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Elastic avalanches reveal marginal behaviour in amorphous solids

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Pith's one-line read Even deep in the elastic regime, amorphous solids deform through scale-free avalanches with a universal exponent τ ≈ 1, the mean-field signature of marginal stability.

desk verdict Systematic numerical study of elastic-regime avalanches in Lennard-Jones glasses claims universal τ≈1, but the paper's own tables show α and df/d are inconsistent with τ≈1 for poorly annealed states, leaving the marginal-stability conclusion unproven. read the letter →

arxiv 1908.08820 v3 pith:6V5AB7AE submitted 2019-08-23 cond-mat.dis-nn cond-mat.softcond-mat.stat-mech

Baoshuang Shang , Pengfei Guan , Jean-Louis Barrat This is my paper · ORCID
classification cond-mat.dis-nncond-mat.softcond-mat.stat-mech
keywords amorphoussolidselasticavalanchesmarginalstabilityathermalquasistaticshearavalancheexponentLennard-Jonesglassespseudo-gapfinite-sizescaling
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

Nearly all studies of plasticity in glasses focus on large strains near or past yielding. This paper instead characterises what happens at strains far below yielding, in the nominally elastic part of the stress–strain curve, where energy drops are small but frequent. It finds that in two- and three-dimensional Lennard-Jones glasses the avalanche number density is a power law with exponent $\tau \approx 1$ ($0.98\pm0.01$ in 2D, $1.01\pm0.01$ in 3D), matching the mean-field prediction for marginally stable amorphous packings. After rescaling by system size and thermal history, all distributions collapse onto a single master curve, and the exponents obey three scaling relations that tie avalanches, dissipation, and the pseudo-gap in low-lying excitations together. The authors conclude that marginal stability is systematic in the thermodynamic limit and that the amorphous solid is intrinsically dissipative, so the apparent elastic regime is a finite-size effect.

What carries the argument

The load-bearing object is the avalanche number density $R(S,N,T_{\mathrm{ini}})$, defined as the number of energy-drop avalanches of size $S$ per unit avalanche size and per unit strain. The argument is carried by a scaling ansatz that factors out system-size and thermal-history dependences: $S_c \sim \xi_1 N^{d_f/d}$ and $R \sim \xi_2 N^b \chi^{-\tau} f(\chi)$, so that the exponent $\tau$ is extracted from a data collapse. Three identities anchor the interpretation: energy balance gives $b + 2d_f/d = 1$; extreme-value statistics of the strain to the first plastic event gives $\langle \epsilon_\gamma \rangle \sim N^{-1/(1+\theta)}$, yielding $\theta \approx 1/2$; and the elasto-plastic relation $\alpha = \theta/(1+\theta)$ links the mean-avalanche-size exponent $\alpha$ to the pseudo-gap exponent $\theta$.

What would settle it

Measure the elastic-regime avalanche distribution in a qualitatively different glass former, e.g. a polymer glass or a metallic-glass model, using the same athermal quasistatic protocol; a value of $\tau$ clearly different from 1 would falsify the claim of a universal marginal-stability signature. Alternatively, check the zero-strain pseudo-gap prediction directly: if the first-avalanche strain in a deeply annealed sample does not scale as $N^{-2/3}$, the extreme-value link to $\theta = 1/2$ fails.

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Extended reading notes

Core claim

The central claim is that amorphous solids respond to arbitrarily small shear strain through scale-free avalanche activity whose statistics are those of a marginally stable phase, not the localized, history-dependent events usually assumed for the elastic regime. In athermal quasistatic simple shear over strain interval $\gamma \in [0, 0.02]$, well below the yield strain, the avalanche number density obeys $R(S,N,T_{\mathrm{ini}}) \sim \xi_2 N^b \chi^{-\tau} f(\chi)$ with $\chi = S/S_c$, $S_c \sim \xi_1 N^{d_f/d}$, and $\tau = 0.98\pm0.01$ (2D) and $1.01\pm0.01$ (3D). The same scaling is compatible with the mean-field result that local minima in a hierarchical energy landscape are marginally stable, and with predictions of an elasto-plastic model in which marginal stability appears as a pseudo-gap with exponent $\theta \approx 1/2$ at zero strain. The paper further shows that the scalar exponents satisfy $b + 2d_f/d = 1$, $d_f/d = \alpha$, and $\alpha = \theta/(1+\theta)$, where $\alpha$ governs the subextensive growth of the mean avalanche size with $N$. From the limit ordering $N\to\infty$ versus $\gamma\to0$ it infers that the thermodynamic-limit solid is intrinsically dissipative.

Load-bearing premise

The interpretation assumes that the mean-field result for marginally stable jammed packings in infinite dimensions transfers to dense, attractive Lennard-Jones glasses; if that transfer fails, the measured exponent $\tau \approx 1$ could have a different cause.

Editorial extensions

If this is right

  • Elastic avalanches belong to a universality class distinct from steady plastic flow: $\tau \approx 1$ here versus the larger exponents found in stationary shearing, so transient and steady-state plasticity must be analysed separately.
  • The energy-balance identity $b + 2d_f/d = 1$ holds in the elastic regime, providing a consistency check for any future simulation or experiment that measures avalanche cutoffs.
  • The pseudo-gap exponent starts at a universal $\theta \approx 1/2$ for the first event and falls to a thermal-history-dependent plateau, linking the brittle-to-ductile behaviour of a glass to its preparation.
  • In the thermodynamic limit the amorphous solid is intrinsically dissipative: taking $N\to\infty$ before $\gamma\to0$ leaves a finite dissipation per unit strain, so the purely elastic regime is a finite-size artefact.

Reading between the lines

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

  • If the $\tau \approx 1$ law is universal, acoustic-emission or micro-mechanical experiments on metallic glasses in the pre-yield regime should observe the same exponent, giving a laboratory test beyond simulation.
  • The clean decrease of $\alpha$ and $\theta_{\mathrm{plateau}}$ with annealing leaves open the possibility of a sharp ductile-to-brittle transition at a finite preparation temperature; testing this would require ultrastable samples at larger sizes than those studied here.
  • The master-curve collapse implies that avalanche statistics in the thermodynamic limit can be extrapolated from small systems once $\xi_1$, $\xi_2$, and $d_f/d$ are known, which could make experimental finite-sample data predictive.
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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

2 major / 4 minor

Summary. The paper studies athermal quasistatic shear in two- and three-dimensional Lennard-Jones glasses, focusing on avalanche statistics in the elastic regime (strain intervals up to 0.02, well below yielding). For several system sizes and thermal histories, the authors measure the avalanche number density R(S,N,T_ini) and propose a scaling form with system-size-dependent cutoff S_c ~ N^{d_f/d} and amplitude N^b. They report a universal avalanche exponent τ ≈ 1 in both 2D (0.98 ± 0.01) and 3D (1.01 ± 0.01), compatible with the Franz–Spigler mean-field prediction for marginally stable systems. They further report energy-balance scaling b + 2d_f/d = 1, a mean-size exponent α that they claim equals d_f/d, and a pseudo-gap exponent θ that evolves from an initial value near 1/2 to a plateau correlated with α. The authors interpret the results as evidence that marginal stability is systematic in the thermodynamic limit, despite the common view that such behavior is restricted to jammed systems with short-range repulsions.

Significance. If the central claim holds, the paper would significantly extend the phenomenology of marginal stability from jammed soft spheres in infinite dimensions to finite-dimensional, high-density Lennard-Jones glasses with attractions, suggesting a universal avalanche exponent in the elastic regime. The manuscript is strengthened by direct numerical evidence, a scaling collapse, an energy-balance check (Fig. 3), and internal consistency checks relating exponents (Figs. 4 and 5). The data availability statement permits independent verification. However, the central claim rests on the validity of the scaling collapse and on the transfer of the mean-field prediction to this system class; the latter is an assumption that the authors acknowledge, and the former is challenged by an inconsistency in the reported fit parameters, as detailed in the major comments.

major comments (2)
  1. [Tables S1-S2 and Fig. 4(d)] The tabulated exponents do not satisfy the consistency relation that the paper relies on. For 3D, T_ini = 0.87, the table gives d_f/d = 0.24 ± 0.02 and α = 0.172 ± 0.008. Using the relation α = (2 − τ) d_f/d quoted in the text, one obtains τ = 2 − α/(d_f/d) = 1.28 ± 0.07, several standard deviations above the collapsed value τ = 1.01 ± 0.01. For 2D, T_ini = 1.0, the analogous calculation gives τ = 1.16 ± 0.06. Only the most stable states (e.g., 3D T_ini = 0.479) are consistent with τ ≈ 1. The text claims that Fig. 4(d) confirms α = d_f/d, but the paper's own fit tables contradict this. This is a load-bearing issue: either the collapse-derived τ is biased by the choice of cutoff function or fitting range, or the reported α values do not correspond to the mean size defined in Eqs. (1)–(2), or the relation α = (2 − τ) d_f/d is misapplied. Please resolve this discrepancy explicitly.
  2. [Eqs. (4)–(5) and the relation α = (2−τ) d_f/d] The scaling form (4) directly implies ⟨S⟩ = η/M ∼ N^{d_f/d} for any τ < 2, because M ~ N^{b+d_f/d} and η ~ N^{b+2d_f/d}. The text instead cites Lin et al. for α = (2 − τ) d_f/d and then reduces it to α = d_f/d when τ ≈ 1. These two statements are inconsistent with each other under the paper's own definitions. The relation α = (2 − τ) d_f/d may apply to a different observable (for instance the total dissipated energy per unit strain rather than the mean avalanche size), but as written the manuscript does not define whether α in Fig. 4 and Tables S1-S2 is the exponent of ⟨S⟩ or of another moment. This matters because the paper presents Fig. 4(d) as a consistency check; please derive the relation used, state precisely which observable α governs, and reconcile it with Eq. (4).
minor comments (4)
  1. [Eq. (7)] Equation (7) has a typesetting issue: the terms 'Nγ−1' and 'Nγ−1ρ−1' are ambiguous. They should be written as N^{-1} and N^{-1} ρ^{-1} respectively, or with explicit parentheses, so that the energy-balance identity is readable.
  2. [Fig. 1 caption] The caption states 'The dashed line shows the avalanche exponent −1 predicted by mean field theory (7) near the ground state.' It is unclear whether this line is a fit to the collapsed data or the predicted slope drawn for comparison; please clarify the role of the line in the figure.
  3. [Methods / Eq. (10)] The avalanche size definition S = N(ΔU + Δγ τ_θ / ρ) uses a threshold S > 0.01 without stating the units. Since the system uses reduced units, please specify the unit convention and confirm that the chosen threshold does not affect the reported exponents, beyond the statement that thresholds from 0.01 to 0.1 give 'qualitatively similar results'.
  4. [Scaling analysis text] The text says the parameters are fitted from Fig. 2 and then 'both for 2D and 3D systems', but the fit parameters are given in Tables S1 and S2, not in the main text. Please refer to the tables at first mention and include the fit ranges used for the power-law fits in Fig. 2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the avalanche exponent is measured and compared to an external mean-field benchmark, and the scaling checks are not self-referential.

full rationale

The central claim, that the elastic-regime avalanche number density follows a power law with exponent τ ≈ 1 compatible with the Franz–Spigler mean-field prediction, is not circular. The exponent is extracted from the shape of the rescaled avalanche distribution after separately fitting the cutoff and amplitude parameters (df/d, b, ξ1, ξ2) from Sc(N) and η(N); none of those fitted inputs fixes τ by construction, and the result is explicitly distinguished from the stationary-flow exponent (≈1.3). The scaling relations b + 2df/d = 1, Γ = ξ1²ξ2, α ≈ df/d, and α = θ/(1+θ) are presented as consistency checks between independently measured quantities — stress-strain dissipation, mean avalanche size, and first-event strain statistics — rather than as predictions that reduce to the fitted inputs. The fact that the collapse in Fig. 1(c,d) uses exponents fitted from the same datasets is a standard finite-size-scaling procedure, not a self-definitional reduction: the reported τ is a fitting output that could in principle differ from unity. The internal tension between Tables S1–S2 and the Lin et al. relation α = (df/d)(2−τ) — e.g., inverting the 3D Tini = 0.87 values gives τ ≈ 1.28 rather than 1.01 — is a statistical robustness and correctness concern about how well the collapse determines τ, but it does not make the derivation circular. No load-bearing self-citation or imported uniqueness claim was found; refs. 7, 20, 28, and 29 are external theoretical benchmarks.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central result is an empirical scaling observation with a theoretical interpretation. The quantitative content rests on a set of fitted exponents (df/d, b, ξ1, ξ2, τ, α, θPlateau) that are used in a data collapse and in consistency relations. The bridge from the measured exponent to the physics of marginal stability is an assumption that the infinite-dimensional mean-field result applies to finite-dimensional LJ glasses. No new entities are introduced.

free parameters (6)
  • df/d (fractal dimension ratio) = 0.225(8), 0.166(5), 0.050(2) (2D); 0.24(2), 0.17(2), 0.090(6) (3D)
    Fitted from the power-law Sc = ξ1 N^{df/d} in Fig. 2(a,b); determines the cutoff scaling with system size and is used in the data collapse.
  • b (size-scaling exponent of avalanche number density) = Not listed separately; fixed via b = 1 - 2df/d from energy balance
    Fitted from η(N) = ξ1^2 ξ2 N^{b+2df/d} in Fig. 2(c,d); the energy-balance identity b + 2df/d = 1 is then used as a consistency check.
  • ξ1, ξ2 (thermal-history prefactors) = Tables S1 and S2
    Prefactors depending on Tini, fitted to Sc(N) and η(N) respectively; they absorb preparation dependence in the scaling collapse.
  • τ (avalanche exponent) = 0.98±0.01 (2D), 1.01±0.01 (3D)
    Obtained by fitting the collapsed master curve R ~ χ^{-τ} f(χ) with a Gaussian cutoff after the collapse; this is the central claim and is compared to the mean-field value 1.
  • α (mean avalanche size exponent) = 0.19(1), 0.134(5), 0.040(8) (2D); 0.172(8), 0.152(2), 0.09(1) (3D)
    Fitted from <S> ~ N^α in Fig. 4(a,b); used to test the relation α = df/d.
  • θPlateau (pseudo-gap exponent plateau) = 0.26±0.01, 0.21±0.01, 0.11±0.01 (3D)
    Average of θ(γ) over the strain window [0.005, 0.015] in Fig. 5(c); used to test α = θ/(1+θ).
assumptions (5)
  • domain assumption The avalanche number density obeys the scaling ansatz R(S,N,Tini) ~ ξ2 N^b χ^{-τ} f(χ) with cutoff Sc ~ ξ1 N^{df/d} (Eqs. 3-4).
    Assumed without derivation; standard for finite-size scaling of avalanche distributions, but the form of f and the power-law ansatz are not proven for the transient elastic regime.
  • domain assumption The first plastic event strain follows <εγ> ~ N^{-1/(1+θ)} from extreme value statistics of a pseudogap P(x) ~ x^θ (ref. 33).
    Used to convert the fitted exponent -0.66 into θ ≈ 1/2; relies on the Müller-Wyart extreme-value argument.
  • ad hoc to paper The Franz-Spigler mean-field prediction (τ=1 for marginally stable packings) transfers from infinite-dimensional jammed spheres to finite-dimensional Lennard-Jones glasses with attractions.
    This transferability is the paper's central interpretive assumption; the paper argues it is surprising because prior consensus restricted marginal stability to finite-range interactions near jamming.
  • standard math Energy balance: the dissipated energy NΓ is extensive and equals the total avalanche energy ηN, giving the identity b + 2df/d = 1 (Eqs. 7-8).
    Follows from conservation of energy in athermal quasistatic deformation; used as a consistency check of the fitted exponents.
  • domain assumption The relations α = (df/d)(2-τ) and α = θ/(1+θ) from elasto-plastic descriptions (Lin et al., Lin-Wyart) hold in the transient elastic regime.
    Borrowed from yielding and steady-flow theories; the paper extends them to the transient regime and verifies them a posteriori.

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Pith. "Pith review of Elastic avalanches reveal marginal behaviour in amorphous solids." pith.science (2026). https://pith.science/paper/6V5AB7AE

@misc{pith2026190808820,
  author       = {Pith},
  title        = {Pith review of: Elastic avalanches reveal marginal behaviour in amorphous solids},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/6V5AB7AE}},
  note         = {Machine review of arXiv:1908.08820}
}
read the original abstract

Mechanical deformation of amorphous solids can be described as consisting of an elastic part in which the stress increases linearly with strain, up to a yield point at which the solid either fractures or starts deforming plastically. It is well established, however, that the apparent linearity of stress with strain is actually a proxy for a much more complex behavior, with a microscopic plasticity that is reflected in diverging nonlinear elastic coefficients. Very generally, the complex structure of the energy landscape is expected to induce a singular response to small perturbations. In the athermal quasistatic regime, this response manifests itself in the form of a scale free plastic activity. The distribution of the corresponding avalanches should reflect, according to theoretical mean field calculations (Franz and Spigler, Phys. Rev. E., 2017, 95, 022139), the geometry of phase space in the vicinity of a typical local minimum. In this work, we characterize this distribution for simple models of glass forming systems, and we find that its scaling is compatible with the mean field predictions for systems above the jamming transition. These systems exhibit marginal stability, and scaling relations that hold in the stationary state are examined and confirmed in the elastic regime. By studying the respective influence of system size and age, we suggest that marginal stability is systematic in the thermodynamic limit.

Figures

Figures reproduced from arXiv: 1908.08820 by the authors.

Figure 2
Figure 2. Cutoff of the distribution of avalanche sizes and total avalanche en [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 4
Figure 4. Mean value of the avalanche size < S > (a),(b) < S > versus system size N for different thermal histories, in 2D and 3D systems. The dashed line is a fit by the equation < S >∼ Nα (c) finite size exponent α versus shear to bulk modulus ratio G/B. (d) Correlation between mean value exponent α and cutoff value exponent df /d for various thermal histories and dimension. Avalanche mean size < S > and distribution cutoff… view at source ↗

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Pith tools

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