteristic wavelengths in the late regime; both are presented as direct outputs of the runs rather than quantities fitted or defined in terms of each other. The LAC reference is an external benchmark, not a self-citation or internal ansatz. No equations, fitted parameters, or uniqueness theorems are invoked that reduce the central claims to the simulation inputs by construction. Lattice-type and update-scheme checks are likewise independent numerical tests. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the assumption that the chosen Monte Carlo dynamics faithfully represent the long-time coarsening behavior of the active Potts model without introducing artifacts from finite size or discrete update rules.

axioms (1)
  • domain assumption The active cyclic flipping rule produces stable finite-wavelength waves whose coarsening can be compared to the passive Lifshitz-Allen-Cahn law.
    Invoked when the authors identify the intermediate-time regime and the saturation wavelength.

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