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REVIEW 4 major objections 5 minor 234 references

Physical ageing from generalised time-translation-invariance

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Pith's one-line read Classical ageing's standard scaling laws are shown to follow from a single logarithmic change of representation of the time-translation and dilatation generators.

desk verdict A plausible unifying framework for ageing phenomenology, but the equality λ_C=λ_R is not derived from the two stated symmetries alone—an extra assumption about the response operator is needed. read the letter →

arxiv 2504.16857 v3 pith:W7A2YDDA submitted 2025-04-23 cond-mat.stat-mech hep-thmath-phmath.MP

Malte Henkel This is my paper · ORCID
classification cond-mat.stat-mechhep-thmath-phmath.MP MSC 82C2782C31
keywords ageinggeneralisedtime-translationinvariancedynamicalscalingJanssen-Schaub-Schmittmannrelationtwo-timeauto-correlatorauto-responsefinite-sizecriticaldynamics
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

This paper tries to establish that the standard phenomenology of physical ageing in classical systems is not a set of independent empirical rules but the common consequence of two symmetries: generalised time-translation-invariance and dynamical scaling. The construction is an intertwining change of representation, $X=e^{\xi\ln t}\,X_{\rm equi}\,e^{-\xi\ln t}$, which turns the equilibrium time-translation generator $-\partial_t$ into $-\partial_t+\xi/t$, softly breaking time-translation-invariance while keeping the Lie algebra intact. From this, the paper derives algebraic two-time scaling functions, the exponent equality $\lambda_C=\lambda_R$, the Janssen-Schaub-Schmittmann relation, and new finite-size plateau laws. A sympathetic reader would care because it replaces a list of empirical scaling laws with a single predictive postulate and produces new testable predictions.

What carries the argument

The central object is the intertwining operator $W(t)=\xi\ln t$, used through $X=e^{W(t)}X_{\rm equi}e^{-W(t)}$ to transform equilibrium symmetry generators into out-of-equilibrium ones. It gives the generalised time-translation generator $-\partial_t+\xi/t$ and shifts scaling dimensions by $\xi$; solving the resulting pair of linear first-order covariance equations (2.5a)-(2.5b) yields the two-time scaling form that carries the argument. The same machinery supplies a criterion $2\xi>1$ for irrelevance of cubic non-linearities in the equation of motion, which is what justifies applying linear Schrödinger-invariant response forms to a wide class of non-conserved phase-ordering models.

What would settle it

In a fully finite system quenched to $T\le T_c$ with $s\ll N^z$, the height of the two-time auto-correlator plateau should scale as $N^{-\lambda}$ at fixed waiting time and as $s^{\lambda/z-b}$ at fixed size; measuring a different $N$-dependence or a different $s$-dependence would falsify the derived finite-size scaling, and a simultaneous measurement of the global correlator plateau $N^{d-\lambda}$ would test the extended Janssen-Schaub-Schmittmann relation.

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Extended reading notes

Core claim

On the paper's own terms, the central claim is that the whole generic ageing phenomenology of classical systems follows from the two covariance conditions on two-point functions: generalised time-translation invariance $X_{-1}C=(-\partial_t-\partial_s+\xi_1/t+\xi_2/s)C=0$ and dynamical scaling $X_0C=(-t\partial_t-s\partial_s-\frac{1}{z}r\partial_r-(\delta_1-\xi_1)-(\delta_2-\xi_2))C=0$. The unique covariant solution is $C(t,s;r)=s^{-\delta_1-\delta_2+\xi_1+\xi_2}(t/s)^{\xi_1}(t/s-1)^{-\delta_1-\delta_2}F(r/(t-s)^{1/z})$. Large-argument limits give $f_C(y)\sim y^{-\lambda_C/z}$ and $f_R(y)\sim y^{-\lambda_R/z}$ with $\lambda_C=\lambda_R$, and the same derivation reproduces the Janssen-Schaub-Schmittmann relation $\Theta=d-\lambda/z$ at criticality while extending it to all $T<T_c$ through global correlators and responses. It also produces new finite-size plateau scalings for fully finite systems, which the paper verifies in the exactly solvable spherical model.

Load-bearing premise

The load-bearing premise is the intertwining prescription $X_{\rm equi}\mapsto e^{\xi\ln t}X_{\rm equi}e^{-\xi\ln t}$ with a constant dimensionless rapidity $\xi$; the paper states in Section 5 that this form is not derived and its physical origin remains open, so if actual symmetries require a time-dependent $\xi$ or additional corrections the scaling laws and exponent equalities would not follow.

Editorial extensions

If this is right

  • The equality $\lambda_C=\lambda_R$ follows from the covariance conditions rather than being assumed, so it is predicted to hold in every ageing system with short-ranged initial correlations.
  • The Janssen-Schaub-Schmittmann relation $\Theta=d-\lambda/z$ applies after quenches to all $T\le T_c$, making $Q(t,0)\sim t^{\Theta}$ a practical way to measure $\lambda$ below criticality.
  • In fully finite systems with $s\ll N^z$, plateau heights scale as $N^{-\lambda}$ and $s^{\lambda/z-b}$ for correlators, with analogous response laws, giving independent estimates of $\lambda$, $a$, and $\lambda/z$ from finite samples.
  • Whenever the criterion $2\xi>1$ holds, cubic non-linearities are irrelevant at late times, so the linear Schrödinger-invariant form of the two-time auto-response describes a wide class of phase-ordering and critical models.
  • At criticality the limit fluctuation-dissipation ratio $X_\infty$ is finite and expressed through exponents and the amplitude ratio $f_{\infty,R}/f_{\infty,C}$, consistent with measured finite values in several experimental systems.

Reading between the lines

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

  • If the intertwining postulate is the true symmetry mechanism, allowing $W(t)$ to be a more general function than $\xi\ln t$ should generate logarithmic sub-ageing or multi-scaling regimes; the paper explicitly leaves this generalisation open.
  • The derivation excludes long-ranged initial correlations, and the paper notes $\lambda_C=\lambda_R$ may fail there; a controlled numerical test would quench with power-law correlated initial states and measure $\lambda_C-\lambda_R$ as a function of the initial correlation exponent.
  • In finite samples, the predicted plateau onset offers an alternative explanation for apparent curvature in $f_C(y)$ at large $y$; fits that ignore the plateau could systematically overestimate $\lambda/z$, so comparing $N$-based and $s$-based plateau scalings provides a consistency check.
  • The paper restricts to classical dynamics; if the same logarithmic representation describes quantum quenches, the equalities $\lambda_C=\lambda_R$ and $\Theta=d-\lambda/z$ should appear in quantum ageing as well, which would be a sharp test of the symmetry's physical origin.
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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 proposes that the generic two-time phenomenology of physical ageing in classical systems can be derived from two dynamical symmetries: generalized time-translation invariance and dynamical scaling. The central postulate, Eq. (2.1), replaces the equilibrium generator of time translations by an intertwined form X = e^{ξ ln t} X_equi e^{-ξ ln t}, producing the modified generators (2.2). Solving the resulting covariance equations for two-point functions yields the exact form (2.6), from which the paper derives algebraic autocorrelation and autoresponse scaling, the equality λ_C = λ_R, the Janssen-Schaub-Schmittmann relation, extensions of that relation to T < T_c, finite-size plateau scalings in fully finite systems, and global two-time observables. Section 4 translates the same representation into a criterion for the irrelevance of nonlinear terms in the equation of motion, and compares the resulting response forms with a large body of exact and numerical results. The paper is explicit in Section 5 that the intertwining form W(t)=ξ ln t is a postulate whose physical origin is not derived.

Significance. If the central hypothesis is accepted, the paper offers a valuable unifying perspective: a single representation choice (2.1) with W(t)=ξ ln t reproduces a wide set of known ageing results and produces new, testable predictions, notably the finite-size plateau scalings (3.15), (3.18), (3.21), (3.43), (3.46) and the extension of the JSS relation to all T ≤ T_c. The covariance lemma (2.6) is a simple but correct exact statement, and the paper backs its claims with exact checks in the spherical model, the 1D Glauber-Ising model, and the fully connected spherical spin glass, together with an extensive compilation of numerical results. The significance is moderated by the facts that the central postulate is not derived, that the equality λ_C = λ_R requires an extra assumption about the response operator that is not among the two stated symmetries, and that the response-plateau finite-size scalings are conditional on a convergence assumption that fails in at least one known model. These issues do not invalidate the framework but they do mean the paper's stated claims exceed what is logically established.

major comments (4)
  1. [§3.2, Eq. (3.5), Prop. 2] The equality λ_C = λ_R is not a consequence of the two stated symmetries (2.5a) and (2.5b) alone. Solving (2.6) for the response R = ⟨φ(t)φ̃(s)⟩ with independent parameter pairs (δ,ξ) and (δ̃,ξ̃) gives λ_R/z = δ + δ̃ − ξ, whereas the autocorrelator gives λ_C/z = 2δ − ξ. Equality holds only if δ̃ = δ. Eq. (3.5) imposes δ̃ = δ by an appeal to local scale-invariance ('If that is admissible'), which is an additional dynamical input not present in (2.5a,b). The abstract and Proposition 2 present λ_C = λ_R as a consequence of generalized time-translation-invariance combined with dynamical scaling; this overstates the logical content of the derivation. The authors should either state the extra assumption explicitly whenever the equality is claimed, or prove δ̃ = δ from the stated symmetries.
  2. [§2, Eq. (2.1); §5] The central postulate (2.1) with W(t) = ξ ln t is not derived, and Section 5 explicitly states that its physical origin is open. All propositions that follow are therefore conditional on this specific logarithmic intertwining form; a different W(t), or a time-dependent ξ, would change the scaling forms and exponent relations. The paper should state this conditionality prominently in the abstract and in Propositions 1–7, and should soften the claim that the whole generic phenomenology of ageing is derived from the two dynamical symmetries, because one of those symmetries is itself an unproved representation choice rather than a symmetry that has been shown to hold for the systems under study.
  3. [§3.4, Prop. 4; §3.7, Cor. 6] The finite-size plateau scaling for the auto-response is an assumption, not a consequence of (2.5a,b). Proposition 4 is phrased conditionally ('if the auto-response function converges to a plateau'), and Appendix D shows that in the fully connected p = 2 spherical spin glass the response does not plateau (D.11). The same conditional structure propagates to Corollaries 3 and 6. The paper should clearly separate the derived scaling (3.17) from the additional hypothesis of plateau convergence whenever these finite-size plateau scalings are advertised as consequences of the generalised time-translation-invariance programme.
  4. [Appendix B; §5] Appendix B shows that generalized time-translation covariance cannot accommodate single-time correlators: it forces FC(u) ∼ u^{−2δz}, in contradiction to the known scaling behaviour. The paper excludes single-time correlators (Section 5, assumption 4), but the abstract and Section 1 claim that the whole generic phenomenology of ageing is derived. Since single-time correlators are part of the standard phenomenology of ageing, the claim of scope is overstated. The authors should qualify the claim to the two-time sector, or justify why the failure of the covariance requirement for single-time correlators does not cast doubt on the use of the same requirement for two-time correlators.
minor comments (5)
  1. [Abstract] The abstract contains the typo 'celebrate' for 'celebrated'; this should be corrected.
  2. [Throughout] There are several non-native spellings, e.g., 'littérature' in §3.2 and 'Forth' in Appendix C; a careful language edit is needed.
  3. [Eq. (2.6)] The uniqueness claim in the proof of the lemma relies on the external reference [144]; stating the method-of-characteristics argument would make the derivation more self-contained.
  4. [Tables 2 and 3] The tables mix literature values with values inferred from the present framework; a sentence clarifying which entries are new determinations and which are taken from the cited references would improve transparency.
  5. [§3.3, Fig. 3] The axis label 'rL FC' in panel (b) appears garbled; the figure would be clearer if the abscissa were labelled y = t/s and the ordinate simply C(ys,s).

Circularity Check

1 steps flagged · score 6.0 of 10

The equality λ_C=λ_R is imposed by the auxiliary identification δ̃=δ in eq. (3.5), so the abstract's claim that it follows from generalized time-translation-invariance and dynamical scaling is partly circular; the remaining scaling relations are conditional consequences of the unproved intertwining postulate.

  1. self definitional [Section 3.2, eq. (3.5) and Proposition 2; also abstract]
    "Furthermore, we should eventually require responses to be co-variant under larger algebras of local scale-transformations, notably conformal transformations which make up local scale-invariance [106, 108, 118]. If that is admissible, we have δ =δ1 =δ2 =~δ , ξ =ξ1 , ~ξ =ξ2 ... Comparison of (3.3) and (3.7) implies the exponent equality λ =λC =λR."

    Without the identification δ̃=δ, the general covariance solution (2.6) gives for the auto-response R(t,s) = s^{-δ-δ̃+ξ+ξ̃} (t/s)^{ξ-δ-δ̃} F_R(0), so λ_R/z = δ+δ̃-ξ, while the auto-correlator gives λ_C/z = 2δ-ξ. These two exponents are equal exactly when δ̃=δ. That equality is inserted by hand in eq. (3.5) through an appeal to local scale-invariance, and Proposition 2 then presents λ_C=λ_R as a derived consequence. Thus the celebrated exponent equality is not a consequence of the two stated symmetries (2.5a,b) alone; it is an input encoded in the chosen scaling dimensions of the response operator. The abstract's statement that λ_C=λ_R is an observable consequence of the generalised time-translation-invariance and dynamical scaling therefore overstates what the derivation actually shows.

full rationale

The paper's central derivation is conditional on the intertwining postulate (2.1) with W(t)=ξ ln t, and Section 5 explicitly admits that the form of W(t) is not derived and its physical origin is open. Conditional derivations from a postulate are not themselves circular. The main circular element is the exponent equality λ_C=λ_R: the response operator's scaling dimension is set equal to the order-parameter dimension in eq. (3.5) ("If that is admissible"), and this equality of dimensions is exactly what makes the response exponent equal to the correlator exponent. Proposition 2 drops the conditional caveat and the abstract lists λ_C=λ_R as an outcome of the two dynamical symmetries. That is a specific reduction of a headline prediction to an auxiliary definitional assumption, so a partial circularity score is appropriate. The other results — the algebraic scaling forms, the finite-size plateau laws, and the extension of the Janssen-Schaub-Schmittmann relation below criticality — are genuine consequences of the covariance equations once the representation parameters are admitted, and are not themselves fits renamed as predictions. The paper also repeatedly checks the resulting forms against known exact and numerical results, which supports the empirical content of the framework. The score of 6 reflects that one central equality is imposed rather than derived, while substantial independent content remains in the rest of the derivation chain.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The central claim rests on one new postulate (2.1), one domain restriction (simple ageing with short-ranged initial correlations), and known exponents imported from earlier work. The derived results do not compute exponents from first principles; several scaling functions and amplitudes are left undetermined. The new predictions, namely the extended JSS relation and the finite-size plateau laws, are not folded into these inputs and are therefore testable.

free parameters (4)
  • ξ (rapidity of order-parameter φ) = Not fitted in the paper; in examples read off from known λ, e.g., ξ = d/4 in the spherical model and ξ ≈ 7/8 in the 2D…
    Introduced in the postulate (2.1). Determines the broken time-translation term ξ/t and sets λ/z = 2δ - ξ; its value is neither derived nor predicted by the framework.
  • ξ̃ (rapidity of response operator φ̃) = Not fitted in the paper.
    Introduced in eq. (3.5). Shifts the response exponent a and the difference a' - a through ξ + ξ̃, but cancels in the autocorrelation exponent λ_R; it remains a free parameter.
  • Exponents a, a', λ in response forms (4.7) and (4.26) = Example values: a = 0, a' = -1/2, λ/z = 1/2 in the 1D Glauber-Ising chain; many fitted entries in table 3.
    Proposition 9 explicitly allows a, a' and λ to be free and independent parameters, fitted to data in each system. The framework does not compute their values.
  • Undetermined scaling functions F_C, F_R, F_m, FC(u), FR(u) = Left arbitrary.
    The covariance equations fix exponents but leave the scaling functions unspecified. Asymptotic behaviors such as F_C(u) ~ u^{-λ} for u << 1 are assumed to obtain the plateau scalings.
assumptions (6)
  • domain assumption Simple ageing: two-time functions obey scaling with an algebraically growing length scale ℓ(t) ~ t^{1/z}.
    Assumed throughout, eq. (1.2a) and Section 5 assumption 1. Excludes logarithmic sub-ageing and multi-scaling.
  • domain assumption All initial correlations are short-ranged.
    Section 5 assumption 2. The paper notes that otherwise λ_C = λ_R may fail, citing refs. [34,35,192,79].
  • ad hoc to paper The intertwining postulate (2.1) with W(t) = ξ ln t.
    Central postulate with no derivation; the paper explicitly states in Section 5 that the physical origin of this prescription is open.
  • domain assumption For response functions, φ and φ̃ have equal scaling dimension δ, while their rapidities ξ and ξ̃ remain independent.
    Used in Section 3.2, eq. (3.5), to obtain λ_C = λ_R and the response exponent relations.
  • ad hoc to paper The auto-response function in fully finite systems converges to a plateau.
    Conditional assumption in Proposition 4 and Corollaries 3, 5, 6. Verified in the spherical model in Appendix D, but the paper notes it is unclear whether this is a peculiarity of that model.
  • domain assumption Known values of the exponents b, a, z, λ, η, β, ν are taken from prior literature.
    The derivation does not compute these exponents; they are inputs from known scaling arguments, exact solutions, field theory, or numerical studies.
invented entities (1)
  • Intertwining operator W(t) = ξ ln t (generalised time-translation representation) independent evidence
    purpose: To break time-translation invariance softly by a similarity transformation of the Lie algebra generators of time translations and dilations.
    This is the new formal object that carries the whole framework. It is not derived from microscopics, but it generates falsifiable predictions including the finite-size plateau scalings and the extension of the JSS relation to T < Tc, some of which are tested in the spherical model.

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Pith. "Pith review of Physical ageing from generalised time-translation-invariance." pith.science (2026). https://pith.science/paper/W7A2YDDA

@misc{pith2026250416857,
  author       = {Pith},
  title        = {Pith review of: Physical ageing from generalised time-translation-invariance},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/W7A2YDDA}},
  note         = {Machine review of arXiv:2504.16857}
}
abstract

A generalised form of time-translation-invariance permits to re-derive the known generic phenomenology of ageing, which arises in classical many-body systems after a quench from an initially disordered system to a temperature $T\leq T_c$, at or below the critical temperature $T_c$. Generalised time-translation-invariance is obtained, out of equilibrium, from a change of representation of the Lie algebra generators of the dynamical symmetries of scale-invariance and time-translation-invariance. Observable consequences include the algebraic form of the scaling functions for large arguments of the two-time auto-correlators and auto-responses, the equality of the auto-correlation and the auto-response exponents $\lambda_C=\lambda_R$, the cross-over scaling form for an initially magnetised critical system and the explanation of a novel finite-size scaling if the auto-correlator or auto-response converge for large arguments $y=t/s\gg 1$ to a plateau. For global two-time correlators, the time-dependence involving the initial critical slip exponent $\Theta$ is confirmed and is generalised to all temperatures below criticality and to the global two-time response function, and their finite-size scaling is derived as well. This also includes the time-dependence of the squared global order-parameter. The celebrate Janssen-Schaub-Schmittmann scaling relation with the auto-correlation exponent is thereby extended to all temperatures below the critical temperature. A simple criterion on the relevance of non-linear terms in the stochastic equation of motion is derived, taking the dimensionality of couplings into account. Its applicability in a wide class of models is confirmed, for temperatures $T\leq T_c$. Relevance to experiments is also discussed.

Figures

Figures reproduced from arXiv: 2504.16857 by the authors.

Figure 1
Figure 1. Schematic free energy of a simple ferromagnet (a) b [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Physical ageing illustrated through the characte [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. Finite-size scaling of the two-time auto-correla [PITH_FULL_IMAGE:figures/full_fig_p016_3.png] view at source ↗
Figures from the paper (1 more)
Figure 4
Figure 4. Figure 4: Time-dependence of the non-equilibrium magnetis [PITH_FULL_IMAGE:figures/full_fig_p020_4.png]

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