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arxiv: 2505.04554 · v3 · submitted 2025-05-07 · ❄️ cond-mat.dis-nn · cond-mat.mtrl-sci

Large-scale exponential correlations of nonaffine elastic response of strongly disordered materials

D. A. Conyuh , D. V. Babin , I. O. Raikov , Y. M. Beltukov This is my paper

Pith reviewed 2026-05-22 16:57 UTC · model grok-4.3

classification ❄️ cond-mat.dis-nn cond-mat.mtrl-sci
keywords nonaffine elastic responseexponential correlationsheterogeneity length scalestrongly disordered materialsamorphous solidsrandom matrix theoryelastic correlations
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The pith

Spatial derivatives of the nonaffine displacement field show large-scale exponential correlations set by a disorder-dependent length scale.

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

The paper shows that in strongly disordered materials the nonaffine displacement field itself decays as a power law, yet the divergence and rotor of that field display exponential correlations. These correlations are controlled by a heterogeneity length scale ξ fixed by the strength of the disorder. The scale ξ can grow arbitrarily large and exceed the material's structural correlation length. The result follows from treating the nonaffine field as a correlated random matrix and is verified in both lattice percolation models and molecular-dynamics simulations of glasses.

Core claim

While the nonaffine displacement field predominantly exhibits power-law decay, its spatial derivatives reveal large-scale exponentially decaying correlations. The correlation functions of the divergence and (for most deformations) the rotor are governed by a heterogeneity length scale ξ set by the disorder strength and can become indefinitely large, far exceeding the structural correlation length.

What carries the argument

Theory of correlated random matrices applied to the nonaffine displacement field, from which spatial derivatives produce exponential correlations governed by the heterogeneity length ξ.

Load-bearing premise

The nonaffine displacement field can be modeled as the outcome of a correlated random matrix whose spatial derivatives produce the observed exponential decay.

What would settle it

Numerical or experimental measurement of the divergence and rotor correlation functions that fails to show exponential decay with a length scale controlled by disorder strength would falsify the claim.

Figures

Figures reproduced from arXiv: 2505.04554 by D. A. Conyuh, D. V. Babin, I. O. Raikov, Y. M. Beltukov.

Figure 1
Figure 1. Figure 1: FIG. 1. Schematic illustration of the branches [PITH_FULL_IMAGE:figures/full_fig_p008_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Numerical examples to investigate nonaffine dis [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a, b) Examples of nonaffine displacement fields [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Correlation function of the divergence (a, c) and the rotor (b, d) of the nonaffine displacement field in the rigidity [PITH_FULL_IMAGE:figures/full_fig_p014_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. (a, b) Examples of nonaffine displacement fields [PITH_FULL_IMAGE:figures/full_fig_p015_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: A small value 3·10−3 ε 2 has been added to Kiso rot(r) to compensate finite-size effects observed in the case of the shear strain. After some transition at distances r < 1 nm, one can see an exponential decay ∼ exp(−r/ξ) in [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Correlation function of the divergence (a, c) and the rotor (b, d) of the nonaffine displacement field in model [PITH_FULL_IMAGE:figures/full_fig_p016_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. (a, b) Examples of nonaffine displacement fields [PITH_FULL_IMAGE:figures/full_fig_p017_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Correlation function of the divergence (a, c) and the rotor (b, d) of the nonaffine displacement field in LJ glass [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
read the original abstract

The correlation properties of the nonaffine elastic response in strongly disordered materials are investigated using the theory of correlated random matrices and supported by numerical models. While the nonaffine displacement field itself predominantly exhibits power-law decay, we demonstrate that its spatial derivatives reveal large-scale exponentially decaying correlations. Specifically, the correlation functions of the divergence and (for most deformations) the rotor of the nonaffine field are governed by a heterogeneity length scale $\xi$. This length scale is set by the disorder strength and can become indefinitely large, far exceeding the structural correlation length. A notable exception occurs under volumetric deformation, where the rotor correlations lack the exponential tail with the length scale $\xi$. The theory also predicts that the rotor correlations may have small power-law tails. We directly observe the exponential decay, characterized by $\xi$, in numerical studies of a rigidity percolation model and in molecular dynamics simulations of amorphous polystyrene and the Lennard-Jones glass. The latter example also confirms the existence of the power-law tail in the rotor correlation function at large distances.

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.

Referee Report

2 major / 2 minor

Summary. The manuscript claims that in strongly disordered materials the nonaffine displacement field u_na exhibits predominantly power-law spatial decay, yet the correlation functions of its divergence and (for most strain deformations) its rotor display large-scale exponential decay governed by a heterogeneity length scale ξ that is set by the disorder strength and can become arbitrarily large, exceeding the structural correlation length. This is derived via the theory of correlated random matrices; an exception is noted for volumetric deformation where rotor correlations lack the ξ-governed exponential tail, and small power-law tails are predicted for the rotor. The analytic results are supported by direct numerical observation in a rigidity-percolation model and in molecular-dynamics simulations of amorphous polystyrene and the Lennard-Jones glass.

Significance. If the central claim is substantiated, the work would be significant for the mechanics of disordered solids: it identifies a tunable, disorder-controlled length scale that governs exponential elastic correlations far beyond structural scales, potentially unifying observations across glasses and networks. The combination of an analytic route from correlated random-matrix theory with multi-model numerical confirmation (including the reported power-law tail) constitutes a clear strength and supplies falsifiable predictions for future experiments.

major comments (2)
  1. [Theory section (correlated-random-matrix construction)] The modeling step that treats the nonaffine field as a correlated random matrix whose spatial derivatives directly produce the observed exponential decay with ξ is load-bearing for the claim that ξ is set by disorder strength rather than fitted. The manuscript does not explicitly derive the required matrix-element statistics from the microscopic definition u_na = −K^{−1} f_affine (K the Hessian) under applied strain; mechanical-equilibrium constraints imposed by K may alter the correlation structure when ξ becomes large.
  2. [Numerical results (rigidity-percolation and MD sections)] In the numerical sections, the extracted values of ξ are reported without quantitative error bars, finite-size scaling analysis, or systematic variation of system size at fixed disorder strength. This weakens the assertion that ξ can become 'indefinitely large' and that the exponential form persists in the thermodynamic limit.
minor comments (2)
  1. [Abstract] The abstract states that the rotor correlations 'may have small power-law tails' but does not indicate whether this tail is observed in all three numerical models or only in the Lennard-Jones glass.
  2. [Theory section] Notation for the divergence and rotor operators is introduced without an explicit definition in terms of the displacement components; a short equation would remove ambiguity.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation of the work's significance and for the constructive major comments. We address each point below and indicate planned revisions to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Theory section (correlated-random-matrix construction)] The modeling step that treats the nonaffine field as a correlated random matrix whose spatial derivatives directly produce the observed exponential decay with ξ is load-bearing for the claim that ξ is set by disorder strength rather than fitted. The manuscript does not explicitly derive the required matrix-element statistics from the microscopic definition u_na = −K^{−1} f_affine (K the Hessian) under applied strain; mechanical-equilibrium constraints imposed by K may alter the correlation structure when ξ becomes large.

    Authors: We thank the referee for identifying this foundational aspect. The correlated random-matrix model is constructed to reproduce the known statistical properties of the Hessian in strongly disordered systems, with off-diagonal correlations set by the disorder strength that directly determine ξ. While the current manuscript motivates this from the general form of u_na rather than deriving every matrix-element statistic step-by-step from the inverse-Hessian expression, the leading exponential decay of the derivative correlations follows from the assumed correlation kernel of the random matrix. We will add an explicit derivation subsection linking the microscopic u_na = −K^{−1} f_affine to the required matrix statistics under strong disorder, and we will show that the mechanical-equilibrium constraints (already built into the nonaffine definition) preserve the exponential form for the divergence and rotor correlations even as ξ grows large. This revision will be incorporated. revision: yes

  2. Referee: [Numerical results (rigidity-percolation and MD sections)] In the numerical sections, the extracted values of ξ are reported without quantitative error bars, finite-size scaling analysis, or systematic variation of system size at fixed disorder strength. This weakens the assertion that ξ can become 'indefinitely large' and that the exponential form persists in the thermodynamic limit.

    Authors: We agree that error bars and finite-size checks would make the numerical support more robust. In the revised manuscript we will report quantitative error bars on all fitted ξ values, obtained from ensemble averages over independent realizations. For the rigidity-percolation model we will add a finite-size scaling study at fixed disorder strength, confirming that the exponential decay persists and that ξ grows with decreasing disorder as predicted analytically. For the MD simulations we will include a discussion of the system sizes employed and argue that the observed scale separation supports the thermodynamic-limit behavior; full additional scaling runs for the MD cases are computationally demanding but the analytic result that ξ diverges with disorder strength independently establishes that ξ can become arbitrarily large. We will therefore make a partial revision that strengthens the numerical sections while retaining the analytic foundation for the indefinite growth claim. revision: partial

Circularity Check

0 steps flagged

No significant circularity in derivation chain

full rationale

The paper applies the theory of correlated random matrices to the nonaffine displacement field obtained from the Hessian inverse under affine strain, deriving that spatial derivatives (divergence and rotor) exhibit exponential correlations governed by a heterogeneity length ξ set by disorder strength. This is supported by direct numerical observation in rigidity percolation and MD simulations of glasses. No quoted step reduces a claimed prediction to a fitted parameter by construction, nor does any load-bearing premise collapse to a self-citation chain or self-definition; the random-matrix treatment supplies an independent modeling route whose outputs are then checked against microscopic simulations rather than being tautological with the inputs. The derivation remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on the applicability of correlated random matrix theory to the nonaffine field and on the background observation that the displacement field itself decays as a power law; ξ is introduced as the scale set by disorder strength rather than as a free fit parameter.

axioms (1)
  • domain assumption The nonaffine displacement field predominantly exhibits power-law decay
    Invoked in the abstract as the contrasting behavior before focusing on derivatives.

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