Logarithmic critical slowing down in complex systems: from statics to dynamics

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These findings are tied to a public preprint: arXiv:2403.07565 · cond-mat.dis-nn

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What a reviewer will push on first

The most severe issues a reviewer will raise, named. Each jumps to the full critique below.

The catch a generic AI would miss

Internal consistencyEqs. 57, 58

Eq. (57) and Eq. (58) disagree on the leading prefactor by a factor of four

Eq. (57) gives the Laplace-transform expansion as $z {LT [G^{2} (t)] + z G^^{2} (z)} = - ζ (2) (g^{'})^{2} + \dots$ with $ζ (2) = π^{2} /6$ . Combining this with Eq. (51) at $w_{1} = w_{2}$ yields $0 = - (π^{2} /6) (g^{'})^{2} + μ g^{3}$ , which produces the leading coefficient $g (y) = 2 π^{2} / (3 μ y^{2})$ shown in Eq. (16).

Editorial verdict

Substantial revisions required first

Likely outcome if submitted today: major revision if sent to referees; nontrivial desk-reject risk if the PRB editor reads the current presentation as too broad ("complex systems") rather than a sharply condensed-matter glass / spin-glass contribution. The theoretical spine is strong, and the static-to-dynamic mapping for an $A_{3}$ singularity is the kind of result PRB readers expect. But the present version contains concrete errors that sit in or near the main result and would damage reviewer confidence quickly.

Read the full editorial recommendation

The single most important fix this week is a coordinated main-result integrity pass from Eq. (16) through Eq. (18), Eq. (50)–(58), and Eq. (120)–(126). Do not treat this as copyediting: rebuild the sign convention for $μ$ , the Laplace-transform expansion, and the Legendre-transform inversion in one consistent notation, then make every appearance of $μ, y_{1} + y_{2} - y_{4}, υ_{1} + υ_{2} - υ_{4}$ agree. Add one short paragraph after Eq. (18) saying explicitly: "with our sign convention, physical logarithmic decay requires $μ > 0$ , corresponding to …" — or explain why no such positivity statement is possible.

Recommended target: Physical Review B Regular Article, reframed around glassy condensed matter, spin glasses, structural glasses, and MCT $A_{3}$ singularities rather than generic "complex systems." Cascade fallback: Physical Review E or Journal of Statistical Mechanics if PRB fit becomes the sticking point. Time to submission readiness: 2–4 weeks of focused revision — mostly equation audit, derivation clarification, and PRB positioning.

Quality scorecard

Overall · 3.6/5

Originality

4/5

The extension of the static-to-dynamic replica mapping to the λ=1 (A3) case and the derivation of a quantitative, measurable formula for μ in terms of overlap cumulants (Eq. 18) is genuinely new, though the broader program of connecting replicated free energies to MCT dynamics was established by Parisi and Rizzo (2013).

Importance of research question

4/5

Logarithmic relaxation at A3 singularities is experimentally observed in colloids, porous-media liquids, and pinned-particle systems, and a static route to the prefactor μ has practical value for numerical simulations that can equilibrate systems faster than real dynamics.

Claims are well-supported

4/5

The central analytical claims (Eqs. 16, 17, 18, 25) are derived in full and the algebra is internally consistent, though no numerical verification against a solvable model is provided and the sign convention for μ is not explicitly checked against known MCT results.

Experimental soundness

3/5

The work is purely analytical; the absence of any benchmark against an exactly solvable model (e.g., the 4-state Potts spin glass, where λ=1 is explicitly stated) leaves the quantitative formula for μ unvalidated.

Clarity of writing

3/5

The structure is logical and the main results are clearly stated in Section II, but the paper contains several typographical issues (inconsistent spelling of "Götze" vs. "Göetze", a missing term in Eq. 114 compared to Eq. 70, and reference [60] is listed as "to be published" with no arXiv identifier), and Figures 1 and 2 lack self-contained legends explaining the diagrammatic conventions.

Value to community

4/5

The formula μ = -r(υ1+υ2-υ4)/(3ω) is directly actionable in equilibrium Monte Carlo or swap-MC simulations of glass-formers and spin glasses, and the universality argument covers a wide range of systems.

Prior work contextualization

3/5

The paper is well-grounded in the MCT and replica literature it directly builds on, but it does not engage with the body of work on higher-order MCT singularities in colloidal systems (Dawson et al., 2000) or with multipoint susceptibility measurements in glass formers (Berthier et al., 2007), both of which are directly relevant to the observability claims in Section VI.

Detailed findings

High severity.
#1MethodologySec. III.A; Eqs. 39–58
The static-to-dynamic mapping needs a justification of why Fast Motion is the right dynamical completion
obtained in the so-called Fast Motion (FM) limit that corresponds to an infinitely fast microscopic dynamics
The static replicated free energy admits many possible dynamical completions. The Fast Motion (FM) limit, in which off-diagonal correlations relax instantaneously and only the order-parameter sector survives at long times, is one specific choice — but the manuscript treats it as the natural one without saying why. A reviewer thinking along the Cugliandolo-Kurchan / Crisanti-Sommers axis will ask: what is being adiabatically eliminated, on which timescale separation, and why is the result of that elimination the schematic MCT equation rather than (say) a memory-kernel form that retains finite-frequency information about the eliminated modes?
This matters for the claim that "static replicated theory implies logarithmic dynamics." The static theory by itself does not imply any particular dynamics; it constrains the dynamics through whatever consistency condition the reader brings with them. The paper currently leans on Ref. [24] for that consistency condition and on the FM limit for the asymptotic reduction. The conceptual bridge — why these two together yield exactly the schematic $A_{3}$ equation rather than something less universal — is the part a careful reader will want spelled out.
Suggested fix
Add a short subsection between Eq. (38) and Eq. (39) titled "Why the Fast Motion limit." Explicitly state: which modes are being adiabatically eliminated, the timescale-separation assumption that makes that elimination valid, and why the residual equation reduces to the single-mode schematic-MCT form rather than retaining a finite-frequency kernel. This is 200-400 words and reframes the central claim from "we derive logarithmic decay" to "we derive logarithmic decay under explicitly stated dynamical assumptions" — a much stronger position to defend in review.
High severity.
#2MethodologyAbstract; Sec. III.A; Sec. V
Connection to schematic MCT is weaker than to full MCT — the asymptotic exponent matches but the equation structure does not
The abstract says the dynamical equations "turn out to be those predicted by schematic Mode Coupling Theory for supercooled viscous liquids at an $A_{3}$ singularity, where the parameter exponent is $λ = 1$ ." This is a strong claim: schematic MCT is a closed equation (Götze's $F_{12}$ model and its variants), and matching it requires both the exponent $λ$ and the kernel structure that produces it. What the present derivation actually delivers is the leading asymptotic $1/ ln^{2} (t / t_{1})$ decay and an explicit formula for the prefactor. That is consistent with schematic MCT at an $A_{3}$ point, but it is not the same as deriving the schematic-MCT equation from first principles.
Phrased as "the asymptotic dynamics matches that of schematic MCT at an $A_{3}$ fixed point" — which is what the math actually supports — the result remains genuinely interesting and the cumulant formula for $μ$ is its own contribution. Phrased as "we derive the schematic MCT equation," the result is overstated and a Mode-Coupling-aware referee will push back.
Suggested fix
Tighten the abstract and Sec. V wording from "the dynamical equations turn out to be those predicted by schematic MCT" to "the leading asymptotic dynamics matches the schematic-MCT prediction at an A₃ singularity (parameter exponent λ=1), with the prefactor µ given in cumulant language by Eq. (18)." This is closer to what the derivation supports and removes the principal claim a Mode-Coupling reviewer would attack first.
High severity.
#3MethodologySec. I; Sec. V
Universality of the µ formula across the cited tricritical settings is asserted but not tested
The Introduction lists the $A_{3}$ tricritical point as appearing in attractive colloids, hypernetted-chain approximations, liquid models with pinned particles, and a "variety of mean-field models." The derivation, however, specializes to a particular cubic+quartic structure with vertices ${w_{1}, w_{2}, y_{1}, y_{2}, y_{3}, y_{4}}$ . Two questions follow: first, are the same cumulant identities $υ_{1} + υ_{2} - υ_{4}$ the natural combination across all of those settings, or is that an artifact of the truncation chosen here? Second, in any one of the cited systems — repulsive colloids, attractive colloids, pinned-particle systems — does Eq. (18) yield a numerically testable prediction for $μ$ ?
Either is fine to address briefly. A single paragraph that picks one of the cited systems (the attractive-colloid HNC of Refs. [13, 14] is the obvious one) and says "in that setting, the cumulant combination evaluates to … and corresponds to the regime …" gives the reader a concrete handle on whether the formula is universal or scheme-dependent. As written, the claim of generality is asserted but not exercised on a single example.
Suggested fix
Add one paragraph in Sec. V evaluating Eq. (18) for one of the cited tricritical realizations (the attractive-colloid HNC system in Refs. [13,14] is a natural choice). Even if the result is qualitative — "the formula predicts µ in the range … for this system; explicit numerical estimates are deferred to future work" — the reader leaves with a concrete connection between the cumulant identity and the listed physical examples.
Medium severity.
#4MethodologyEq. 18; Sec. IV
Physical interpretation of µ in cumulant language is the paper's claim to relevance — it deserves more than half a sentence
Eq. (18) — $μ = - r (υ_{1} + υ_{2} - υ_{4}) / (3 ω_{1})$ — is, on the face of it, the most useful equation in the paper. It connects an asymptotic decay exponent to overlap cumulants that are in principle measurable. But the manuscript treats Eq. (18) as a quick rewrite of Eq. (17), without saying what the cumulant combination $υ_{1} + υ_{2} - υ_{4}$ represents physically.
Several questions a reader will ask, none answered in the current text: (i) Why does $υ_{4}$ enter with a minus sign — what is it competing against in $υ_{1} + υ_{2}$ ? (ii) In experiments, are these three cumulants independently measurable, or only ever measurable in this combination? (iii) Does the sign of $μ$ have a physical interpretation — does $μ < 0$ correspond to a phase where the asymptotic logarithmic decay is preempted by a sharper transition?
A short subsection — even half a page — answering these would significantly strengthen the abstract's claim that $μ$ is "physically observable." Right now that claim is true at the level of the formula; it is not yet operational at the level of an experimentalist or simulator who would actually try to measure the combination.
Suggested fix
Add a short subsection after Eq. (18) — "Physical content of the cumulant formula" — answering: what each of υ₁, υ₂, υ₄ measures about the overlap distribution; whether they are independently measurable or always entangled; what sign(µ) physically distinguishes; and what experimental or simulation observable would deliver each cumulant. 300-500 words. This is the bridge between the formal result and the manuscript's claim to relevance.
Medium severity.
#5Novelty assessmentSec. I; Sec. V
The novel contribution relative to Götze MCT and to Ref. [24] should be stated explicitly in the Introduction
Götze's schematic MCT predicts the $1/ ln^{2} (t / t_{1})$ decay at an $A_{3}$ singularity (Refs. [1-3] in the manuscript). Ref. [24] establishes the static-replica framework the present derivation builds on. What this paper adds — and a careful reader of either Götze MCT or the static-replica literature will want to know — is the closed-form expression for $μ$ as a function of the underlying quartic vertices (Eq. 17) and equivalently as a cumulant identity (Eq. 18), plus the demonstration that the static-derivation route reproduces the MCT exponent.
The manuscript currently states that "we extend the standard analysis" but doesn't separate out the three contributions: (a) reproducing the MCT exponent from a static-replica route, which is mostly a confirmation of consistency between two existing frameworks; (b) the closed-form $μ$ formula in terms of vertices; and (c) the cumulant rewrite that gives $μ$ a measurable interpretation. Each of (b) and (c) is a real contribution; reviewers will want them flagged distinctly so the paper's claim to novelty does not lean on (a) alone.
Suggested fix
Add a short paragraph at the end of Sec. I explicitly listing three contributions: (1) cross-confirmation between static-replica and MCT routes for the A₃ exponent; (2) closed-form Eq. (17) for the prefactor µ in terms of quartic vertices; (3) cumulant identity Eq. (18) connecting µ to physically measurable observables. Frame (2) and (3) as the paper's primary contributions, and (1) as the consistency check that motivates them.
High severity.
#6Internal consistencyEqs. 17, 50, 125
Sign convention for μ changes between Eq. (17), Eq. (50) and Eq. (125)
The Outline gives the central definition as $μ = - (y_{1} + y_{2} - y_{4}) / (3 w)$ (Eq. 17). After Eq. (50), Sec. III A re-introduces the parameter as $μ \equiv (y_{1} + y_{2} - y_{4}) / (3 w_{1})$ — the opposite sign. Eq. (125) of the appendix returns to the negative-sign form. $μ$ is the parameter advertised in the abstract and in Eq. (18); a referee will not regard the disagreement across these three equations as a typo.
A second issue compounds this. Eq. (17) as set in the manuscript reads $μ = - y_{1} + y_{2} - y_{4} /3 w$ . Even if the intended meaning is $- (y_{1} + y_{2} - y_{4}) / (3 w)$ , the parenthesization must be unambiguous. As typeset, the expression is read by a careful referee as a different function of the couplings.
Suggested fix
Choose one sign convention and propagate it through Eq. (17), Eq. (18), Eq. (50), Eq. (51), Eq. (58), and Eq. (125). Add explicit parentheses to Eq. (17). If Eq. (50) is retained as written, then the definition of μ after it requires the negative sign — make that adjustment and verify Eqs. (51), (58), (125) are consistent under the chosen sign.
High severity.
#7Internal consistencyEqs. 57, 58
Eq. (57) and Eq. (58) disagree on the leading prefactor by a factor of four
Eq. (57) gives the Laplace-transform expansion as $z {LT [G^{2} (t)] + z G^^{2} (z)} = - ζ (2) (g^{'})^{2} + \dots$ with $ζ (2) = π^{2} /6$ . Combining this with Eq. (51) at $w_{1} = w_{2}$ yields $0 = - (π^{2} /6) (g^{'})^{2} + μ g^{3}$ , which produces the leading coefficient $g (y) = 2 π^{2} / (3 μ y^{2})$ shown in Eq. (16).
But Eq. (58) is set as $- (2 π^{2} /3) (g^{'})^{2} + μ g^{3} = 0$ . That prefactor is larger by a factor of four relative to Eq. (57). Used literally, Eq. (58) does not reproduce the leading coefficient in Eq. (16). The derivation between Eq. (57) and Eq. (58) is therefore visibly algebraically inconsistent, and the chain Eq. (51) → Eq. (57) → Eq. (58) → Eq. (16) is the central derivation of the paper.
Suggested fix
Replace Eq. (58) with the equation actually implied by Eq. (57) — namely 0 = -(π²/6)(g')² + μg³ — or, if the discrepancy reflects a normalization of g, G, or the Laplace transform between Eq. (57) and Eq. (58), state that normalization explicitly in a one-line remark before Eq. (58). As written, the algebra cannot be followed.
High severity.
#8Internal consistencySec. IV; Eqs. 114, 119; Appendix B
υ₄ summed over (i,j,k) but contains index ℓ; Eq. (114) duplicates ω₁; Eq. (119) uses δQ_ab³ where δQ_ab² is required
Three equation-level errors in the appendix are catchable on a careful referee read. First, $υ_{4}$ is summed over $(i, j, k)$ but the summand contains the index $ℓ$ ; the sum and the integrand do not share their dummy indices, which makes the expression undefined.
Second, Eq. (114) carries a duplicated $ω_{1}$ and a $λ_{e f}$ index structure that contradicts the contraction pattern set up earlier in the section. Third, Eq. (119) appears to use $δ Q_{ab}^{3}$ where $δ Q_{ab}^{2}$ is required by the cubic-in-fluctuations expansion the section is doing. None of these three is fatal individually, but together they signal that the appendix did not have a fresh-eyes pass before submission.
Suggested fix
Re-derive Eq. (114), Eq. (119), and the υ₄ sum on a clean sheet, then verify each against the contraction pattern established in Sec. IV.A. A single sweep through the appendix with a referee's eye for index discipline should resolve all three.
Medium severity.
#10Internal consistencySec. IV; cross-references to Eq. (5)
Eq. (5) referenced as "Gibbs free energy" — but Eq. (5) is the spin autocorrelation C(t)
Sec. IV opens with: "we retained only two cubic diagrams and four quartic diagrams in the Gibbs free energy (5)." Eq. (5) in this manuscript is the spin autocorrelation $C (t)$ , not the Gibbs free energy. The intended reference is Eq. (4). The same swap recurs a few lines later: "we could have also ignored the term proportional to $y_{3}$ in expression (5)…" — again, the term lives in Eq. (4).
A reviewer following the derivation will jump back to Eq. (5), see $C (t)$ , and lose confidence in the equation accounting for the rest of the section. This is the kind of small error that makes the rest of a strong paper read as careless.
Suggested fix
Replace both occurrences of "Gibbs free energy (5)" with "Gibbs free energy (4)". Run a global cross-reference pass — papers with this kind of slip usually have one or two more.
Medium severity.
#11Internal consistencySec. V (Conclusions)
Compilation-level invalid citation: "[58?]" appears in the Conclusions
can be connected to off-equilibrium dynamics as well [58? ]
In the Conclusions: "static replicated theories can be connected to off-equilibrium dynamics as well [58?]." The literal "?" inside the citation marker means the BibTeX key did not resolve at compile time and LaTeX inserted a placeholder. This is not subtle — it is the kind of thing a desk editor sees in the first 30 seconds and uses as a signal that the manuscript was submitted without a final compile.
Suggested fix
Replace "[58?]" with the intended citation. If no citation was intended, remove the brackets and rephrase. Re-run pdflatex twice with bibtex in between to confirm no other unresolved keys remain.
Medium severity.
#12Editorial framingAbstract; Introduction; Eq. 16
Title-claim "logarithmic decay" is used for two different asymptotic regimes
The abstract says "the static replicated theory implies slowing down with a logarithmic decay in time." The Introduction reinforces this by noting that $- ln t$ "appears to be the correct fitting law for about a decade or two." But Eq. (16) — the central result — gives an inverse-log-squared decay,
$C (t) - C (\infty) = \frac{2 π ^{2}}{3 μ ln ^{2} ( t / t _{1} )} + \frac{24 ζ ( 3 )}{μ ln ^{3} ( t / t _{1} )} ln ln (t / t_{1}) + \dots$
These are not the same scaling. Near an $A_{3}$ singularity, preasymptotic $- ln t$ fits and the exact asymptotic $1/ ln^{2} t$ law are both legitimate, but the manuscript currently blurs them. A careful reader will lock onto Eq. (16), match it to the abstract, and conclude that one of the two is wrong.
Suggested fix
Distinguish the two regimes explicitly. Use "logarithmic" for the asymptotic 1/ln²(t) decay (which is the Eq. 16 result), and "preasymptotic −ln t" for the experimental fitting law. One paragraph after Eq. (16) framing the relationship — and a corresponding sentence in the abstract — resolves the apparent conflict.
Medium severity.
#14Reporting guidelineSec. III.A; Sec. V
No statement of which limits and assumptions make the static-to-dynamic mapping rigorous
The static-to-dynamic mapping (Sec. III.A) leans on Ref. [24] but does not make explicit which assumptions select the Fast Motion regime, where the result is asymptotic, mean-field, equilibrium, and replica-symmetric. PRB referees will ask:
— Does the result require the thermodynamic limit $N \to \infty$ before the long-time limit $t \to \infty$ , or the reverse?
— Is replica symmetry assumed throughout, or broken in some regime?
— Does the FM limit assume timescale separation between fast modes and the order parameter? If so, where does that assumption first enter?
Without one tight paragraph stating these assumptions explicitly, a reviewer reading Sec. III.A backward from Eq. (16) cannot tell whether the result is exact, leading-order, or schematic.
Suggested fix
Add a one-paragraph "Assumptions and regime of validity" remark immediately after Eq. (50), explicitly stating: (1) the order of limits, (2) replica symmetry assumption, (3) timescale separation, and (4) which terms are kept and which are dropped at each step of the static-to-dynamic mapping.
Medium severity.
#15Submission readinessSubmission package
Four critical structural elements missing from the submission package
Format-compliance audit flagged four critical blockers: Conflict of Interest statement, Funding declaration / sponsor statement, Author contributions statement (per ICMJE), and Data availability statement. PRB will return the package without review until these are present. Three are one-line fixes; the data-availability statement requires a sentence about whether the supporting symbolic-computation outputs underlying Eqs. (50)-(126) are available on request or in a public repository.
Suggested fix
Add COI ("The authors declare no competing interests"), funding (CNR / Sapienza affiliations imply institutional support — name the grant numbers), author contributions (e.g. "L.L. and T.R. contributed equally to the derivation; T.R. drafted the manuscript"), and data availability ("Symbolic-computation notebooks underlying Eqs. 50-126 are available from the corresponding author on reasonable request") as a single end-matter block.

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