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REVIEW 4 major objections 5 minor 1 cited by

Finite-Size Effects in Aging can be Interpreted as Sub-Aging

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read Finite-size effects can transform simple aging into the phenomenology of sub-aging, and the fitted sub-aging exponent then carries no thermodynamic meaning.

desk verdict A clean exact-model demonstration that finite-size effects can masquerade as sub-aging, with honest caveats; worth reviewing despite the transfer to real glasses being conditional. read the letter →

arxiv 2501.04843 v1 pith:MSVXKH2Q submitted 2025-01-08 cond-mat.stat-mech physics.comp-ph

Henrik Christiansen , Suman Majumder , Wolfhard Janke , Malte Henkel This is my paper · ORCID
classification cond-mat.stat-mechphysics.comp-ph
keywords agingsub-agingfinite-sizeeffectsdynamicalscalingsphericalmodelIsingphase-orderingkineticstwo-timeautocorrelator
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 argues that finite-size effects can take a system whose true behavior is simple aging—the two-time autocorrelator collapses as a function of $t/t_w$—and make it look as if it sub-ages, with best-fit exponents $0<\mu<1$. The core demonstration uses the exactly solved spherical model in $2

What carries the argument

The load-bearing object is the exact finite-size two-time autocorrelator of the spherical model, Eq. (4), obtained in the double scaling limit $t_w\to\infty$, $L\to\infty$, with $Z=L^2/(y t_w)$ fixed. It is the bulk autocorrelator times a ratio of Jacobi $\theta$ functions, $\vartheta_3$, classical special functions that sum over the discrete finite-size modes, and this ratio encodes the finite-size modification appearing when $Z\lesssim 1$. The argument proceeds by showing that this closed form is incompatible with the sub-aging form $C=F_C(h(t)/h(t_w))$, so any successful sub-aging collapse of it must be produced by the fitting procedure; the numerical collapse measure $S$, a reduced $\chi^2$-like distance from a master curve, is used to locate the best $\mu$ in the spherical and Ising data.

What would settle it

Take the exact correlator (4) and ask whether any fixed $\mu<1$ exists for which $C(t,t_w)$ depends only on $h(t)/h(t_w)$ for all $t_w$; a numerically exact, $t_w$-independent collapse at fixed $\mu<1$ would refute the claim that Eq. (4) rules out sub-aging.

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Extended reading notes

Core claim

For a ferromagnet quenched from a disordered state to $T<T_c$, the paper claims that the exact finite-size autocorrelator (4) is analytically inconsistent with true sub-aging, yet fitting the sub-aging form to it produces apparently good collapses with $\mu<1$. The optimal $\mu$ shifts with system size—about 0.98 for $L=50$ and 0.92 for $L=16$ in the spherical model, with analogous shifts in the two Ising models—so it has no objective thermodynamic meaning and instead reflects the presence of a second length scale, the system size, alongside the growing domain size. In the infinite-size limit $\mu\to 1$ and simple aging is recovered. The same 'sub-aging' signature can be produced by any additional small length scale, such as stray magnetic or electric fields.

Load-bearing premise

Everything rests on the double scaling limit in which the spherical-model correlator is exactly known ($t_w\to\infty$ and $L\to\infty$ with $Z=L^2/(y t_w)$ fixed); if real or simulated data are dominated by the non-scaling finite-size and finite-time corrections that this limit removes by construction, the apparent sub-aging could be stronger, weaker, or absent.

Editorial extensions

If this is right

  • For the exactly solved spherical model, the fitted sub-aging exponent has no thermodynamic content: values such as $\mu\simeq0.98$ for $L=50$ and $\mu\simeq0.92$ for $L=16$ simply track the system size.
  • In the two-dimensional nearest-neighbor Ising model, the same size-dependent shift ($\mu\simeq0.97$ for $L=256$, $\mu\simeq0.91$ for $L=128$) appears in numerical data, so the effect is not an artifact of the exact solution.
  • For the long-range Ising model with $\sigma=0.6$, even $L=4096$ gives a best collapse at $\mu\simeq0.98$, explaining a previously published sub-aging-looking collapse as a finite-size effect.
  • Any additional length scale competing with the growing domain size, such as stray magnetic or electric fields, should produce the same artificial sub-aging signature.
  • Reported sub-aging exponents should be accepted only after checking that the fitted value is stable as the system size approaches the thermodynamic limit.

Reading between the lines

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

  • A testable consequence the authors leave implicit: for a fixed material or simulation protocol, the fitted $\mu$ should drift monotonically upward with sample size; controlled grain-size or sample-thickness experiments on colloids or soft glasses could detect this drift.
  • The theta-function expression provides a quantitative calibration: one could precompute $\mu_\mathrm{eff}(L)$ for the spherical model and use it to subtract finite-size contamination in disorder-free candidates for sub-aging.
  • The paper does not settle whether spin-glass sub-aging is intrinsic: it acknowledges rigorous sub-aging in activated hopping models, so the artifact mechanism applies to systems with a second small length scale, and intrinsic and finite-size contributions would have to be disentangled in real glasses.
  • Reanalyzing published spin-glass data with sample size or grain size as an explicit parameter might reveal an $L$-dependence analogous to the one shown here, strengthening or weakening the case for true sub-aging in each material.
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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

4 major / 5 minor

Summary. The paper argues that finite-size effects in aging systems can produce the phenomenology of sub-aging, even when the infinite system would exhibit simple aging. For the exactly solvable spherical model in 2<d<4, the authors use an exact finite-size autocorrelator in a double-scaling limit (Eq. (4)) to show that simple aging is violated at finite system size, and that fitting the sub-aging form (2) yields an effective exponent mu<1 that decreases with L. They corroborate this trend with Monte Carlo simulations of the two-dimensional nearest-neighbor and long-range Ising models. They conclude that the fitted mu has no thermodynamic meaning and that apparent sub-aging can be a finite-size artifact.

Significance. If the central claim holds, the paper provides an important caution for the interpretation of sub-aging in spin glasses and other glassy systems, where mu is routinely extracted from finite-size or finite-time data. The exactly solvable spherical-model result is a valuable analytical example, and the numerical evidence spans three distinct model classes (spherical, NN Ising, and long-range Ising), strengthening the message that apparent sub-aging can be a finite-size artifact. The paper also makes an explicit, falsifiable prediction that the optimal mu tends to 1 as L increases. The main weakness is that the exact demonstration is for an effective scaling-limit model, and the simulation-based evidence is noisy and relies on a collapse analysis with several free parameters.

major comments (4)
  1. [Spherical model, Eq. (4) and SM] The SM (Spherical Model section, after Eq. (S.16)) explicitly states that Eq. (S.16) "is not the one of the original lattice model" but describes an effective model where non-scaling finite-size and finite-time corrections have been subtracted off. The main text presents Eq. (4) as "the two-time spin-spin autocorrelator" without this caveat. Since the title and abstract make a general claim about finite-size effects, the authors should either move this caveat into the main text or provide evidence that the subtracted non-scaling corrections do not change the qualitative conclusion. As written, the exact demonstration applies to the scaling-limit effective model, not to a real finite lattice at finite L and tw.
  2. [SM, Collapse Analysis, Eq. (S.21)] The collapse quality S is defined in terms of standard errors dy_ij and dY_ij, but for the exact spherical-model data the authors assume a relative error of 3% without justification. The reported optimal mu values (e.g., mu_L=50 ≈ 0.98 vs mu_L=16 ≈ 0.92) and the apparent monotonic trend with L may depend on this arbitrary error choice, as well as on the number of bins (100) and the truncation at t_max. A sensitivity analysis varying these choices is needed to establish that the size-dependent mu trend is robust rather than an artifact of the collapse measure.
  3. [Section 1, last paragraph] The statement that the exact autocorrelator (4) "does rule out any sub-aging" is a nontrivial claim that is not demonstrated. While it is plausible that Eq. (4) cannot be written exactly in the form (2) for any 0<mu<1, the paper should provide a brief argument or a citation to a proof. Without this, the strong conclusion that sub-aging is ruled out by the exact spherical-model result is unsupported.
  4. [Section 2 and SM, Collapse Analysis] The 2D Ising simulations are genuine finite-lattice data, but they are noisy and the collapse analysis truncates the data at a maximum t_max, "to remove clearly finite-size effected data". This selection removes the regime where finite-size saturation is strongest, which is precisely the regime of interest. The authors should show how the fitted mu values change when t_max is varied, or plot the raw data with the truncation indicated, so that the reader can judge whether the reported mu values are an artifact of this data selection.
minor comments (5)
  1. [SM, Eq. (S.21)] The symbol L is used both for the system size and for the number of waiting times in the normalization of S. Please use a different symbol (e.g., N_tw) for the latter.
  2. [Figure 2] The S(mu) curves are presented without error bars or a formal criterion for locating the minimum. It would be helpful to state explicitly how mu_opt is read off from the curves and how sensitive the location is to the binning details.
  3. [Abstract] The abstract states "Here it is shown that finite-size effects modify the dynamical scaling behavior" without noting that the exact derivation is for a finite-size scaling limit. Consider rephrasing to "in the finite-size scaling limit" or "for the models studied" to avoid overgeneralization.
  4. [SM, Eq. (S.4)] Equation (S.4) contains the expression "c(1)age", which appears to be a typographical artifact; it should likely be a constant times t_w^zeta. Please correct this.
  5. [General] The central exact result Eq. (4) is quoted from reference [45], which is a self-citation. This is acceptable, but the text should indicate more clearly that the derivation is not repeated in the main text and that the result relies on published work.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the central exact correlator is a self-cited but independent input, and all sub-aging exponents are explicit fits rather than predictions.

full rationale

The central claim — finite-size effects can turn simple aging into apparent sub-aging — rests chiefly on Eq. (4)/(S.16), the exact finite-size autocorrelator of the spherical model, imported from the authors' own ref. [45]. This self-citation is load-bearing but not circular: the formula is derived from the spherical-model Langevin dynamics in the double scaling limit (3)/(S.14)-(S.15), and those assumptions do not include the sub-aging form (2). The paper asserts the opposite direction — 'the exact autocorrelator (4) ... does rule out any sub-aging' — and supplies internal consistency checks (bulk factor giving λ = d/z = d/2; finite-size factor saturating for Z ≪ 1). The formula is parameter-free, externally checkable evidence under hard rule 4, not an unverified self-justification. The sub-aging exponents are explicitly fitted, not predicted: Fig. 2 shows 'the values of µ that provide the optimal data collapse' from the S-statistic master-curve analysis, and the conclusion is that the optimum varies with L and hence 'cannot have an objective thermodynamic meaning'. Fitting a phenomenological form to an exact correlator and finding an L-dependent best-fit μ demonstrates a fitting degeneracy; it is not equivalent to the input by construction. Two self-flagged limitations are scope conditions, not circularity. The SM concedes that Eq. (S.16) 'is not the one of the original lattice model but rather describes an effective model where the non-scaling finite-size and finite-time corrections have been subtracted off,' so the exact demonstration is conditional on the scaling limit; the NNIM/LRIM simulations are genuine-lattice evidence beyond it. The collapse analysis also 'truncate[s] the data at a maximum value t_max' to remove 'clearly finite-size effected' data before fitting μ; this is transparent and makes the μ<1 finding conservative for the remaining regime. No uniqueness theorem from prior author work is invoked, no ansatz is smuggled in via citation (the sub-aging form is the experimental literature's own, and the paper's point is that it fits only imperfectly), and the same-data master curve is standard collapse methodology. The derivation is self-contained against the sub-aging hypothesis; score 1 reflects self-citations in the support chain with no circular step among them.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The analysis adds no hidden physical entities. It relies on a published exact result for the spherical model, three modeling assumptions about the scaling regime and fitting family, and a small number of analysis choices (bins, t_max, assumed 3% error). The fitted mu values are the object of study, not hidden inputs, but the S-collapse procedure has genuine tunable choices whose influence is not fully quantified.

free parameters (4)
  • Sub-aging exponent mu = 0.98 (spherical L=50), 0.92 (spherical L=16); 0.97/0.91 (NNIM L=256/128); 0.98/0.93 (LRIM L=4096/1024)
    Fitted by the S-collapse analysis; the paper's conclusion is that the fitted value depends on L and has no thermodynamic meaning. The values are outcomes of the analysis, not hidden inputs.
  • t_max truncation = not stated
    SM collapse analysis excludes data beyond a maximum time t_max, chosen independent of t_w, to remove clearly finite-size-effected data; this choice affects S and the location of its minimum.
  • Assumed relative error for spherical data = 3%
    SM collapse analysis assigns a 3% relative error to the exact spherical correlator, which has no statistical error; this weights the S sum and can shift the optimal mu.
  • Number of bins in collapse analysis = 100
    SM uses 100 logarithmically increasing bins for the piecewise linear master curve; the binning is a tunable analysis parameter.
assumptions (5)
  • domain assumption Equation (4) is the exact two-time autocorrelator of the finite spherical model in the double-scaling limit (3).
    Imported from ref. [45], whose author overlaps with the present paper. The paper does not rederive it; the central demonstration of apparent sub-aging starts from this formula.
  • domain assumption The double-scaling limit (3) with Z = L^2 / (y tw) fixed is the relevant finite-size regime.
    Main text Eq. (3). The claim that finite-size effects produce sub-aging is established inside this limit, not for arbitrary fixed L and large tw.
  • domain assumption The experimental sub-aging form Eq. (2) with h(t) = exp((t^(1-mu) - 1)/(1-mu)) is the correct family for fitting.
    SM derives the form from two independent arguments; the paper's demonstration is that this form, when forced onto finite-size data, gives size-dependent mu.
  • domain assumption The 2D NNIM and LRIM simulations probe the same finite-size mechanism as the exact spherical model.
    Main text Secs. 2 and 3; without an exact solution for the Ising models, the analogy is supported only by the similar behavior of S(mu).
  • standard math Standard Jacobi theta identities connect the two forms of the correlator, Eqs. (S.16a) and (S.16b).
    SM; no proof is given in the paper, but these are standard modular transformations for theta functions.

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Cite this review

Pith. "Pith review of Finite-Size Effects in Aging can be Interpreted as Sub-Aging." pith.science (2026). https://pith.science/paper/MSVXKH2Q

@misc{pith2026250104843,
  author       = {Pith},
  title        = {Pith review of: Finite-Size Effects in Aging can be Interpreted as Sub-Aging},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/MSVXKH2Q}},
  note         = {Machine review of arXiv:2501.04843}
}
abstract

Systems brought out of equilibrium through a rapid quench from a disordered initial state into an ordered phase undergo physical aging in the form of phase-ordering kinetics, with characteristic dynamical scaling. In many systems, notably glasses, dynamical scaling is often described through sub-aging, where a phenomenological sub-aging exponent $0<\mu< 1$ is empirically chosen to achieve the best possible data collapse. Here it is shown that finite-size effects modify the dynamical scaling behavior, away from simple aging with $\mu=1$ towards $\mu<1$, such that phenomenologically it would appear as sub-aging. This is exemplified for the exactly solved dynamical spherical model in dimensions $2<d<4$ and numerical simulations of the two-dimensional Ising model, with short-ranged and long-ranged interactions.

Figures

Figures reproduced from arXiv: 2501.04843 by the authors.

Figure 1
Figure 1. FIG. 1. Simple and sub-aging in the two-time autocorrelator [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. (a) for the spherical model, (b) for the NNIM, and (c) for the LRIM with σ = 0.6, in each case for the sys￾tem sizes mentioned in the figure key. The minimum of S implies the value of µ for optimal data collapse. The µopt we obtained through S agree well with what we observe visually. In particular, for the spherical model with L = 50, µL=50 ≈ 0.98 should be chosen for the best data collapse and for L = 16, µL=16 ≈ … view at source ↗
Figure 3
Figure 3. FIG. 3. Simple and sub-aging in the two-time autocorrelator [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Simple and sub-aging in the two-time autocorrelator [PITH_FULL_IMAGE:figures/full_fig_p004_4.png]

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Forward citations

Cited by 1 Pith paper

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