Self-Organized Criticality Below The Glass Transition

Elizabeth A. Baker; Katharina Vollmayr-Lee

REVIEW

Self-Organized Criticality Below The Glass Transition

Katharina Vollmayr-Lee, Elizabeth A. Baker

Disordered Systems and Neural Networks · Submitted 28 Feb 2006 · 19 references

Pith's verdict: A plausibly real power-law cluster-size distribution across temperatures below Tg, but the evidence stops short of proving that the binning and jump-threshold choices didn't create the apparent scale invariance.

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CitationElizabeth A. Baker, Katharina Vollmayr-Lee. "Self-Organized Criticality Below The Glass Transition." arXiv:cond-mat/0602669 (2006). https://pith.science/paper/E2HDXNSP

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Editor's note

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Colleague,

The new thing here is simple and worth knowing: Vollmayr-Lee and Baker take the Donati et al. string-cluster analysis of the Kob–Andersen binary LJ mixture and push it below Tg, down to T=0.15. They report that the cluster size distribution P(s) of simultaneously jumping particles is a power law with tau ~2.2–2.7 for every temperature they tried, for two extended cluster definitions, and across time windows. If that is true, it is a real observation: criticality frozen in below the glass transition, consistent with an extension of mode-coupling theory. The geometry check showing string-like clusters via coordination number is a nice complement, not just another plot of the same claim.

What earns credit: the quantity is measured directly, not fit from a model; the jump threshold, bin length, and neighbor cutoff were fixed in earlier work; the paper varies the cluster connectivity rule and checks waiting-time dependence; the uncertainty on the average tau is reported. The authors also cite the relevant prior work (Donati, Kob–Andersen, MCT freeze-in papers). This is honest, reproducible work in the sense that the method is transparent.

The soft spot is the one the stress-test note hits. The power-law claim rests almost entirely on the straightness of log-log plots with no error bars and no comparison to an alternative like an exponential cutoff or a null model in which the same jump events are shuffled in time. The 800-unit bin is not just a coarse-graining convenience; it sets the time scale over which simultaneous jumps are defined, and the sqrt(20) sigma_n threshold sets the amplitude scale. The extended cluster definitions change the spatial/temporal connectivity rules but not those two scales. So the central question — is the apparent scale invariance a property of the dynamics or of the cluster-construction parameters? — is not fully answered. This is not a fatal flaw; it is the difference between a strong suggestive observation and a demonstrated one. Also, system size is fixed at 1000 particles, so the range of power-law scaling is modest (roughly two decades at best).

My own take: the observation is probably real. Reversible jumps are included and the tau values are stable across definitions, which argues against pure artifact. But the paper overclaims slightly by calling it self-organized criticality without a null control. The mode-coupling freeze-in interpretation is plausible but not uniquely forced by the data. I would still send this to a serious referee; the result is important enough that the community should see it with the analysis cleaned up. For citation in my own work, I'd cite it cautiously as an empirically observed power-law cluster distribution below Tg, noting the missing null-model test.

Recommendation: engage with it; ask for error bars, fit ranges, and at least one threshold/bin variation or a shuffled-event control. If those come back as expected, this becomes a solid paper.

Referee report

Significance

If the power law in P(s) is genuine, the observation would be significant: it would indicate that cooperative dynamics in a structural glass below Tg display scale invariance without fine-tuning, connecting glassy relaxation to self-organized criticality. The paper's strengths are that the cluster size distribution is a direct, simulation-based observable (not a fitted model parameter), that the authors check robustness against time-window partitioning and against two modifications of the cluster connectivity definition, and that the coordination-number analysis provides a geometric characterization of the clusters. However, the key evidence is currently incomplete: there is no statistical validation of the power-law fits, no null-model comparison to exclude artifacts from the binning/thresholding procedure, and no system-size dependence test. These gaps are load-bearing for the central claim.

Major objections (3)

[Jump detection and binning (paragraph beginning 'For the definition of jump events...')] The power-law result depends on three parameters that are not varied: the 800-time-unit bin width, the four-bin displacement (equivalent to 3200 time units), and the jump criterion Delta r_n > sqrt(20) sigma_n. The manuscript tests the robustness of P(s) to the cluster connectivity rule (Definitions I and II) and to a partition into five time windows, but it does not test whether the apparent scale invariance survives a change in the bin width or in the jump threshold. In particular, random coincidences of independent jump events in the same bin, combined with the spatial neighbor cutoff, could in principle produce a broad cluster size distribution whose shape is controlled by the bin size. A null model with reshuffled jump times (preserving the single-particle jump statistics) would quantify this, but no such control is provided. The inset of Fig. 1 is not a null control; it merely compares five windows of the same correlated data. Because the central claim is scale invariance, the absence of this control is a major gap.
[Fig. 1 and Eq. (1)] The plotted P(s) is shown without error bars, so it is not possible to judge whether the deviations from a power law at small and large s are significant. The fitting range, the fitting procedure (e.g., least squares in log-log space), and the per-temperature uncertainties are not given. The average exponent tau = 2.50 +/- 0.05 is stated, but how the average is taken over ten temperatures and how the uncertainty is computed is not described; the reported range tau = 2.2-2.7 suggests a spread larger than the quoted error. No goodness-of-fit test or comparison with alternative distributions (such as s^{-tau} e^{-s/s0} or a stretched exponential) is reported. Without such tests, a two-decade straight line in a log-log plot is not sufficient to establish scale invariance, especially for a finite-size system.
[System size (Fig. 1 and Discussion)] The simulation contains 1000 particles, and clusters with up to about 250 particles are reported in the inset of Fig. 1. The largest clusters therefore constitute a substantial fraction of the system, so the tail of P(s) is likely influenced by the finite system size. No test with a different system size is performed, and no argument is given for why finite-size effects are negligible. A power law over only two decades, with the upper range approaching 25% of the particle number, could be a finite-size crossover rather than genuine scale invariance. The authors should either provide a larger-system check or state clearly the range over which the scaling holds and how it depends on system size.

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The paper

The paper claims

Below the glass transition, cluster sizes keep a power law

The reading

This paper reports a distinctive statistical signature in the slow, cooperative motion of a glassy material. Using computer simulations of a standard binary mixture of Lennard-Jones particles, the researchers find that clusters of particles that jump out of their cages at nearly the same time have sizes $s$ distributed as a power law $P(s)\sim s^{-\tau}$, with an exponent near $\tau\approx 2.5$, at every temperature from $T=0.15$ up to $0.43$, all below the glass transition $T_g\approx 0.435$. The power law holds for every waiting time after the quench and for two different ways of defining a cluster, which the authors read as evidence of self-organized criticality: the critical, scale-free state does not require tuning to a special temperature but is frozen in below $T_g$. If true, this links the cooperative dynamics of supercooled liquids to the nonequilibrium aging of glasses and gives a concrete observable for the glassy state.

What carries the argument

The central object is a cluster of simultaneously jumping particles. A jump is defined operationally: each trajectory is coarse-grained into 800-time-unit bins, a particle's thermal fluctuation $\sigma_n$ is computed, and a jump is recorded when the distance between average positions four bins apart exceeds $\sqrt{20}\,\sigma_n$. Particles that jump in the same bin are then linked if their initial average positions are closer than the first minimum of the partial radial distribution functions ($r_{\min}=1.4$ for AA, $1.2$ for AB, $1.07$ for BB), and connected components form the clusters whose sizes enter the distribution $P(s)$. The power-law shape of $P(s)$, together with its invariance under two extended cluster definitions and under waiting time, is the load-bearing observation of the paper.

Extended reading notes

Core claim

On the paper's own terms, the central claim is that the cooperative dynamics of a glass below its transition is critical in the sense of percolation theory. The authors track every particle in a quenched binary Lennard-Jones system, identify jumps when the time-averaged position changes by more than $\sqrt{20}$ times the particle's average thermal fluctuation $\sigma_n$, and group jumps that occur in the same 800-time-unit window and within a characteristic neighbor distance. The resulting cluster size distribution is a straight line in log-log coordinates for all ten temperatures investigated, with an average exponent $\tau=2.50\pm 0.05$, close to the mean-field percolation value $\tau=2.5$. The distribution is unchanged by dividing the run into time windows, meaning it is independent of waiting time, and it remains a power law, with somewhat smaller exponents, when the cluster definition is extended to allow jumps in neighboring time bins. The interpretation is that critical fluctuations develop as $T_c$ is approached from above and then freeze in, so that the system exhibits self-organized criticality over the whole temperature range below $T_g$.

Load-bearing premise

The power law must survive changes to the jump threshold ($\sqrt{20}\,\sigma_n$), the 800-time-unit averaging, and the system size; the paper tests two extended cluster definitions but not these parameters.

Editorial extensions

If this is right

Aging does not wash out the critical cluster structure: the power law appears already in the first of five equal time windows and continues unchanged, so the criticality is a property of the glassy state rather than a transient.
The clusters are string-like rather than compact at all sizes and temperatures, so the same cooperative motif found above $T_g$ extends below it.
The measured exponents are close to percolation theory's mean-field value $\tau=2.5$, placing the phenomenon near the class of random percolation at its critical point.
The authors expect the same self-organized criticality to occur in other structural glasses out of equilibrium, making the cluster-size distribution a candidate universal diagnostic.

Reading between the lines

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

A finite-size scaling test is the natural next step: if the power law survives larger and smaller simulation boxes while the large-$s$ cutoff shifts systematically with box size, it is a genuine critical phenomenon rather than a small-system artifact.
The same jump-cluster analysis could be applied to particle-tracking data from colloidal or granular glasses below their transitions, turning the predicted power law into a directly testable experimental signature.
A stronger test of the self-organized-criticality interpretation would be to look at the temporal structure of jump events, such as waiting-time distributions or aftershock statistics, which should also be scale-free if the clusters are truly self-organized critical; the paper does not report such an analysis.

Original abstract

We obtain evidence that the dynamics of glassy systems below the glass transition is characterized by self-organized criticality. Using molecular dynamics simulations of a model glass-former we identify clusters of cooperatively jumping particles. We find string-like clusters whose size is power-law distributed not only close to T_c but for ALL temperatures below T_c, indicating self-organized criticality which we interpret as a freezing in of critical behavior.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,927 input tokens, $0.0001 reported.

What the abstract itself states, as probabilities: claims first 0.03 · states limitation 0.08 · human participants 0.01 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.17 · compares with prior results 0.07 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

We obtain evidence that the dynamics of glassy systems below the glass transition is characterized by self-organized criticality. Using molecular dynamics simulations of a model glass-former we identify clusters of cooperatively jumping particles. We find string-like clusters whose size is power-law distributed not only close to T_c but for ALL temperatures below T_c, indicating self-organized criticality which we interpret as a freezing in of critical behavior.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	We obtain evidence that the dynamics of glassy systems below the glass transition is characterized by self-organized criticality.	result 0.90	0.92	0.94	asserted 0.76
s1	Using molecular dynamics simulations of a model glass-former we identify clusters of cooperatively jumping particles.	method 0.95	0.96	0.03	asserted 1.00
s2	We find string-like clusters whose size is power-law distributed not only close to T_c but for ALL temperatures below T_c, indicating self-organized criticality which we interpret as a freezing in of critical behavior.	result 0.84	0.96	0.55	asserted 0.83

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v2

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

References

Works cited (19)

[1]

07 for BB independent of temperature). This analysis gives us for each time Kl ≥ 0 distinct clusters. The clus- ters are numbered by k = 1 , 2, . . . , K l and we denote by Nl,k the set of particle labels composing the kth cluster in time bin l with Nl,k particles (i.e. Kl∑ k=1 Nl,k = Nl). We now look at the size distribution P (s) of all clusters Nl,k , ...

View on Pith
[2]

5. found. Close to Tc the distribution of cluster sizes fol- lows a power law. Below the glass transition this critical behavior gets frozen in. Independent of details of the cluster deﬁnition and independent of waiting time, we ﬁnd string-like clusters with a cluster size distribution which follows a power law for all investigated tempera- tures. We expe...

View on Pith
[3]

K. Binder and W. Kob, “Glassy Materials and Disor- dered Solids: An Introduction to Their Statistical Me- chanics,” Word Scientiﬁc, Singapore 2005

View on Pith 2005
[4]

G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965)

View on Pith 1965
[5]

C. Donati, J. F. Douglas, W. Kob, S. J. Plimpton, P. H. Poole, and S. C. Glotzer, Phys. Rev. Lett. 80, 2338 (1998); C. Donati, S. C. Glotzer, P. H. Poole, W. Kob, and S. J. Plimpton, Phys. Rev. E 60, 3107 (1999); M. Vogel, B. Doliwa, A. Heuer, and S. C. Glotzer, J. Chem. Phys. 120, 4404 (2004)

View on Pith 1998
[6]

E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schoﬁeld and D. A. Weitz, Science 287, 627 (2000). 4

View on Pith 2000
[7]

H. Miyagawa, Y. Hiwatari, B. Bernu, and J. Hansen. J. Chem. Phys. 88, 3879 (1988); G. Wahnstr¨ om, Phys. Rev. A 44, 3752 (1991); D. N. Perera and P. Harrowell, J. Chem. Phys. 111, 5441 (1999); Y. Gebremichael, M. Vo- gel, and S. C. Glotzer, Mol. Simul. 30, 281 (2004); Y. Ge- bremichael, M. Vogel, and S. C. Glotzer, J. Chem. Phys. 120, 4415 (2004) and refe...

View on Pith 1988
[8]

G. A. Appignanesi, J. A. Rodriguez Fris, R. A. Montani, and W. Kob, e-print cond-mat/0506577 (2005)

View on Pith Pith review arXiv 2005
[9]

D. Stauﬀer, Phys. Rep. 54, 1 (1979)

View on Pith 1979
[10]

W. G¨ otze and L. Sj¨ ogren, Rep. Prog. Phys. 55, 241 (1992)

View on Pith 1992
[11]

A. Latz, preprint cond-mat/0106086

View on Pith Pith review arXiv
[12]

A. Latz, J. Phys.: Condens. Matter 12, 6353 (2000)

View on Pith 2000
[13]

J.-P. Bouchaud, L. Cugliandolo, J. Kurchan, M. M´ ezard , Physica A 226, 243 (1996)

View on Pith 1996
[14]

W. Kob and H. C. Andersen, Phys. Rev. Lett. 73, 1376 (1994); Phys. Rev. E 51, 4626 (1995); Phys. Rev. E 52, 4134 (1995)

View on Pith 1994
[15]

K. Vollmayr-Lee, J. Chem. Phys. 121, 4781 (2004)

View on Pith 2004
[16]

As reported in [13] σ n and ∆ rn are of the order of 0 . 1 and 0 . 7 respectively

View on Pith
[17]

For more details of the jump deﬁnition see [13]. Therein we distinguish irreversible and reversible jumps. In this Letter we include for all presented results both irre- versible and reversible jumps

View on Pith
[18]

P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59,381 (1987)

View on Pith 1987
[19]

C. Oligschleger and H. R. Schober, Phys. Rev. B 59, 811 (1999); H. Teichler, J. Non-Cryst. Solids 312-314, 533 (2002); R. E. Courtland and E. R. Weeks, J. Phys.: Condens. Matter 15, S359 (2003); M. Kluge and H. R. Schober, Phys. Rev. B 70, 224209 (2004); H. Teichler, Phys. Rev. E 71, 031505 (2005)

View on Pith 1999

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [1]

07 for BB independent of temperature). This analysis gives us for each time Kl ≥ 0 distinct clusters. The clus- ters are numbered by k = 1 , 2, . . . , K l and we denote by Nl,k the set of particle labels composing the kth cluster in time bin l with Nl,k particles (i.e. Kl∑ k=1 Nl,k = Nl). We now look at the size distribution P (s) of all clusters Nl,k , ...

View on Pith

[2] [2]

5. found. Close to Tc the distribution of cluster sizes fol- lows a power law. Below the glass transition this critical behavior gets frozen in. Independent of details of the cluster deﬁnition and independent of waiting time, we ﬁnd string-like clusters with a cluster size distribution which follows a power law for all investigated tempera- tures. We expe...

View on Pith

[3] [3]

K. Binder and W. Kob, “Glassy Materials and Disor- dered Solids: An Introduction to Their Statistical Me- chanics,” Word Scientiﬁc, Singapore 2005

View on Pith 2005

[4] [4]

G. Adam and J. H. Gibbs, J. Chem. Phys. 43, 139 (1965)

View on Pith 1965

[5] [5]

C. Donati, J. F. Douglas, W. Kob, S. J. Plimpton, P. H. Poole, and S. C. Glotzer, Phys. Rev. Lett. 80, 2338 (1998); C. Donati, S. C. Glotzer, P. H. Poole, W. Kob, and S. J. Plimpton, Phys. Rev. E 60, 3107 (1999); M. Vogel, B. Doliwa, A. Heuer, and S. C. Glotzer, J. Chem. Phys. 120, 4404 (2004)

View on Pith 1998

[6] [6]

E. R. Weeks, J. C. Crocker, A. C. Levitt, A. Schoﬁeld and D. A. Weitz, Science 287, 627 (2000). 4

View on Pith 2000

[7] [7]

H. Miyagawa, Y. Hiwatari, B. Bernu, and J. Hansen. J. Chem. Phys. 88, 3879 (1988); G. Wahnstr¨ om, Phys. Rev. A 44, 3752 (1991); D. N. Perera and P. Harrowell, J. Chem. Phys. 111, 5441 (1999); Y. Gebremichael, M. Vo- gel, and S. C. Glotzer, Mol. Simul. 30, 281 (2004); Y. Ge- bremichael, M. Vogel, and S. C. Glotzer, J. Chem. Phys. 120, 4415 (2004) and refe...

View on Pith 1988

[8] [8]

G. A. Appignanesi, J. A. Rodriguez Fris, R. A. Montani, and W. Kob, e-print cond-mat/0506577 (2005)

View on Pith Pith review arXiv 2005

[9] [9]

D. Stauﬀer, Phys. Rep. 54, 1 (1979)

View on Pith 1979

[10] [10]

W. G¨ otze and L. Sj¨ ogren, Rep. Prog. Phys. 55, 241 (1992)

View on Pith 1992

[11] [11]

A. Latz, preprint cond-mat/0106086

View on Pith Pith review arXiv

[12] [12]

A. Latz, J. Phys.: Condens. Matter 12, 6353 (2000)

View on Pith 2000

[13] [13]

J.-P. Bouchaud, L. Cugliandolo, J. Kurchan, M. M´ ezard , Physica A 226, 243 (1996)

View on Pith 1996

[14] [14]

W. Kob and H. C. Andersen, Phys. Rev. Lett. 73, 1376 (1994); Phys. Rev. E 51, 4626 (1995); Phys. Rev. E 52, 4134 (1995)

View on Pith 1994

[15] [15]

K. Vollmayr-Lee, J. Chem. Phys. 121, 4781 (2004)

View on Pith 2004

[16] [16]

As reported in [13] σ n and ∆ rn are of the order of 0 . 1 and 0 . 7 respectively

View on Pith

[17] [17]

For more details of the jump deﬁnition see [13]. Therein we distinguish irreversible and reversible jumps. In this Letter we include for all presented results both irre- versible and reversible jumps

View on Pith

[18] [18]

P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. Lett. 59,381 (1987)

View on Pith 1987

[19] [19]

C. Oligschleger and H. R. Schober, Phys. Rev. B 59, 811 (1999); H. Teichler, J. Non-Cryst. Solids 312-314, 533 (2002); R. E. Courtland and E. R. Weeks, J. Phys.: Condens. Matter 15, S359 (2003); M. Kluge and H. R. Schober, Phys. Rev. B 70, 224209 (2004); H. Teichler, Phys. Rev. E 71, 031505 (2005)

View on Pith 1999