Aging of glass-forming materials following a temperature jump

Ayata Ueno; Takashi Odagaki; Tomoko Mizuguchi

REVIEW 4 major objections 5 minor 52 references

Aging of glass-forming materials following a temperature jump

T0 review · 4 major / 5 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read This paper claims that after a temperature increase, the temporal relaxation time of a glass former can first decrease and then increase—producing a minimum—when the free-energy landscape responds with an intermediate delay, and that this s

desk verdict Plausible model prediction of a non-monotonic relaxation time after a T-up jump, but the key figure has no error bars and the conceptual mapping in Section V is more asserted than demonstrated. read the letter →

arxiv 2509.03022 v2 pith:UTHPYSWO submitted 2025-09-03 cond-mat.stat-mech

Ayata Ueno , Tomoko Mizuguchi , Takashi Odagaki This is my paper · ORCID

classification cond-mat.stat-mech

keywords physicalagingglassformerstemperaturejumpfreeenergylandscapetrapmodeltwo-timerelaxationfunctiontemporaltimefictive

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

Aging means observable properties keep evolving long after a temperature change. This paper studies a glass model with two causes of aging at once: traps that make escape slower as the system settles deeper, and a free-energy landscape that only gradually adjusts to the new temperature. The model confirms that the first alone gives Type-I aging—relaxation time always grows with waiting time—while the second alone gives Type-II aging—relaxation time grows after cooling but shrinks after heating. The central result is that when both operate, a temperature-up jump makes the temporal relaxation time dip to a minimum and then rise, provided the landscape responds on an intermediate timescale. This short-time non-monotonicity is proposed as a fingerprint that distinguishes the two mechanisms, and the paper reinterprets material time and fictive temperature as descriptions of the landscape lag.

What carries the argument

The load-bearing object is a single scalar internal temperature T_int(t), which starts at the old bath temperature and exponentially relaxes to the new one with timescale tau_F, rescaling every basin's jump rate uniformly through W_n(t)=w_0 exp[-epsilon_n (T_g-T_K)/(T_int(t)-T_K)]. This separates the two aging mechanisms: basin-specific depths epsilon_n produce trapping, while the common rescaling by T_int produces the delayed landscape response. The readout is the two-time relaxation function and the temporal relaxation time tau_tmp(t',tw), whose zero-waiting-time shape after a T-up jump carries the predicted minimum.

What would settle it

Measure the temporal relaxation time from the two-time correlation function immediately after a T-up jump in a glass former and look for the minimum: the model predicts a dip for intermediate landscape response times, but not for zero or very long response times. Data showing monotonic increase for all response times, or a dip when the landscape is known to respond instantly, would refute the claim. A numerical variant replacing the exponential delay with a stretched exponential, or letting T_int vary across lattice sites, would test the single-clock assumption directly.

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Extended reading notes

Core claim

The central claim is that the extended trapping diffusion model—a random walk on a one-dimensional lattice whose jump rates are power-law distributed through basin depths and uniformly rescaled by a delayed internal temperature T_int(t)—produces a characteristic non-monotonic temporal relaxation time after a temperature increase. For T-up protocol and tau_F=5, tau_tmp(t'*,0) first decreases and then increases, with a minimum near t'* about 6; the same minimum appears for tau_F=2 and 10 but disappears for tau_F=0 (no delay) and tau_F=30 (delay too slow). The position of the minimum is approximately the crossing point between the temporal relaxation time of the pure delayed random walk and the

Load-bearing premise

The load-bearing premise is that the entire free-energy landscape can be represented by one internal temperature that relaxes exponentially and rescales every basin uniformly; if different regions lag differently or nonexponentially, the predicted minimum and clean Type-I/Type-II separation may not survive.

Editorial extensions

If this is right

If the model is right, a T-up jump in a glass former with an intermediate landscape response time should show a temporal relaxation time that first decreases, then increases, with a minimum; pure trapping or an instantaneous landscape would give only a monotonic rise.
The short-time behavior of the temporal relaxation time at zero waiting time can separate Type-I and Type-II aging: a dip is the fingerprint of the delayed landscape mechanism.
The model identifies material time and internal clock with the scaled time integral of the delayed jump rate, giving these phenomenological concepts a concrete microscopic definition.
The fictive temperature used in aging analyses is reinterpreted as the internal temperature describing delayed free-energy-landscape response, not a purely structural parameter.
In this model the relaxation function is KWW-like only over about two to three decades of time, so full-domain KWW fits should not be assumed valid.

Reading between the lines

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

The paper leaves implicit that the position and depth of the predicted minimum could be used to estimate the landscape response time tau_F quantitatively, provided the trap distribution is known; analyzing several T-up jumps at different final temperatures could map tau_F as a function of temperature.
If the single-clock assumption is relaxed to a distribution of local internal temperatures, the minimum should broaden; this would connect the model to dynamical heterogeneity and memory effects, which the paper names as future directions but does not model.
The same competition between uniform rate rescaling and trapping should appear in other two-time observables, such as dielectric permittivity or stress relaxation, not only the self-intermediate scattering function, because only the trap distribution and the common rate factor enter the argument.
A direct experimental test is to measure the two-time correlation function immediately after a T-up jump and look at even shorter elapsed times than usual: the model predicts an apparent speeding-up that is later overtaken by the trapping slowdown.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The take: this is a clean model study with one genuinely new prediction, but the prediction is less solid than the abstract suggests. The new thing is combining the trap model with a delayed free-energy-landscape response and showing, within that combined model, that the temporal relaxation time after a T-up jump can develop a minimum for intermediate FEL response times. The authors also connect this to material time and fictive temperature, which is an interesting interpretive step but not proven.

What the paper does well: the trap-free case in Appendix B is exactly solvable and the numerics match the exact expression. The model distinctions between Type-I (trapping) and Type-II (delayed FEL) aging are clean and reproduce prior expectations. The authors are honest about the KWW fit being limited to 2–3 decades. They also ship code and data on Zenodo, which makes the numerical work checkable.

Where it gets soft: the central Fig. 8 has no error bars and averages over only 30 disorder realizations. The minimum is shallow—maybe 10–20% of the relaxation time—and the temporal relaxation time is a derivative of a ratio of averaged SISFs, so sample-to-sample fluctuations could plausibly wash it out or shift it. I think the stress-test concern about finite-sample artifact is real and the authors should have addressed it with bootstrap confidence bands or more realizations. This is the load-bearing point of the paper, so it matters.

The other soft spot is Section V. Identifying the scaled time of the exact solution with "material time" and T_int with "fictive temperature" is a mapping of definitions, not a derivation. It is reasonable and may be useful, but the paper sells it as an explanation. The broader claim that the minimum will show up in experiments and simulations is speculative; the model has several hand-inserted ingredients, and the single-exponential, spatially uniform T_int is a strong assumption that the authors themselves flag.

Overall, the numerical core of the model is sound and the combination is new as a model prediction. The paper deserves a serious referee—an editor should not desk reject it. But the referee should ask for error bars on Fig. 8, more disorder samples, and a sharper separation between what the model actually shows and what the interpretive discussion claims.

My advice: engage with it if you work on aging or trap models; it is a useful reference for the combined-mechanism picture, but don't take the non-monotonic minimum as established until the finite-sample issue is checked.

Referee Report

4 major / 5 minor

Summary. The paper extends a trapping diffusion model of glass-forming materials by adding a delayed response of the free energy landscape (FEL) to temperature changes, parameterized by an internal temperature T_int(t) that relaxes exponentially with time constant τ_F (Eqs. (5)-(8)). The trapping mechanism alone produces Type-I aging (relaxation time increases with waiting time for both T-up and T-down), while the delayed FEL response alone produces Type-II aging (waiting-time dependence changes sign with protocol), the latter being exactly solvable in the uniform-trap case (Appendix B). In the combined 'extended trapping diffusion model', the authors report that for a T-up protocol and intermediate τ_F, the temporal relaxation time τ_tmp(t'*,0) is non-monotonic in the observation time t'*, with a minimum (Fig. 8). They interpret this as a competition between the two aging mechanisms and argue that the material time/internal clock and fictive temperature are understood as consequences of the delayed FEL response.

Significance. If the reported non-monotonicity is robust, it offers a practical signature for disentangling trapping-dominated from landscape-response-dominated aging in T-up experiments, and it provides a concrete microscopic-ish interpretation of the TNM fictive temperature and material time. The paper is transparent: the uniform-trap case is solved exactly in Appendix B, and the data/code are openly available (Ref. [46]). The model is simple and the separation of mechanisms is instructive. However, the significance is conditional: the central prediction rests on a shallow minimum in a quantity with no reported uncertainties, and the conceptual mapping to material time/fictive temperature goes beyond what the model can currently prove.

major comments (4)

[§IV, Fig. 8] The central claim—that τ_tmp(t'*,0) is non-monotonic for intermediate τ_F—is based on Fig. 8, which shows averages over only 30 disorder realizations (Eq. (12)) and no uncertainty estimates. τ_tmp is a logarithmic derivative of a ratio of ensemble-averaged SISFs (Eqs. (14)-(15)), so sample-to-sample fluctuations propagate directly into the depth and location of the apparent minimum. The minimum is shallow and its position is estimated by a crossing of two noisy quantities. Without bootstrap confidence intervals, more realizations, or a quantitative criterion, the reader cannot distinguish a robust model feature from a finite-sample artifact. Because this minimum is the paper's proposed experimental signature, this is load-bearing.
[Abstract, §IV, §V] The abstract and Section V state that the temporal relaxation time has a minimum 'as a function of waiting time,' but Fig. 8 and the surrounding text (e.g., 'at t'_w=0') show τ_tmp as a function of the elapsed observation time t'* at fixed waiting time t_w=0. These are different observables: the former is a dependence on t_w, the latter on t'. The reported simulation demonstrates the latter. If the minimum is intended to be in t_w, it is not shown; if it is intended to be in t', the abstract and discussion should be reworded to avoid a misleading experimental prescription.
[§V and Appendix B] The statement that material time/internal clock are identical to the scaled time introduced in Eq. (B1) is justified only for the trapping random walk with ϵ_n=1, in which all jump rates share a common time-dependent factor. In the extended trapping diffusion model, W_n(t)=w0 exp[-ϵ_n (T_g-T_K)/(T_int(t)-T_K)] (Eq. (7)); for ϵ_n≠ϵ_m, the ratio W_n/W_m varies as T_int(t) changes, so there is no single time reparametrization that makes the master equation (9) time-homogeneous. The identification in Section V therefore does not follow for the central model. The authors should either restrict this conceptual claim to the uniform case or demonstrate that an approximate material time exists in the heterogeneous case.
[§II.A, §V] The predicted minimum and the clean Type-I/Type-II separation depend on the specific assumption that the FEL responds as a single scalar internal temperature with a simple exponential delay (Eqs. (5)-(8)). The authors acknowledge in Section V that local heat-transfer differences could make T_int position-dependent, but they do not test the robustness of Fig. 8 against straightforward generalizations, such as a stretched exponential φ(t) or a distribution of τ_F. Since the abstract and Section V generalize beyond the particular functional form, a sensitivity study (even in the uniform-trap case) would materially strengthen the claim that the non-monotonicity is generic.

minor comments (5)

[Appendix B] 'SSIF' appears to be a typo for 'SISF'.
[§V] 'fotT-up' should be 'for T-up'.
[§IV] In Figure 7 and the surrounding text, the notation 't'_w=0' should probably be 't^*_w=0' or 't_w=0'; as written it is confusing.
[Eq. (12)] The averaging procedure over 30 samples is described, but no lattice size, boundary conditions, or statistical precision (e.g., standard errors) are reported. Including these details would improve reproducibility.
[References] Minor formatting typos in references, e.g., 'J. Am .Ceram. Soc.' in Ref. [35].

Circularity Check

2 steps flagged · score 4.0 of 10

Partial circularity in the interpretive claims: material time and fictive temperature are identified with the paper's own constructs by self-citation and renaming, while the numerical non-monotonicity prediction is independent.

self citation load bearing [Appendix B, Eq. (B1); Section V Discussion]
"Note that this time is identical to the material time [41]. ... Meaning of the material time and internal clock are identical to the scaled time introduced in Appendix B to incorporate the relaxation of the FEL."

The identification of the scaled time t̃(t)=∫W(t')dt' with the phenomenological 'material time' is not derived in this paper; the only support given is the citation to [41], whose author overlaps with the present paper. Section V then uses this identification as the basis for one of the paper's advertised conceptual conclusions ('Meaning of the material time and internal clock are identical to the scaled time...'). Thus the conceptual claim about material time reduces to a self-citation rather than an independent derivation. The quantitative result (B2)-(B4) is self-contained, so this circularity affects the interpretive conclusion, not the numerical prediction.
renaming known result [Section V Discussion]
"The internal temperature introduced in Eqs. (5)–(7) is a parameter that describes the delayed change in the depth of the FEL after a temperature change, although the TNM model introduced T_fic focusing on the delayed response of structure. Therefore, the internal temperature is essentially the same concept as the TNM fictive temperature..."

The internal temperature T_int is introduced by assumption (Eqs. 5-8) as a delayed response of the FEL to temperature change. The conclusion that 'the TNM fictive temperature can be understood as a parameter describing the delayed response of the FEL' is the same statement with the label 'fictive temperature' attached. No independent evidence or chain of derivation connects the empirically motivated TNM fictive temperature to T_int; the claim is an equality by definition/renaming. This does not affect the numerical minimum prediction, but it makes the advertised 'understanding' of fictive temperature tautological.

full rationale

The paper's central quantitative result—the non-monotonic temporal relaxation time with a minimum for intermediate τ_F in the T-up protocol (Fig. 8)—is a genuine numerical consequence of the model. It is not fitted to data, and it follows from combining two mechanisms that are explicitly put into the model and solved (trapping diffusion and delayed FEL response). No external benchmark is invoked to produce the minimum, so the core prediction is self-contained and not circular. The circularity arises only in the interpretive claims advertised in the abstract and discussion: the assertion that 'material time' is identical to the scaled time of Appendix B is supported solely by a self-citation ([41]), and the assertion that the internal temperature 'is essentially the same concept as' the TNM fictive temperature is a renaming of the model's own definition rather than an independent result. These steps make part of the paper's conceptual conclusion definitional, but they do not undermine the numerical non-monotonicity prediction. Uncertainty in Fig. 8 is a statistical robustness concern, not a circularity concern. Score 4 reflects partial circularity in the interpretive overlay with an independent central numerical claim.

Assumptions & free parameters 3 free parameters · 6 assumptions · 1 invented entities

The model imports trapping statistics and an FEL delay assumption from earlier work, and adds an exponential scalar delay by hand. The central prediction depends on these choices; no experimental data are used to set or validate them.

free parameters (3)

tau_F (FEL response time) = 0, 2, 5, 10, 30 (reduced units)
Controls whether the predicted minimum appears; chosen by hand over a range, not determined from data.
k* = k a = 0.79
Observation wavevector for the self-intermediate scattering function; chosen once, affects quantitative relaxation times but not the classification.
Temperature protocol (T_g/T_K=1.25; T_i/T_f=1.15/1.24) = T_i*=1.15, T_f*=1.24
Selected to keep both states below T_g; direction and magnitude of the jump enter the Type-I/Type-II comparison.

assumptions (6)

domain assumption Jump rates follow a power-law distribution with exponent rho=(T-T_g)/(T_g-T_K), equivalent to exponential basin depths Eq. (4)
Imported from prior FEL/trap work (refs 30,31,42,43); the trapping Type-I results depend on it.
ad hoc to paper The FEL responds to the bath temperature through an exponential delay phi(t)=exp(-t/tau_F) (Eq. 8)
This is the paper's central added assumption; the predicted minimum appears only for intermediate tau_F.
ad hoc to paper At any time T_int(t) rescales every basin depth uniformly via Eq. (7)
Uniform rescaling makes the delay a scalar; heterogeneous landscape relaxation is explicitly deferred to future work.
domain assumption Dynamics follows a one-dimensional nearest-neighbor master equation (Eq. 9)
Representative walk on a 1D reaction coordinate; the SISF is computed from this walk.
domain assumption The system starts in the steady state at T_i with weights W_n^{-1} (Eq. 13)
Defines the waiting-time protocol; different initial conditions could change the transient shape.
domain assumption The SISF at k*=0.79 is the observable used for all aging analysis
A single wavevector is chosen; the authors do not demonstrate robustness of the minimum across k*.

invented entities (1)

Internal temperature T_int(t)
purpose: A single scalar representing the delayed depth of FEL basins after a temperature jump, used to define time-dependent jump rates (Eqs. 5-7).
Introduced as a model construct. The paper identifies it with the TNM fictive temperature, but gives no direct experimental measurement or independent falsifiable handle beyond the untested prediction of nonmonotonic tau_tmp.

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Cite this review

Pith. "Pith review of Aging of glass-forming materials following a temperature jump." pith.science (2026). https://pith.science/paper/UTHPYSWO

@misc{pith2026250903022,
  author       = {Pith},
  title        = {Pith review of: Aging of glass-forming materials following a temperature jump},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UTHPYSWO}},
  note         = {Machine review of arXiv:2509.03022}
}

read the original abstract

Physical aging is one of the non-equilibrium phenomena where physical properties change over time due to structural relaxation. Aging in spin glass systems has been explained by a trap model on the temperature-independent energy landscape. Meanwhile, in the free energy landscape (FEL) approach to aging phenomena, it is assumed that the FEL responds to temperature changes with a time delay. In this paper, aging in a glass forming model in which both the trapping effect and the delayed response of the FEL exist is studied after the temperature is changed. It is confirmed that the trapping effect gives rise to Type-I aging where the relaxation time increases with waiting time regardless of the direction of temperature change, and that the delayed response of the FEL produces Type-II aging where the waiting-time dependence of the relaxation time depends on the direction of temperature change. When both effects exist and the response time of the FEL is appropriate, these effects can be differentiated in the short-time behavior of the temporal relaxation time. It is argued that the material time or the internal clock and the fictive temperature introduced phenomenologically are understood as the concepts describing the delayed response of the FEL to temperature change.

Figures

Figures reproduced from arXiv: 2509.03022 by the authors.

**Figure 2.** FIG. 2. Waiting time dependence of TTRF for the trapping diffusion model: (a) [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗

**Figure 3.** FIG. 3. Temporal relaxation time [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗

Figures from the paper (7 more)

**Figure 4.** Figure 4: FIG. 4. Waiting time dependence of TTRF for the trapping random walk with the delayed response of the [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: FIG. 5. The [PITH_FULL_IMAGE:figures/full_fig_p010_5.png]

**Figure 6.** Figure 6: FIG. 6 [PITH_FULL_IMAGE:figures/full_fig_p010_6.png]

**Figure 7.** Figure 7: FIG. 7 [PITH_FULL_IMAGE:figures/full_fig_p011_7.png]

**Figure 8.** Figure 8: FIG. 8 [PITH_FULL_IMAGE:figures/full_fig_p011_8.png]

**Figure 9.** Figure 9: FIG. 9. SISF for the trapping diffusion model can be fitted by the KWW function in a limited area about [PITH_FULL_IMAGE:figures/full_fig_p014_9.png]

**Figure 10.** Figure 10: FIG. 10. The temporal relaxation time of [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]

Discussion (0). Continue with ORCID to comment.

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2024

Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.

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