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Glauber dynamics phase transitions in athermal random field Blume-Capel and Blume-Emery-Grifitths models

T0 review · 3 major / 4 minor · reviewed 2026-08-01 · deepseek-v4-flash

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Pith's one-line read Zero-temperature Glauber dynamics in random-field spin-1 models is initial-state dependent below a critical disorder variance and matches equilibrium above it; phase boundaries follow from local stability of the equilibrium free energy.

desk verdict Blume-Capel half is solid and worth your time; the RFBEGM stability analysis has a load-bearing gap because the Glauber map's Jacobian, not the equilibrium Hessian, controls the transition. read the letter →

arxiv 2607.16561 v1 pith:MABOXCFV submitted 2026-07-18 cond-mat.stat-mech cond-mat.dis-nn

Sumedha , Aldrin B E This is my paper · ORCID
classification cond-mat.stat-mechcond-mat.dis-nn
keywords GlauberdynamicsrandomfieldBlume-CapelmodelBlume-Emery-Griffithstricriticalpointhysteresisfixedstabilitycompletegraph
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper solves the random-field Blume-Capel and Blume-Emery-Griffiths models on a complete graph at zero temperature, both in equilibrium and under Glauber dynamics. It shows that the variance R of the Gaussian random field acts like temperature: above a critical value, the non-equilibrium steady state forgets its initial condition and coincides with the equilibrium state, while below it the steady state depends on where the dynamics started. The transition lines for the Glauber steady state are obtained exactly by requiring that a fixed point of the equilibrium free energy loses local stability, yielding closed-form equations such as Δ_c = sqrt(−R_c² log(πR_c²/2)). For repulsive biquadratic coupling, R_c can vanish, and in some regimes the model crosses over from R_c = 0 to the random-field Ising value sqrt(2/π). In a magnetic field, the same stability condition gives analytic hysteresis-loop shapes and coercive-field equations.

What carries the argument

The engine of the argument is the identity between the fixed-point equations of the zero-temperature equilibrium rate function and the self-consistent steady-state equations of zero-temperature Glauber dynamics. Since each allowed Glauber move lowers energy, the dynamics is argued to remain in the basin of a local minimum of the equilibrium free energy; the transition under dynamics is therefore exactly where that fixed point changes from a local minimum to a maximum or saddle. For the Blume-Capel model this reduces to the second derivative g(m) = 1 − (1/√(2πR²))[exp(−(m−Δ)²/2R²) + exp(−(m+Δ)²/2R²)]; for the Blume-Emery-Griffiths model, the Hessian matrix of f(m,q) supplies two conditions, 1

What would settle it

Simulate Glauber dynamics on a complete graph for the random-field Blume-Capel model at fixed R=0.2, starting from a state with small nonzero magnetization, and measure the quasi-static transition value of Δ; the claim fails if the system leaves the basin while g(m)>0 or if the transition occurs at a Δ different from the predicted Δ_c. For the Blume-Emery-Griffiths model with K<0 and z≠0, the claim R_c=0 is falsified if a nonzero steady-state magnetization persists for arbitrarily small R or if a transition appears at finite R away from the z=0 line.

Watch

Extended reading notes

Core claim

The central claim is that for spin-1 random-field models on a complete graph, the zero-temperature Glauber steady state is controlled by the local minima of the zero-temperature equilibrium free energy f(m,q). Whenever f has multiple minima (small R), the dynamics keeps the system in the basin of its starting minimum, so the steady state is initial-state dependent; when R is large enough that only one minimum remains, equilibrium and non-equilibrium coincide. The dynamical transition occurs exactly where a fixed point ceases to be a local minimum: for one order parameter, the condition is g(m) = 0, giving Δ_c = sqrt(−R_c² log(πR_c²/2)); for two order parameters, the Hessian conditions 1−A−B

Load-bearing premise

The whole Glauber phase diagram rests on the assertion that zero-temperature energy-lowering dynamics keeps the system trapped in the basin of a local minimum of the equilibrium free energy, so a dynamical transition happens exactly when that minimum loses stability; the paper motivates this heuristically rather than deriving it from the Jacobian of the actual dynamical map.

Editorial extensions

If this is right

  • For R above the tricritical variance, the Glauber steady state and the equilibrium phase diagram coincide, so measurements of the steady state cannot distinguish equilibrium from athermal dynamics once disorder is strong enough.
  • Below that threshold, phase boundaries are path-dependent: increasing Δ from an m=0 start and decreasing Δ from an m=1 start give different transition points, so any protocol must specify the initial state and the direction of drive.
  • For repulsive biquadratic coupling, R_c can be zero, meaning the ordered state is unstable to arbitrarily weak disorder; increasing R can then induce a crossover to a transition at R_c = sqrt(2/π), giving the system a finite disorder threshold it did not initially have.
  • The analytic hysteresis-loop shapes (rectangular, wasp-waisted, parallelogram, hexagonal, double) are set by the R=0 dynamics; disorder shrinks the loop area and sets the coercive field through the same local-stability equation.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A natural extension is to test whether the local-stability criterion survives at finite but small temperature; the paper notes the finite-temperature steady state is equilibrium-like, so initial-state memory would appear only as slow relaxation rather than in the steady state.
  • The R-as-temperature analogy suggests that response or avalanche statistics near R_c in these spin-1 models might inherit random-field Ising critical behavior, a prediction the paper does not develop.
  • Because the coercive-field equation is derived on a complete graph, comparing it with finite-dimensional simulations would show how sensitive the mechanism is to mean-field assumptions.
  • For the frustrated regime with negative K, where the dynamics is non-abelian, the coexistence of continuous and first-order segments inside hysteresis loops suggests avalanche statistics could differ qualitatively from the abelian random-field Ising model; the paper reports the shapes but does not analyze avalanche distributions here.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper studies zero-temperature Glauber dynamics and equilibrium statistical mechanics of the random-field Blume-Capel (RFBCM) and Blume-Emery-Griffiths (RFBEGM) models on a fully connected graph. For each model it derives self-consistent mean-field equations for the magnetization m and quadrupole moment q, and an equilibrium rate function via large deviations. It claims that the Glauber steady-state and zero-temperature equilibrium fixed-point equations coincide, but that below a critical disorder strength R_c the steady state depends on the initial condition, while for R≥R_c the two descriptions coincide. Analytic phase boundaries are given for both models, including a criterion for the RFBEGM based on the Hessian of the equilibrium free energy. The paper also derives hysteresis-loop shapes and coercive fields in the presence of a uniform field, and reports numerical simulations on N=1000 complete graphs supporting several of the analytical predictions.

Significance. If the central claims are correct, the paper provides a substantial exact result: for two spin-1 random-field models, the non-equilibrium zero-temperature Glauber steady state is exactly solvable and agrees with the equilibrium state above a disorder-controlled critical value, with interesting exceptions such as R_c=0 and a disorder-induced crossover to RFIM-like behavior. The large-deviations derivation of the zero-temperature rate function and the exact analytic treatment of hysteresis loop shapes are valuable. The RFBCM part, where the stability criterion reduces to a scalar condition, is convincing and is corroborated by simulations. However, the two-variable stability criterion for RFBEGM is the load-bearing element for the paper's most novel claims, and it is asserted rather than derived from the actual Glauber map. The correctness of that criterion is doubtful for K≠1, so the RFBEGM phase diagrams and the R_c=0/crossover conclusions require substantial revision or re-derivation.

major comments (3)
  1. [Sec. VI, Eqs. (49)-(53)] The stability criterion for RFBEGM is asserted as positive-definiteness of the Hessian L of the equilibrium free energy f(m,q), with the transition at 1−A−B=0 and D=0 simultaneously. This is not derived from the Jacobian of the Glauber mean-field map (18)-(19), which is J=[[A+B, K(A−B)], [A−B, K(A+B)]]. For K≠1, L is not I−J and the vector field is not a gradient unless (K−1)(A−B)=0. At m=0 the Jacobian has eigenvalues c and Kc, where c=√(2/(πR^2))e^{−z^2/(2R^2)}. For K>1 the q-direction eigenvalue crosses +1 at c=1/K, before the m-direction crossing at c=1. Therefore Eq. (53) (c=1) does not give the loss of stability of the m=0 fixed point for K>1; the correct linear-stability boundary is c=1/K for that eigenvalue. This directly affects the phase boundaries derived for K>0 in Sec. VI and the claimed R_c values in that region.
  2. [Sec. VI, case K<0 and the R_c=0 statements] The argument that for K<0, z≠0, D≥0 requires c≥1 while 1−A−B≥0 requires c≤1, and therefore R_c=0, uses the Hessian positive-definiteness conditions rather than the eigenvalues of the actual Glauber map. For K<0 the q-direction eigenvalue is Kc<0; stability of the fixed point requires only |Kc|<1, which can hold for c>1 when |K|c<1. Thus the determinant condition of the equilibrium Hessian is not the correct linear-stability condition for the map. The conclusion R_c=0 for K<0, z<0 is therefore not established. The simulation evidence in Fig. 4(c) is at a single very small R and does not rule out a transition at finite R for the actual dynamics.
  3. [Sec. VI, crossover claim in case 2 (K<Δ<0)] The crossover from R_c=0 to R_c=√(2/π) is argued on the basis of the special z=0 line and Eq. (55). However, the boundary at which the system crosses from the z<0 regime to z=0 is not derived from the Jacobian of the two-variable map. The statement 'once q≤Δ/K the z becomes greater than 0... the system has a continuous order-disorder phase transition at R_c=√(2/π)' implicitly assumes that the m=0, z≥0 branch is stable according to the correct dynamics. Since the Hessian-based criterion is not the stability criterion for K≠1, the location and even existence of this crossover are not rigorously established by the paper's analysis.
minor comments (4)
  1. [Throughout] There are several typographical errors: the title and text alternate between 'Grifitths' and 'Griffiths'; the abstract contains 'We also. consider' with a stray period; the Introduction says 'Gluaber dynamics'; Sec. IV has 'performming' and 'subsusbtituting'. These should be corrected.
  2. [Sec. VI, Eqs. (51)-(53)] The notation 'ccosh' and 'cc=1' is unclear; presumably it means c·cosh and a condition involving c. Please define all symbols explicitly and avoid the ambiguous two-letter combination.
  3. [Sec. V.B, Fig. 1 caption] The caption for Fig. 1(b) appears to refer to both 'increasing Δ' and 'decreasing Δ' but does not clearly separate the two branches; consider splitting or clarifying the quasi-static protocol.
  4. [Sec. VII] The derivation of hysteresis shapes at R=0 is clear, but the statement that for R≠0 'the shape of the hysteresis loop is retained until R_c' is only illustrated, not proven; a brief argument that the nullcline structure is preserved for small R would strengthen the claim.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Glauber steady-state equations and the equilibrium rate function are derived independently, and their equality is a derived identity benchmarked by simulation.

full rationale

The central derivation chain is self-contained. The Glauber steady-state equations (18)–(19) are obtained from the spin-flip energy rules (4)–(8), while the zero-temperature equilibrium rate function f(m,q) (Eq. 35) is obtained from a large-deviations calculation (Eqs. 20–30). The equality of the Glauber fixed-point equations and the equilibrium fixed-point equations (Eqs. 36–37 vs. 18–19) is derived algebraically, not assumed. The paper then identifies the Glauber phase boundary with the loss of local stability of f (Eqs. 46–47, 49–53). This is an unproved physical heuristic rather than a circular reduction: for K≠1 the mean-field Glauber map is not obviously gradient, so the Hessian of f need not control the stability of the dynamics. That concern is a correctness risk, not a circularity—it is not an input that has been renamed as a prediction. The self-citations [21–23,25] provide methodology and prior equilibrium results, but the key equations are rederived in the paper and the predicted phase boundaries are checked against explicit complete-graph simulations, so these citations are not load-bearing in a circular way. No fitted parameter is renamed as a prediction; the transition criteria are analytic conditions on derived equations. Overall, the derivation is independent of its conclusions, and no circular step satisfying the hard rules can be exhibited.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper introduces no new free parameters fitted to data; all parameters (J, K, Delta, R, H) are model inputs. The primary axioms are standard large-deviations tools, the mean-field complete-graph assumption, and the zero-temperature Glauber update rule. The most fragile axiom is the equivalence between Glauber-map stability and the Hessian of the equilibrium free energy, which is asserted heuristically rather than derived.

assumptions (5)
  • standard math Gartner-Ellis theorem and the large-deviation principle for the quenched random-field measure
    Used in Sec. IV to derive the equilibrium free energy functional I(m,q); citations [26-28].
  • domain assumption On a complete graph, local fields L1 and L2 self-average and the steady-state spin probabilities factorize
    Needed in Sec. III to write P_s as functions of m and q alone.
  • domain assumption Zero-temperature Glauber dynamics allows only energy-lowering spin flips, and the steady state is described by the fixed point of the mean-field map
    Defines the dynamics in Sec. III and is used throughout.
  • ad hoc to paper The stability of a Glauber-dynamics fixed point is governed by the positive-definiteness of the Hessian of the equilibrium free energy f(m,q)
    Asserted in Secs. V and VI (Eqs. 46-53) without a derivation from the Jacobian of the Glauber map; this is the main load-bearing heuristic for the two-order-parameter phase boundaries.
  • domain assumption The quenched random fields are independent Gaussians with mean 0 and variance R
    The model definition in Sec. II.

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Cite this review

Pith. "Pith review of Glauber dynamics phase transitions in athermal random field Blume-Capel and Blume-Emery-Grifitths models." pith.science (2026). https://pith.science/paper/MABOXCFV

@misc{pith2026260716561,
  author       = {Pith},
  title        = {Pith review of: Glauber dynamics phase transitions in athermal random field Blume-Capel and Blume-Emery-Grifitths models},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MABOXCFV}},
  note         = {Machine review of arXiv:2607.16561}
}
abstract

We solve the two models for Glauber dynamics and in equilibrium, both in the presence and absence of the external magnetic field on a complete graph. We compare the steady state of the Glauber dynamics with equilibrium and find that for low values of variance $R$ of the Gaussian random field, the steady state of the Glauber dynamics depends on the initial state. Beyond a critical value $R_{c}$ the equilibrium and non equilibrium steady states coincide. The variance $R$ in random field models behaves similar to the temperature. The location of both the continuous and first order transitions can be obtained exactly for the Glauber dynamics steady state. The frustration is introduced by considering repulsive bi-quadratic interaction for Blume-Emery-Griffiths model. We also. consider repulsive bi-quadratic interaction and show that $R_c$ can become zero depending on the value of the crystal field. Interestingly, we also find that even when a system has $R_c=0$ at the start of quasi-static evolution with Glauber dynamics, with increasing $R$, in some regime of the couplings, the model undergoes a crossover to a random field Ising model universaility with $R_c$ changing from $0$ to $\sqrt{\frac{2}{\pi}}$. In the presence of uniform magnetic field, regions of first order transition exhibit hysteresis under Glauber dynamics. These models exhibit rectangular, hexagonal, parallelogram, wasp-waisted, and double hysteresis loops. We derive the shapes of hysteresis loops analytically giving the equation for the value of the coercive field and show that while the area under the hysteresis loop depends on $R$, the shape is determined by the behavior of the models at $R=0$. In particular, in the case of Blume-Emery-Griffiths model the hysteresis plots have regions of continuous and first order transitions both, resulting in a rich phase diagram that depends non-trivially on the initial state.

Figures

Figures reproduced from arXiv: 2607.16561 by the authors.

Figure 1
Figure 1. FIG. 1: Plot of steady state magnetisation( [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: Phase diagram of RFBCM under Glauber dynamics with initial state [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: For [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (8 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Plots for RFBEGM Glauber steady state with [PITH_FULL_IMAGE:figures/full_fig_p013_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5: Different shapes of hysteresis loops in [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6: The [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Different shapes of [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8: The equilibrium phase diagrams of the Blume-Emery-Griffiths model are given in this figure. The regions of different [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9: Different shapes of hysteresis loops in [PITH_FULL_IMAGE:figures/full_fig_p017_9.png]
Figure 10
Figure 10. Figure 10: FIG. 10: Phase diagram for a) [PITH_FULL_IMAGE:figures/full_fig_p020_10.png]
Figure 11
Figure 11. Figure 11: FIG. 11: The [PITH_FULL_IMAGE:figures/full_fig_p020_11.png]

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

  1. Disorder induced time crystal in athermal random field Ising model with non-reciprocal interactions

    cond-mat.stat-mech 2026-07 conditional novelty 5.0 of 10

    In an athermal random-field Ising model with non-reciprocal couplings, a spontaneous time-oscillatory (time-crystal) phase exists for intermediate disorder strength on complete graphs and in 3D, but not in 2D.

Reference graph

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Pith tools

Reviewed August 1, 2026 · model on record in the stance chip above.