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arxiv: 2507.07933 · v1 · pith:4K3ROMBEnew · submitted 2025-07-10 · ❄️ cond-mat.stat-mech

Finite-time and Finite-size scalings of coercivity in dynamic hysteresis

Miao Chen , Xiu-Hua Zhao , Yu-Han Ma This is my paper

Pith reviewed 2026-05-22 00:45 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech
keywords coercivitydynamic hysteresisfinite-size scalingfinite-time scalingstochastic phi^4 modelCurie-Weiss modelphase transitions
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0 comments X

The pith

The coercivity in dynamic hysteresis exhibits a plateau at a characteristic driving rate that separates distinct scaling regimes inaccessible in the thermodynamic limit.

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

This paper shows that for the stochastic φ⁴ model under periodic driving, the coercivity H_c as a function of driving rate v_H develops a plateau at a specific rate v_P with coercivity H_P. Below this plateau, the expected linear scaling H_c ~ v_H from near-equilibrium conditions cannot be reached when the system size goes to infinity. Above the plateau, the fast-driving regime follows H_c ~ v_H to the power 1/2, but the slow-driving regime right after the plateau follows a different power law H_c minus H_P proportional to (v_H minus v_P) to the power 2/3. The authors demonstrate this with analytical proofs and simulations, and extend it to the Curie-Weiss model where finite-size effects make scaling behavior universal except in the fast-driving regime.

Core claim

For the stochastic φ⁴ model, the coercivity landscape H_c(v_H) exhibits plateau features at a characteristic rate v_P with the corresponding coercivity H_P. Below this plateau (v_H < v_P), the H_c ~ v_H scaling obtained in the near-equilibrium regime becomes inaccessible in the thermodynamic limit. Above the plateau (v_H > v_P), scaling in the fast-driving regime, H_c ~ v_H^{1/2}, is completely different from that, H_c - H_P ~ (v_H - v_P)^{2/3}, in the post-plateau slow-driving regime. The emergence of the plateau with a finite-size scaling reflects the competition between the thermodynamic limit and the quasi-static limit. Finite-time coercivity scaling shows model-specific behavior only in

What carries the argument

The coercivity landscape H_c(v_H) featuring a plateau at characteristic rate v_P due to competition between thermodynamic and quasi-static limits

If this is right

  • In the thermodynamic limit, the near-equilibrium linear scaling for coercivity becomes inaccessible.
  • The fast-driving regime universally follows square-root scaling with driving rate.
  • The post-plateau slow-driving regime follows a two-thirds power law scaling relative to the plateau values.
  • In concrete systems like the Curie-Weiss model, coercivity scaling is universal except in the fast-driving regime where it is model-specific.

Where Pith is reading between the lines

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

  • This framework could guide experiments on magnetic materials to identify the plateau rate to access different scaling behaviors.
  • Similar plateau phenomena might appear in other periodically driven systems exhibiting hysteresis, such as in ferroelectric or mechanical systems.
  • Finite-size scaling analysis around the plateau could provide a method to extrapolate coercivity behavior to infinite system sizes.

Load-bearing premise

The stochastic φ⁴ model under periodic driving of rate v_H accurately represents the essential dynamics of the interacting systems studied.

What would settle it

Numerical simulations or experiments showing the absence of a plateau in H_c(v_H) or the presence of the linear scaling in the thermodynamic limit for slow driving rates would falsify the central claim.

Figures

Figures reproduced from arXiv: 2507.07933 by Miao Chen, Xiu-Hua Zhao, Yu-Han Ma.

Figure 1
Figure 1. Figure 1: FIG. 1. Potential curves and probability distributions of the [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Hysteresis loops for [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Coercivity values and (b) their scaling exponents [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Geometric method to explain the scaling exponent [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Dynamic hysteresis loops of the Curie–Weiss model [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Coercivity [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Hysteresis loops and coercivity panorama of the [PITH_FULL_IMAGE:figures/full_fig_p009_6.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Coercivity delay behavior of the Curie-Weiss model [PITH_FULL_IMAGE:figures/full_fig_p010_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. (a) and (b) present numerical observations of two [PITH_FULL_IMAGE:figures/full_fig_p010_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10. Coercivity of the Curie-Weiss model as a function [PITH_FULL_IMAGE:figures/full_fig_p011_10.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Finite-size scaling collapse of coercivity plateau with [PITH_FULL_IMAGE:figures/full_fig_p012_11.png] view at source ↗
read the original abstract

The coercivity panorama for characterizing the dynamic hysteresis in interacting systems across multiple timescales is proposed by Chen et al. in a companion paper. For the stochastic $\phi^4$ model under periodic driving of rate $v_H$, the coercivity landscape $H_c(v_H)$ exhibits plateau features at a characteristic rate $v_P$ with the corresponding coercivity $H_P$. Below this plateau ($v_H<v_P$), the $H_c\sim v_H$ scaling obtained in the near-equilibrium regime becomes inaccessible in the thermodynamic limit. Above the plateau ($v_H>v_P$), scaling in the fast-driving regime, $H_c\sim v_H^{1/2}$, is completely different from that, $H_c-H_P\sim (v_H-v_P)^{2/3}$, in the post-plateau slow-driving regime. The emergence of the plateau with a finite-size scaling reflects the competition between the thermodynamic limit and the quasi-static limit. In this paper, we provide detailed analytical proofs and numerical evidence supporting these results. Moreover, to demonstrate the coercivity panorama in concrete physical systems, we study the magnetic hysteresis in the Curie-Weiss model and analyze its finite-size effects. We reveal that finite-time coercivity scaling shows model-specific behavior only in the fast-driving regime, while exhibiting universal characteristics elsewhere.

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 paper examines finite-time and finite-size scalings of coercivity in dynamic hysteresis for the stochastic φ⁴ model under periodic driving of rate v_H. It claims that the coercivity landscape H_c(v_H) features a plateau at a characteristic rate v_P with corresponding coercivity H_P. Below the plateau (v_H < v_P), the near-equilibrium scaling H_c ∼ v_H becomes inaccessible in the thermodynamic limit. Above the plateau, the fast-driving regime yields H_c ∼ v_H^{1/2}, while the post-plateau slow-driving regime follows H_c − H_P ∼ (v_H − v_P)^{2/3}. Detailed analytical proofs and numerical evidence are provided for these scalings, which arise from competition between thermodynamic and quasi-static limits. The work also analyzes the Curie-Weiss model to illustrate finite-size effects, finding model-specific behavior only in the fast-driving regime and universal characteristics elsewhere.

Significance. If the results hold, the manuscript strengthens the coercivity panorama framework by supplying explicit analytical derivations and numerical support for regime-dependent scalings, including the plateau's finite-size scaling. The demonstration of universal vs. model-specific behaviors across regimes in the Curie-Weiss model adds concrete physical insight. Credit is due for the parameter-free scaling relations and the focus on ordering of limits (thermodynamic vs. quasi-static), which addresses a key aspect of dynamic hysteresis in interacting systems.

major comments (2)
  1. [finite-size scaling section] Finite-size scaling analysis of v_P (near the discussion of thermodynamic limit and Eq. defining v_P(N)): the inaccessibility of the H_c ∼ v_H regime for v_H < v_P as N → ∞ requires explicit demonstration that v_P(N) remains finite or approaches a positive constant; if v_P(N) ∼ N^{-α} with α > 0, then for any fixed v_H > 0 the system enters the post-plateau regime at large N and the linear scaling is recovered. The current analysis does not independently verify this limiting behavior.
  2. [analytical proofs section] § on analytical proofs for post-plateau scaling: the derivation of H_c − H_P ∼ (v_H − v_P)^{2/3} assumes a specific ordering of limits that should be cross-checked against the numerical data for consistency in the slow-driving regime above v_P.
minor comments (2)
  1. [introduction] Clarify the notation for the driving rate v_H versus the characteristic rate v_P in the introduction and abstract to avoid potential confusion with the companion paper.
  2. [figures] Figure captions for the coercivity landscape plots should explicitly label the plateau region and the different scaling regimes for easier cross-reference with the text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful and constructive review of our manuscript. The comments highlight important points regarding the finite-size behavior of v_P and the consistency of the post-plateau scaling derivation with numerics. We address each major comment below and have revised the manuscript to incorporate additional verification and discussion.

read point-by-point responses

  1. Referee: [finite-size scaling section] Finite-size scaling analysis of v_P (near the discussion of thermodynamic limit and Eq. defining v_P(N)): the inaccessibility of the H_c ∼ v_H regime for v_H < v_P as N → ∞ requires explicit demonstration that v_P(N) remains finite or approaches a positive constant; if v_P(N) ∼ N^{-α} with α > 0, then for any fixed v_H > 0 the system enters the post-plateau regime at large N and the linear scaling is recovered. The current analysis does not independently verify this limiting behavior.

    Authors: We agree that an explicit demonstration of the limiting behavior of v_P(N) is required to rigorously establish the inaccessibility of the linear regime in the thermodynamic limit. In the revised manuscript we have added new numerical data for v_P(N) across a wider range of system sizes. These results show that v_P(N) approaches a positive finite constant as N increases, rather than vanishing as a power law. This is now presented in an extended figure with accompanying text that directly addresses the ordering of limits and confirms that the H_c ∼ v_H scaling remains inaccessible for fixed v_H below the plateau in the large-N limit. revision: yes

  2. Referee: [analytical proofs section] § on analytical proofs for post-plateau scaling: the derivation of H_c − H_P ∼ (v_H − v_P)^{2/3} assumes a specific ordering of limits that should be cross-checked against the numerical data for consistency in the slow-driving regime above v_P.

    Authors: We thank the referee for pointing out the need to cross-check the assumed ordering of limits against the numerical evidence. The analytical derivation is performed with the thermodynamic limit taken before the quasi-static limit, which is the ordering that produces the plateau. In the revision we have added a short subsection that re-examines the numerical data specifically in the slow-driving regime immediately above v_P. The effective exponent extracted from the data approaches 2/3 as v_H approaches v_P from above, consistent with the analytical prediction under the stated limit ordering. This consistency check has been included together with a clarifying remark on the relevant ordering of limits. revision: yes

Circularity Check

1 steps flagged

Self-citation to companion paper frames the coercivity panorama and plateau features

specific steps
  1. self citation load bearing [Abstract]
    "The coercivity panorama for characterizing the dynamic hysteresis in interacting systems across multiple timescales is proposed by Chen et al. in a companion paper. For the stochastic φ⁴ model under periodic driving of rate v_H, the coercivity landscape H_c(v_H) exhibits plateau features at a characteristic rate v_P with the corresponding coercivity H_P. Below this plateau (v_H < v_P), the H_c ~ v_H scaling obtained in the near-equilibrium regime becomes inaccessible in the thermodynamic limit."

    The load-bearing premise—the existence and properties of the plateau at v_P together with the claimed inaccessibility of the near-equilibrium scaling once the thermodynamic limit is taken—is introduced solely by citation to a companion paper whose authors overlap with the present work. The subsequent analytical proofs and numerics support scalings conditional on that framing rather than deriving the panorama independently.

full rationale

The manuscript explicitly attributes the coercivity panorama, including the plateau at characteristic rate v_P with coercivity H_P, to a companion paper by overlapping authors (Chen et al.). This paper then supplies analytical proofs and numerical evidence for the scalings in the stochastic φ⁴ model and demonstrates the panorama in the Curie-Weiss model. The central claims about inaccessibility of H_c ~ v_H for v_H < v_P in the N→∞ limit therefore inherit their framing from the self-citation, yet retain independent content via the finite-size scaling analysis and model-specific comparisons. This matches a moderate self-citation load-bearing pattern without reducing the entire derivation to a fit or definition by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard stochastic φ⁴ model and periodic driving protocol without introducing new free parameters or invented entities; the plateau is an emergent feature rather than an added construct.

axioms (1)
  • domain assumption The stochastic φ⁴ model under periodic driving of rate v_H captures the essential dynamics of interacting systems for the purpose of coercivity scaling.
    Invoked to derive the coercivity landscape and its plateau features.

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Coercivity Landscape Characterizes Dynamic Hysteresis

    cond-mat.stat-mech 2025-06 unverdicted novelty 5.0

    In the stochastic φ⁴ model, coercivity exhibits v_H scaling, a plateau at the first-order transition field H*, then v_H^{1/2} scaling, with finite-size scalings v_P ~ σ² and (H* - H_P) ~ σ^{4/3} from renormalization-g...

Reference graph

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