{"id":"c6a6f0d9-d04b-4f5c-be82-ca5adabc17cc","arxiv_id":"2501.04843","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":4,"one_line_summary":"Finite-size effects in an exactly solved spherical model and in 2D Ising simulations make simple aging look like sub-aging, with fitted sub-aging exponents that depend on system size.","lead":"This paper shows that the apparent \"sub-aging\" behavior seen in many glassy materials can be produced by finite-size effects in systems that actually follow simple aging. It matters because measured sub-aging exponents, often treated as material properties, may partly be artifacts of sample size.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The exact spherical-model support is derived in a double-scaling limit that explicitly subtracts non-scaling finite-size corrections, so the general claim is conditional on those corrections being negligible.","rationale":"The paper aims to show that finite-size effects can lead to apparent sub-aging in phase-ordering kinetics. The strongest and most original evidence is the exact solution of the spherical model in the double-scaling limit, Eq. (4)/(S.16). My central concern is that this exact result is explicitly not the autocorrelator of the original finite lattice model: the SM states it 'describes an effective model where the non-scaling finite-size and finite-time corrections have been subtracted off.' The visual demonstrations in Figs. 1 and S2–S4, the collapse analysis, and the reported μ(L) values all follow from this effective scaling function. The paper's title and abstract make a general claim about finite-size effects; that claim requires that the subtracted corrections do not change the qualitative phenomenon. The 2D Ising simulations do use the real lattice model, but they are noisy, and the collapse analysis itself truncates away the 'clearly finite-size effected' saturation regime, so they provide only partial, indirect support. Hence the central claim is conditional on the non-scaling corrections being negligible. This is a load-bearing concern because if the corrections are not negligible, the apparent sub-aging exponent extracted from experiments could differ substantially from the predicted μ(L) or could even be absent. The proposed test—computing the exact finite-lattice (non-scaling-limit) autocorrelator for the spherical model, or the leading corrections to Eq. (S.16)—would directly measure how much the results shift. If the shift is small, the concern is resolved and the claim stands. If not, the paper's title claim would need to be restricted to the scaling-limit effective model or to regimes where non-scaling corrections are negligible. I therefore agree with the reader's conditional assessment.","tokens_in":22095,"tokens_out":10571,"duration_ms":101398,"concrete_test":"Simulate the 3D spherical model on finite lattices (L=16, 50) at T<Tc using the exact Langevin mode dynamics without taking the double-scaling limit, compute C(t,tw) for the same tw values, and repeat the S(μ) collapse analysis, or equivalently compute the leading 1/tw and 1/L corrections to Eq. (S.16) analytically. If the fitted μ(L) from the finite-lattice data (or from the corrected formula) departs by more than ~0.02 from the scaling-limit values (μ=0.92 for L=16, μ=0.98 for L=50), then non-scaling finite-size corrections are significant and the transfer of the claim to real systems is not justified without further evidence.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim is that finite-size effects can change simple aging into sub-aging phenomenology. The strongest support is the exact spherical-model autocorrelator, Eq. (4)/(S.16). However, the Supplemental Material explicitly says this expression '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' (SM, Spherical Model section, after Eq. S.16). Eq. (4) is therefore valid only in the limit tw→∞, L→∞ with Z = L^2/(y tw) fixed, not for a real finite lattice at finite tw and L. Since the figures, the collapse analysis, and the fitted μ(L) values are all based on this effective scaling function, the paper's title claim—as a statement about finite-size effects in general—requires that the subtracted corrections not change the qualitative result. The 2D Ising simulations are genuine lattice data, but they are noisy and the collapse analysis truncates data at a maximum tmax to remove 'clearly finite-size effected' data, so they do not fully close the gap. Thus, the general claim is conditionally supported at best; it is conclusively demonstrated only for the scaling-limit effective model.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":22330,"tokens_out":6996,"duration_ms":69674,"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":[{"comment":"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.","section":"Spherical model, Eq. (4) and SM"},{"comment":"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.","section":"SM, Collapse Analysis, Eq. (S.21)"},{"comment":"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.","section":"Section 1, last paragraph"},{"comment":"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.","section":"Section 2 and SM, Collapse Analysis"}],"minor_comments":[{"comment":"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.","section":"SM, Eq. (S.21)"},{"comment":"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.","section":"Figure 2"},{"comment":"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.","section":"Abstract"},{"comment":"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.","section":"SM, Eq. (S.4)"},{"comment":"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.","section":"General"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is interesting and likely publishable after revision. The main concern is the gap between the exact scaling-limit result and the general claim about finite-size effects; the SM already acknowledges this gap, but it should be reflected in the main text. A sensitivity analysis of the collapse measure (assumed errors, binning, t_max truncation) would substantially strengthen the paper. The reliance on a self-cited reference for the central exact result is acceptable but should be stated transparently."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The short version: this paper shows, in an exactly solvable spherical model, that finite-size effects can turn simple aging into the phenomenology of sub-aging, with the optimal fitted mu decreasing as L shrinks. The demonstration is new and mostly convincing; the generalization to real experiments is conditional, and the authors largely say so themselves.\n\nWhat is new: Eq. (4) is imported from one of the authors' earlier papers, but the size-dependent mu and the claim that finite-size effects can fake sub-aging are not in that earlier work. The S-based collapse analysis applied to aging data is a useful addition, and the 2D Ising results, both nearest-neighbor and long-range, point the same way as the spherical model. For the spherical model the argument is mathematically clean: Eq. (4) is exact in the stated double-scaling limit, and the fitted mu genuinely has no thermodynamic meaning, which is the point.\n\nSoft spots, in proportion: the exact result lives in a double-scaling limit that explicitly subtracts non-scaling finite-size and finite-time corrections. The Supplemental Material says this plainly—Eq. (S.16) describes an effective model, not the original lattice model. So the title claim, read as a statement about finite-size effects in general, is strictly demonstrated only for that scaling-limit effective model. The Ising simulations are genuine lattice data and reduce that worry, but they are noisy, and the collapse analysis truncates data at t_max to remove 'clearly finite-size effected' points, which is a little circular if the goal is to characterize finite-size effects. The S measure also assumes a 3% relative error for exact spherical data; that is arbitrary but unlikely to move the minima. No code or data is shipped, which makes the Ising analysis harder to audit.\n\nThese are addressable, not fatal. The central argument holds: in these models, fitting sub-aging to finite-size data gives mu values that depend strongly on system size, so mu cannot be a material-independent exponent in that regime. Whether real glasses' sub-aging is explained this way remains open; the paper does not claim to close that question, only to urge caution.\n\nWho is this for: anyone fitting aging data in spin glasses or domain-growth simulations. It deserves a serious referee. I would send it out, asking for data/code and for a clearer statement of where the scaling-limit result ends and the lattice result begins.","headline":"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.","tokens_in":22881,"tokens_out":2570,"would_cite":true,"duration_ms":26593,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"Finite-size effects can transform simple aging into the phenomenology of sub-aging, and the fitted sub-aging exponent then carries no thermodynamic meaning.","keywords":["aging","sub-aging","finite-size effects","dynamical scaling","spherical model","Ising model","phase-ordering kinetics","two-time autocorrelator"],"falsifier":"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.","tokens_in":21881,"feed_emoji":"⏳","tokens_out":9990,"duration_ms":87659,"temperature":0.7,"pith_summary":"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 $2<d<4$, whose closed-form finite-size autocorrelator shows that the apparent sub-aging is an artifact: the fitted $\\mu$ depends on system size and approaches 1 as $L$ grows. Monte Carlo simulations of the two-dimensional Ising model with short-range and long-range interactions show the same size-dependent phenomenology. The paper concludes that an observed sub-aging signature should not be accepted at face value unless a genuinely intrinsic mechanism, such as activated hopping in a rugged energy landscape, can be identified.","feed_headline":"Finite size can fake sub-aging in aging systems","feed_subtitle":"An exactly solved model shows fitted sub-aging exponents may be size effects, not real physics.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the exact finite-size two-time autocorrelator (Eq. (4)) in the double scaling limit, the central object of the argument.","marker":"[45]"},{"why":"Defines the spherical model of a ferromagnet whose exact dynamics are used as the primary example.","marker":"[22]"},{"why":"Co-defines the spherical model and its basic thermodynamics, needed for the quench to $T<T_c$.","marker":"[23]"},{"why":"Provides the finite-size analysis method for the spherical model that underlies the scaling limit (3).","marker":"[37]"},{"why":"Standard reference for finite-size scaling theory used to justify the double scaling limit and removal of non-scaling corrections.","marker":"[43]"},{"why":"Spin-glass experiments showing that cooling protocols can raise effective sub-aging exponents toward full aging, the empirical contrast the paper explains as a finite-size artifact.","marker":"[12]"},{"why":"Documents that similar protocols do not raise $\\mu$ in other glasses, motivating the search for an alternative, here finite-size, mechanism.","marker":"[15]"},{"why":"Rigorous example of true sub-aging in an activated hopping model, used as the contrast showing that not all sub-aging reports are finite-size artifacts.","marker":"[19]"},{"why":"Earlier simulation study of aging in the long-range Ising model whose $\\mu\\approx0.976$ collapse is reinterpreted here as a finite-size effect.","marker":"[49]"}],"fun_headline_variants":["Finite-size effects masquerade as sub-aging","Sub-aging may be a finite-size illusion","Finite-size effects explain apparent sub-aging","Apparent sub-aging traced to finite-size effects","When sub-aging is just a finite-size effect"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Finite-size effects masquerade as sub-aging","Sub-aging may be a finite-size illusion","Finite-size effects explain apparent sub-aging","Apparent sub-aging traced to finite-size effects","When sub-aging is just a finite-size effect"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000669,"raw_usage":{"total_tokens":3003,"prompt_tokens":848,"completion_tokens":2155,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":464,"completion_tokens_details":{"reasoning_tokens":2094}},"tokens_in":464,"tokens_out":2155,"duration_ms":16425,"temperature":1.0,"reasoning_tokens":2094,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-10T21:24:39.306566+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Allen and R","cited_arxiv_id":null,"evidence_quote":"Supplies the exact finite-size two-time autocorrelator (Eq. (4)) in the double scaling limit, the central object of the argument."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the spherical model of a ferromagnet whose exact dynamics are used as the primary example."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Co-defines the spherical model and its basic thermodynamics, needed for the quench to $T<T_c$."},{"cited_title":"Annibale and P","cited_arxiv_id":null,"evidence_quote":"Provides the finite-size analysis method for the spherical model that underlies the scaling limit (3)."},{"cited_title":"Singh and R","cited_arxiv_id":null,"evidence_quote":"Standard reference for finite-size scaling theory used to justify the double scaling limit and removal of non-scaling corrections."},{"cited_title":"Andreanov and A","cited_arxiv_id":null,"evidence_quote":"Spin-glass experiments showing that cooling protocols can raise effective sub-aging exponents toward full aging, the empirical contrast the paper explains as a finite-size artifact."},{"cited_title":"Rodriguez, G","cited_arxiv_id":null,"evidence_quote":"Documents that similar protocols do not raise $\\mu$ in other glasses, motivating the search for an alternative, here finite-size, mechanism."},{"cited_title":"Kawashima and N","cited_arxiv_id":null,"evidence_quote":"Earlier simulation study of aging in the long-range Ising model whose $\\mu\\approx0.976$ collapse is reinterpreted here as a finite-size effect."}],"review_version":1}