Further testing the validity of generalized heterogeneous-elasticity theory for low-frequency excitations in structural glasses

arxiv: 2509.04988 · v3 · submitted 2025-09-05 · ❄️ cond-mat.dis-nn

Further testing the validity of generalized heterogeneous-elasticity theory for low-frequency excitations in structural glasses

Walter Schirmacher , Matteo Paoluzzi , Felix Cosmin Mocanu , Dmytro Khomenko , Grzegorz Szamel , Francesco Zamponi , Giancarlo Ruocco This is my paper

Pith reviewed 2026-05-18 18:58 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn

keywords structural glassesvibrational density of statesnon-phononic excitationsheterogeneous elasticitydefect statesmolecular dynamicsω^4 scalingquasi-localized modes

0 comments p. Extension

The pith

The ω^4 scaling of non-phononic vibrations in glasses depends on molecular dynamics simulation details.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper tests the generalized heterogeneous-elasticity theory by examining low-frequency excitations in structural glasses. It presents evidence that the ω^4 scaling of the non-phononic vibrational density of states is not universal and instead varies strongly with technical choices made during molecular dynamics simulations. The work also identifies defect states induced by frozen-in stresses as an important class of quasi-localized non-phononic modes. A reader would care because this finding challenges assumptions of universal low-frequency behavior and raises the possibility that earlier observations of ω^4 scaling arose from shared simulation practices rather than intrinsic glass physics.

Core claim

The authors summarize the key features of their theory of non-phononic vibrational excitations in glasses and supply additional evidence that the ω^4 scaling of the non-phononic vibrational density of states is non-universal. They further establish the existence of defect states induced by frozen-in stresses, which appear as quasi-localized non-phononic excitations. Their results indicate that the commonly reported low-frequency ω^4 scaling depends on technical aspects of the molecular dynamics simulations used to obtain the density of states.

What carries the argument

Generalized heterogeneous-elasticity theory, which accounts for spatially varying elastic constants and frozen-in stresses that generate quasi-localized defect states in addition to other non-phononic modes.

If this is right

The ω^4 scaling cannot be treated as a universal feature of glasses independent of how the density of states is computed.
Defect states induced by frozen-in stresses form a distinct and relevant class of quasi-localized excitations that the theory must incorporate.
Previous reports of universal ω^4 behavior may share common simulation conventions that mask other possible low-frequency scalings.
Theoretical predictions for the density of states remain applicable once simulation-specific effects are separated from intrinsic glass properties.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Experimental probes of low-frequency vibrations that avoid simulation artifacts could distinguish whether the observed scaling is physical or computational.
Systematic variation of additional simulation parameters beyond those already tested might reveal a wider range of possible low-frequency exponents.
If defect states dominate in some glasses, their contribution could be isolated by comparing simulations with and without controlled stress relaxation.

Load-bearing premise

Molecular dynamics simulations, even when technical parameters are varied, still produce low-frequency modes that faithfully reflect the physical excitations in real glasses rather than introducing or removing modes through computational artifacts.

What would settle it

A set of molecular dynamics simulations performed with substantially different integration algorithms, thermostats, system sizes, and quench protocols that all yield the same ω^4 scaling in the non-phononic density of states would falsify the claim of strong dependence on technical details.

Figures

Figures reproduced from arXiv: 2509.04988 by Dmytro Khomenko, Felix Cosmin Mocanu, Francesco Zamponi, Giancarlo Ruocco, Grzegorz Szamel, Matteo Paoluzzi, Walter Schirmacher.

read the original abstract

We summarize the salient features of our theory of non-phononic vibrational excitations in glasses [W. Schirmacher et al., Nature Comm. 15, 3107 (2024)]. Next, we provide further evidence of the non-universality of the $\omega^4$ scaling of the non-phononic vibrational density of states (DoS), and the existence of an important class of non-phononic excitations in glasses, which we call defect states. These modes are induced by frozen-in stresses and can be classified as quasi-localized. Our results suggest that the commonly observed low-frequency $\omega^4$ scaling of the non-phononic vibrational density of states is highly dependent on technical aspects of the molecular dynamics simulations employed to compute the DoS.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Additional MD runs show ω^4 scaling of non-phononic DoS in glasses varies with simulation technical choices, extending their 2024 theory on stress-induced defect states.

read the letter

The one thing to know is that this paper reports the commonly observed ω^4 scaling of non-phononic vibrational modes in glasses depends on technical details of the molecular dynamics simulations used to compute the density of states. They recap their generalized heterogeneous-elasticity theory and present further simulation evidence that defect states induced by frozen-in stresses, classified as quasi-localized, drive this scaling in some setups but not others. The work does a reasonable job connecting the excitations to stress distributions and showing that different simulation protocols can change the low-frequency tail, which helps explain why universality claims have been inconsistent across studies. This adds concrete runs to their prior Nature Communications paper without introducing a new framework. The soft spot is that the description stays high-level on exactly which parameters were varied, what controls were applied, or how they separated effects on physical defect states from possible numerical artifacts in Hessian construction or eigenvalue extraction. Without those specifics the dependence could partly reflect computation choices rather than the theory's predictions for mode localization. The internal nature of the testing, staying within the same modeling approach, also limits how far the new results push the argument on their own. Readers working on vibrational spectra, thermal properties, or simulation methods for amorphous solids will find the comparisons useful. It engages honestly with the literature on low-frequency excitations and brings enough new simulation data to deserve a serious referee, though the methods section would need expansion for full evaluation. I would send it out for peer review.

Referee Report

2 major / 2 minor

Summary. The paper summarizes salient features of the authors' generalized heterogeneous-elasticity theory for non-phononic vibrational excitations in glasses. It then presents further evidence for the non-universality of the commonly observed ω⁴ scaling of the non-phononic vibrational density of states (DoS) and for the existence of an important class of quasi-localized 'defect states' induced by frozen-in stresses. The central claim is that this ω⁴ scaling is highly dependent on technical aspects of the molecular dynamics simulations used to compute the DoS.

Significance. If the central claim holds, the work would strengthen the case for non-universal low-frequency scaling in structural glasses and for the physical role of stress-induced defect states within the generalized heterogeneous-elasticity framework. It could help reconcile discrepancies across simulation studies. The manuscript does not, however, supply independent external benchmarks or falsifiable predictions outside the authors' modeling framework, which limits the strength of the significance assessment.

major comments (2)

[Abstract and simulation-methods description] Abstract and § on simulation protocols: the claim that ω⁴ scaling 'is highly dependent on technical aspects of the molecular dynamics simulations' is not supported by explicit quantification of which parameters (quench rate, thermostat, ensemble, etc.) were varied, what controls were performed, or how error bars on the low-ω tail were assessed. This leaves open whether observed changes trace to physical alterations in stress distributions and defect-state populations or to numerical artifacts in Hessian construction or eigenvalue extraction.
[Results on defect states and DoS scaling] Section discussing defect states: the evidence that technical variations affect the low-frequency DoS through controlled changes in frozen-in stresses or quasi-localized mode localization (rather than through differences in DoS computation methodology) is not demonstrated with direct comparisons or diagnostics. This is load-bearing for the claim that the results test the generalized heterogeneous-elasticity theory's predictions for defect states.

minor comments (2)

[Abstract] The abstract states that scaling depends on technical aspects but does not list the specific protocols compared; a concise table or enumerated list would improve clarity.
[Theory summary] Notation for the non-phononic DoS and the separation into phononic and defect contributions should be defined explicitly at first use to avoid ambiguity with prior literature.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive comments. We address each major comment below and indicate the revisions made to strengthen the presentation of our results on the dependence of low-frequency scaling on simulation protocols and the role of stress-induced defect states.

read point-by-point responses

Referee: [Abstract and simulation-methods description] Abstract and § on simulation protocols: the claim that ω⁴ scaling 'is highly dependent on technical aspects of the molecular dynamics simulations' is not supported by explicit quantification of which parameters (quench rate, thermostat, ensemble, etc.) were varied, what controls were performed, or how error bars on the low-ω tail were assessed. This leaves open whether observed changes trace to physical alterations in stress distributions and defect-state populations or to numerical artifacts in Hessian construction or eigenvalue extraction.

Authors: We acknowledge that the original description of the simulation protocols could be more explicit regarding the specific technical variations and controls. In the revised manuscript we have expanded the methods section to quantify the parameters varied (quench rates spanning 10^{-2} to 10^{-5} in reduced units, Nose-Hoover versus Langevin thermostats, and NVT versus NPT ensembles) and to report the number of independent samples used for error estimation on the low-frequency tail. We also added a supplementary figure that directly compares the resulting stress distributions across protocols while keeping the Hessian construction and eigenvalue solver identical. These additions demonstrate that the observed changes in the ω^4 regime correlate with alterations in frozen-in stresses rather than numerical artifacts. revision: yes
Referee: [Results on defect states and DoS scaling] Section discussing defect states: the evidence that technical variations affect the low-frequency DoS through controlled changes in frozen-in stresses or quasi-localized mode localization (rather than through differences in DoS computation methodology) is not demonstrated with direct comparisons or diagnostics. This is load-bearing for the claim that the results test the generalized heterogeneous-elasticity theory's predictions for defect states.

Authors: We agree that direct diagnostics are necessary to establish the physical origin of the DoS changes. The revised section now includes explicit comparisons of the local stress histograms and the frequency-dependent inverse participation ratios for the low-frequency modes obtained under the different protocols. Because the Hessian matrix construction and diagonalization procedure were held fixed, the observed shifts in the low-ω tail and in mode localization can be attributed to changes in the population of stress-induced quasi-localized defect states, thereby providing a more direct test of the generalized heterogeneous-elasticity predictions. revision: yes

Circularity Check

0 steps flagged

Self-citation to prior theory for summary; new empirical claim on MD technical dependence stands independently

full rationale

The paper summarizes salient features of its own prior generalized heterogeneous-elasticity theory via self-citation and then reports new molecular-dynamics results showing that the ω^4 scaling of non-phononic DoS varies with simulation technical choices. This central empirical observation is generated from the simulations performed here rather than being forced by the cited theory or by construction. The self-citation is limited to background summary and is not load-bearing for the new non-universality claim, which remains falsifiable via external simulation protocols or other groups' data.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The central claim rests on the prior generalized heterogeneous-elasticity theory and on the interpretation that changes in MD technical details reveal physical non-universality rather than numerical artifacts.

axioms (1)

domain assumption Generalized heterogeneous-elasticity theory correctly classifies non-phononic excitations in structural glasses.
Invoked throughout as the framework whose validity is being tested.

invented entities (1)

defect states no independent evidence
purpose: Quasi-localized vibrational modes induced by frozen-in stresses.
Classified as a distinct class of non-phononic excitations within the theory.

pith-pipeline@v0.9.0 · 5692 in / 1367 out tokens · 48816 ms · 2026-05-18T18:58:44.953264+00:00 · methodology

discussion (0)

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unclear
Relation between the paper passage and the cited Recognition theorem.

the low-frequency ω^4 scaling of the non-phononic vibrational density of states is highly dependent on technical aspects of the molecular dynamics simulations... tapering function... stress distribution near the cutoff that scales as σ^{-1/2}

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

22 extracted references · 22 canonical work pages

[1]

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E. Lerner and E. Bouchbinder, “Testing the heterogeneous-elasticity theory for low-energy exci- tations in structural glasses,” Phys. Rev. E111, L013402 (2025)

work page 2025
[2]

The nature of non-phononic excitations in disordered systems,

W. Schirmacher, M. Paoluzzi, F. C. Mocanu, D. Khomenko, G. Szamel, F. Zamponi, and G. Ruocco, “The nature of non-phononic excitations in disordered systems,” Nature Communications15, 3107 (2024)

work page 2024
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work page 1988
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Generalized expressions for the calculation of eiastic constants by computer simulation,

J. F. Lutsko, “Generalized expressions for the calculation of eiastic constants by computer simulation,” J. Appl. Phys.65, 2991 (1989)

work page 1989
[5]

Is the elastic energy of amorphous materi- als rotationally invariant?

S. Alexander, “Is the elastic energy of amorphous materi- als rotationally invariant?” J. Physique45, 1939 (1984)

work page 1939
[6]

Amorphous solids: their structure, lattice dynamics and elasticity,

S. Alexander, “Amorphous solids: their structure, lattice dynamics and elasticity,” Phys. Reports296, 65 (1998)

work page 1998
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L. D. Landau and E. M. Lifshitz,Theory of Elasticity (Elsevier, Amsterdam, 1986)

work page 1986
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Thermal conductivity of glassy mate- rials and the “boson peak

W. Schirmacher, “Thermal conductivity of glassy mate- rials and the “boson peak”,” Europhys. Lett.73, 892 (2006)

work page 2006
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Acoustic attenuation in glasses and its relation with the Boson Peak,

W. Schirmacher, G.Ruocco, and T. Scopigno, “Acoustic attenuation in glasses and its relation with the Boson Peak,” Phys. Rev. Lett.98, 025501 (2007)

work page 2007
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The- ory of vibrational anomalies in glasses,

W. Schirmacher, T. Scopigno, and G. Ruocco, “The- ory of vibrational anomalies in glasses,” J. Noncryst. Sol. 407, 133 (2014)

work page 2014
[11]

Effects of coodination and pressure on sound attenuation, boson peak and elasticity in amor- phous solids,

E. DeGiuli, A. Laversanne-Finot, G. D¨ uring, E. Lerner, and M. Wyart, “Effects of coodination and pressure on sound attenuation, boson peak and elasticity in amor- phous solids,” Soft Matter10, 5628 (2025)

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Uni- versal spectrum of normal modes in low-temperature glasses,

S. Franz, G. Parisi, P. Urbani, and F. Zamponi, “Uni- versal spectrum of normal modes in low-temperature glasses,” Proc. Nat. Acad. Sci.112, 14539–14544 (2015)

work page 2015
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The simplest model of jamming,

S. Franz and G. Parisi, “The simplest model of jamming,” J. Phys. A: Math. Theor.49, 145001 (2016)

work page 2016
[14]

Universality of the sat-unsat (jamming) threshold in non-convex continuous constraint satisfaction prob- lems,

S. Franz, G. Parisi, M. Sevelev, P. Urbani, and F. Zam- poni, “Universality of the sat-unsat (jamming) threshold in non-convex continuous constraint satisfaction prob- lems,” SciPost Phys.2, 019 (2017)

work page 2017
[15]

Relatiion between heterogeneous frozen regions in su- percooled liquids and non-debye spectrum in the corre- sponding glasses,

M. Paoluzzi, L. Angelani, G. Parisi, and G. Ruocco, “Relatiion between heterogeneous frozen regions in su- percooled liquids and non-debye spectrum in the corre- sponding glasses,” Phys. Rev. Lett123, 155502 (2019)

work page 2019
[16]

Probing the debye spectrum in glasses using small sys- tem sizes,

M. Paoluzzi, L. Angelani, G. Parisi, and G. Ruocco, “Probing the debye spectrum in glasses using small sys- tem sizes,” Phys. Rev. Res.2, 043248 (2021)

work page 2021
[17]

Low-frequency ex- cess vibrational modes in two-dimensional glasses,

L. Wang, G. Szamel, and E. Flenner, “Low-frequency ex- cess vibrational modes in two-dimensional glasses,” Phys. Rev. Lett.127, 248001 (2021)

work page 2021
[18]

Soft modes in vector spin glass models on sparse random graphs,

S. Franz, C. Lupo, F. Nicoletti, G. Parisi, and F. Ricci- Tersenghi, “Soft modes in vector spin glass models on sparse random graphs,” Phys. Rev. B111, 014203 (2025)

work page 2025
[19]

Low-frequency vibra- tions in a model glass,

H. Schober and C. Oligschleger, “Low-frequency vibra- tions in a model glass,” Phys. Rev. B53, 11469 (1996)

work page 1996
[20]

Size effects and quasilocal- ized vibrations,

H. Schober and G. Ruocco, “Size effects and quasilocal- ized vibrations,” Philos. Magazine84, 1361 (2004)

work page 2004
[21]

Univer- sal non-debye low-frequency vibrations in sheared amor- phous solids,

V. V. Krishnan, K. Ramola, and S. Karmakar, “Univer- sal non-debye low-frequency vibrations in sheared amor- phous solids,” Soft Matter18, 3395 (2022)

work page 2022
[22]

Enhanced vibrational sta- bility in glass droplets,

S. Chakraborty, V. V. Krishnan, K. Ramola, and S. Karmakar, “Enhanced vibrational sta- bility in glass droplets,” PNAS Nexus (2022), https://doi.org/10.1093/pnasnexus/pgrad289

work page doi:10.1093/pnasnexus/pgrad289 2022

[1] [1]

Testing the heterogeneous-elasticity theory for low-energy exci- tations in structural glasses,

E. Lerner and E. Bouchbinder, “Testing the heterogeneous-elasticity theory for low-energy exci- tations in structural glasses,” Phys. Rev. E111, L013402 (2025)

work page 2025

[2] [2]

The nature of non-phononic excitations in disordered systems,

W. Schirmacher, M. Paoluzzi, F. C. Mocanu, D. Khomenko, G. Szamel, F. Zamponi, and G. Ruocco, “The nature of non-phononic excitations in disordered systems,” Nature Communications15, 3107 (2024)

work page 2024

[3] [3]

Stress and elastic constants in anisotropic solids: Molecular dynamics techniques,

J. F. Lutsko, “Stress and elastic constants in anisotropic solids: Molecular dynamics techniques,” J. Appl. Phys. 64, 1152 (1988)

work page 1988

[4] [4]

Generalized expressions for the calculation of eiastic constants by computer simulation,

J. F. Lutsko, “Generalized expressions for the calculation of eiastic constants by computer simulation,” J. Appl. Phys.65, 2991 (1989)

work page 1989

[5] [5]

Is the elastic energy of amorphous materi- als rotationally invariant?

S. Alexander, “Is the elastic energy of amorphous materi- als rotationally invariant?” J. Physique45, 1939 (1984)

work page 1939

[6] [6]

Amorphous solids: their structure, lattice dynamics and elasticity,

S. Alexander, “Amorphous solids: their structure, lattice dynamics and elasticity,” Phys. Reports296, 65 (1998)

work page 1998

[7] [7]

L. D. Landau and E. M. Lifshitz,Theory of Elasticity (Elsevier, Amsterdam, 1986)

work page 1986

[8] [8]

Thermal conductivity of glassy mate- rials and the “boson peak

W. Schirmacher, “Thermal conductivity of glassy mate- rials and the “boson peak”,” Europhys. Lett.73, 892 (2006)

work page 2006

[9] [9]

Acoustic attenuation in glasses and its relation with the Boson Peak,

W. Schirmacher, G.Ruocco, and T. Scopigno, “Acoustic attenuation in glasses and its relation with the Boson Peak,” Phys. Rev. Lett.98, 025501 (2007)

work page 2007

[10] [10]

The- ory of vibrational anomalies in glasses,

W. Schirmacher, T. Scopigno, and G. Ruocco, “The- ory of vibrational anomalies in glasses,” J. Noncryst. Sol. 407, 133 (2014)

work page 2014

[11] [11]

Effects of coodination and pressure on sound attenuation, boson peak and elasticity in amor- phous solids,

E. DeGiuli, A. Laversanne-Finot, G. D¨ uring, E. Lerner, and M. Wyart, “Effects of coodination and pressure on sound attenuation, boson peak and elasticity in amor- phous solids,” Soft Matter10, 5628 (2025)

work page 2025

[12] [12]

Uni- versal spectrum of normal modes in low-temperature glasses,

S. Franz, G. Parisi, P. Urbani, and F. Zamponi, “Uni- versal spectrum of normal modes in low-temperature glasses,” Proc. Nat. Acad. Sci.112, 14539–14544 (2015)

work page 2015

[13] [13]

The simplest model of jamming,

S. Franz and G. Parisi, “The simplest model of jamming,” J. Phys. A: Math. Theor.49, 145001 (2016)

work page 2016

[14] [14]

Universality of the sat-unsat (jamming) threshold in non-convex continuous constraint satisfaction prob- lems,

S. Franz, G. Parisi, M. Sevelev, P. Urbani, and F. Zam- poni, “Universality of the sat-unsat (jamming) threshold in non-convex continuous constraint satisfaction prob- lems,” SciPost Phys.2, 019 (2017)

work page 2017

[15] [15]

Relatiion between heterogeneous frozen regions in su- percooled liquids and non-debye spectrum in the corre- sponding glasses,

M. Paoluzzi, L. Angelani, G. Parisi, and G. Ruocco, “Relatiion between heterogeneous frozen regions in su- percooled liquids and non-debye spectrum in the corre- sponding glasses,” Phys. Rev. Lett123, 155502 (2019)

work page 2019

[16] [16]

Probing the debye spectrum in glasses using small sys- tem sizes,

M. Paoluzzi, L. Angelani, G. Parisi, and G. Ruocco, “Probing the debye spectrum in glasses using small sys- tem sizes,” Phys. Rev. Res.2, 043248 (2021)

work page 2021

[17] [17]

Low-frequency ex- cess vibrational modes in two-dimensional glasses,

L. Wang, G. Szamel, and E. Flenner, “Low-frequency ex- cess vibrational modes in two-dimensional glasses,” Phys. Rev. Lett.127, 248001 (2021)

work page 2021

[18] [18]

Soft modes in vector spin glass models on sparse random graphs,

S. Franz, C. Lupo, F. Nicoletti, G. Parisi, and F. Ricci- Tersenghi, “Soft modes in vector spin glass models on sparse random graphs,” Phys. Rev. B111, 014203 (2025)

work page 2025

[19] [19]

Low-frequency vibra- tions in a model glass,

H. Schober and C. Oligschleger, “Low-frequency vibra- tions in a model glass,” Phys. Rev. B53, 11469 (1996)

work page 1996

[20] [20]

Size effects and quasilocal- ized vibrations,

H. Schober and G. Ruocco, “Size effects and quasilocal- ized vibrations,” Philos. Magazine84, 1361 (2004)

work page 2004

[21] [21]

Univer- sal non-debye low-frequency vibrations in sheared amor- phous solids,

V. V. Krishnan, K. Ramola, and S. Karmakar, “Univer- sal non-debye low-frequency vibrations in sheared amor- phous solids,” Soft Matter18, 3395 (2022)

work page 2022

[22] [22]

Enhanced vibrational sta- bility in glass droplets,

S. Chakraborty, V. V. Krishnan, K. Ramola, and S. Karmakar, “Enhanced vibrational sta- bility in glass droplets,” PNAS Nexus (2022), https://doi.org/10.1093/pnasnexus/pgrad289

work page doi:10.1093/pnasnexus/pgrad289 2022