Effective-medium theory for elastic systems with correlated disorder

Danilo B. Liarte; Jorge M. Escobar-Agudelo; Rui Aquino

arxiv: 2510.02090 · v1 · submitted 2025-10-02 · ❄️ cond-mat.stat-mech · cond-mat.dis-nn· cond-mat.soft

Effective-medium theory for elastic systems with correlated disorder

Jorge M. Escobar-Agudelo , Rui Aquino , Danilo B. Liarte This is my paper

Pith reviewed 2026-05-18 10:21 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.dis-nncond-mat.soft

keywords effective-medium theorycoherent potential approximationcorrelated disorderelastic networksrigidity percolationcolloidal gels

0 comments

The pith

A generalized coherent potential approximation incorporates spatial correlations into effective-medium theory for elastic networks.

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

The paper develops a multi-purpose generalization of the coherent potential approximation to handle spatially correlated disorder in elastic systems. Traditional effective-medium approaches lacked tools for such correlations, which appear in colloidal gels, bio-polymer networks, and suspensions near jamming. The new framework folds correlations into the self-consistent calculation of an effective medium. When tested on a rigidity-percolation model for gels, it shows that correlations lower the critical packing fraction while the critical coordination number stays exactly isostatic. The method is presented as applicable to many other disordered elastic materials.

Core claim

We propose a multi-purpose generalization of the coherent potential approximation to describe the mechanical behavior of elastic networks with spatially-correlated disorder. We apply our theory to a simple rigidity-percolation model for colloidal gels and study the effects of correlations in both the critical point and the overall scaling behavior. We find that although the presence of spatial correlations (mimicking attractive interactions of gels) shifts the critical packing fraction to lower values, suggesting sub-isostatic behavior, the critical coordination number of the associated network remains isostatic.

What carries the argument

A generalization of the coherent potential approximation that folds spatial correlations of the disorder into the self-consistent effective-medium equations.

If this is right

Correlations that mimic attractions move the rigidity onset to lower packing fractions.
The critical coordination number remains isostatic regardless of correlation strength.
Scaling of elastic moduli near the transition can be extracted for correlated cases.
The same framework applies to bio-polymer networks and suspensions near shear jamming.

Where Pith is reading between the lines

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

The method could be used to tune material rigidity by controlling the length scale of attractive correlations during fabrication.
Extensions to time-dependent or finite-frequency mechanical responses would connect to viscoelastic measurements in gels.
The preserved isostaticity suggests a deeper structural reason that might appear in other effective-medium treatments of jamming.

Load-bearing premise

Spatial correlations can be built into the self-consistent effective-medium framework without invalidating the core approximation or introducing new uncontrolled approximations for the disorder statistics.

What would settle it

Direct comparison of the theory's predicted critical packing fraction versus correlation strength against numerical simulations of the rigidity-percolation model on finite networks.

Figures

Figures reproduced from arXiv: 2510.02090 by Danilo B. Liarte, Jorge M. Escobar-Agudelo, Rui Aquino.

**Figure 1.** Figure 1: FIG. 1. Schematic representation of the Coherent Potential Approximation. (a) Regular networks in which bonds (springs) are [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗

**Figure 2.** Figure 2: FIG. 2. Contrast between correlated and uncorrelated dis [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗

**Figure 3.** Figure 3: FIG. 3. Rigidity-percolation model for gels in the CCPA [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

**Figure 5.** Figure 5: FIG. 5. Scaling collapse plot in terms of [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗

**Figure 4.** Figure 4: FIG. 4. (a) Effective elastic constant [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗

read the original abstract

Correlated structures are intimately connected to intriguing phenomena exhibited by a variety of disordered systems such as soft colloidal gels, bio-polymer networks and colloidal suspensions near a shear jamming transition. The universal critical behavior of these systems near the onset of rigidity is often described by traditional approaches as the coherent potential approximation - a versatile version of effective-medium theory that nevertheless have hitherto lacked key ingredients to describe disorder spatial correlations. Here we propose a multi-purpose generalization of the coherent potential approximation to describe the mechanical behavior of elastic networks with spatially-correlated disorder. We apply our theory to a simple rigidity-percolation model for colloidal gels and study the effects of correlations in both the critical point and the overall scaling behavior. We find that although the presence of spatial correlations (mimicking attractive interactions of gels) shifts the critical packing fraction to lower values, suggesting sub-isostatic behavior, the critical coordination number of the associated network remains isostatic. More importantly, we discuss how our theory can be employed to describe a large variety of systems with spatially-correlated disorder.

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

This extends CPA to spatial correlations in elastic networks and finds correlations lower the critical packing fraction while keeping isostatic coordination, but the approximation's range needs explicit checks.

read the letter

The core advance is a generalization of the coherent potential approximation that folds in the pair-correlation function of the disorder to handle spatial correlations in elastic systems. They apply it to a rigidity-percolation model meant to capture colloidal gels and report that correlations shift the critical packing fraction downward while the critical coordination number stays isostatic. That combination is the main concrete result and it lines up with the intuition that attractions can promote rigidity at lower densities without changing the local isostatic condition. The framework is presented as multi-purpose, which is useful if it really carries over to other systems like bio-polymer networks or shear-jammed suspensions. The derivations appear to replace the usual single-site self-consistency with an effective-medium equation that includes the correlation function, and they discuss how this affects both the critical point and scaling. That is a clear step beyond standard CPA, which the abstract correctly notes lacked this ingredient. The numerics or analytic checks they provide for the percolation model seem to support the shift in packing fraction without breaking isostaticity. The main soft spot is the one the stress-test flags: when correlations become long-ranged, the assumption that a correlated defect is still screened by the same effective medium may not hold, and higher-order cluster effects could enter. The paper does not appear to derive a bound on the error or test against simulations for strong correlations, so the quantitative reliability outside the weak-correlation regime is not fully settled. This is aimed at soft-matter researchers who model gels, jamming, or disordered elastic networks and want a practical effective-medium tool. Readers already comfortable with CPA will see the extension quickly and can judge whether the uncontrolled approximations are acceptable for their systems. It is coherent on its own terms and engages the literature on rigidity percolation, so it deserves a serious referee who can press on the validity range and any numerical benchmarks.

Referee Report

2 major / 2 minor

Summary. The manuscript proposes a generalization of the coherent potential approximation (CPA) for elastic networks with spatially correlated disorder. The central construction replaces the single-site self-consistency condition with an effective-medium equation that incorporates the pair-correlation function of the disorder. Applied to a rigidity-percolation model for colloidal gels, the theory predicts that spatial correlations (mimicking attractive interactions) shift the critical packing fraction to lower values while the critical coordination number remains isostatic. The work presents the approach as a multi-purpose tool for a variety of disordered elastic systems.

Significance. If controlled, the generalization supplies a practical route to include spatial correlations within effective-medium theory, which is relevant for modeling rigidity onset in colloidal gels, biopolymer networks, and shear-jammed suspensions. The observation that isostaticity is preserved at the shifted critical point offers a concrete prediction that could be tested against simulations or experiments on attractive colloids. The multi-purpose framing, if substantiated, would extend CPA utility beyond uncorrelated disorder.

major comments (2)

[§III, Eq. (10)] §III, Eq. (10): the effective-medium self-consistency condition that folds in the pair-correlation function assumes local strain fluctuations remain screened by the same effective medium as in the uncorrelated CPA. No explicit error bound or validity criterion is derived for correlation lengths comparable to elastic wavelengths; this assumption is load-bearing for the quantitative reliability of the reported shift in critical packing fraction.
[§IV, Fig. 4] §IV, Fig. 4 and surrounding text: the critical packing fractions and coordination numbers obtained for varying correlation strengths are shown without direct comparison to independent Monte Carlo or molecular-dynamics simulations of the same correlated network model. This omission leaves open whether the predicted sub-isostatic shift remains accurate beyond the weak-correlation regime.

minor comments (2)

[Abstract] The abstract states that correlations suggest 'sub-isostatic behavior' while simultaneously reporting an isostatic critical coordination number; a brief clarifying sentence would avoid potential misreading.
[§II] Notation for the effective Lamé parameters and the disorder correlation function is introduced piecemeal; a consolidated table of symbols would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below and indicate the changes we will make to the manuscript.

read point-by-point responses

Referee: [§III, Eq. (10)] §III, Eq. (10): the effective-medium self-consistency condition that folds in the pair-correlation function assumes local strain fluctuations remain screened by the same effective medium as in the uncorrelated CPA. No explicit error bound or validity criterion is derived for correlation lengths comparable to elastic wavelengths; this assumption is load-bearing for the quantitative reliability of the reported shift in critical packing fraction.

Authors: We agree that the screening assumption underlying Eq. (10) requires explicit discussion. This assumption is the direct extension of the standard CPA closure and is expected to remain accurate when the spatial correlation length of the disorder is short compared with the wavelength of the relevant elastic modes. Deriving a rigorous, quantitative error bound valid for arbitrary correlation lengths would require a separate, technically involved analysis that lies beyond the scope of the present work. In the revised manuscript we will add a new paragraph immediately after Eq. (10) that states the scaling argument for the validity regime and explicitly notes that the correlation lengths considered in our colloidal-gel application fall well inside this regime. revision: yes
Referee: [§IV, Fig. 4] §IV, Fig. 4 and surrounding text: the critical packing fractions and coordination numbers obtained for varying correlation strengths are shown without direct comparison to independent Monte Carlo or molecular-dynamics simulations of the same correlated network model. This omission leaves open whether the predicted sub-isostatic shift remains accurate beyond the weak-correlation regime.

Authors: We acknowledge that a direct numerical benchmark would be desirable. Generating statistically independent realizations of large correlated networks and performing finite-size scaling analysis near the rigidity threshold is, however, a substantial computational undertaking that is outside the scope of this theoretical paper. In the revised manuscript we will expand the discussion of Fig. 4 to include (i) explicit comparison of the zero-correlation limit with existing Monte Carlo results for uncorrelated rigidity percolation and (ii) qualitative consistency checks against published simulation trends for attractive colloidal gels. We view a full, dedicated simulation study of the correlated model as valuable future work rather than a prerequisite for the present analytic contribution. revision: partial

Circularity Check

0 steps flagged

CPA generalization for correlated disorder is an independent ansatz extension with no reduction to inputs by construction

full rationale

The paper extends the coherent potential approximation by incorporating the pair-correlation function of the disorder into the self-consistency condition for the effective medium. This is presented as a controlled approximation for elastic networks, and the reported shift in critical packing fraction (while preserving isostatic coordination) follows from solving the modified equations on a specific rigidity-percolation model. No equations reduce the output quantities to fitted inputs or prior self-citations by definition; the central construction is a new effective-medium equation rather than a renaming or tautological fit. The derivation is self-contained against the model's assumptions and does not rely on load-bearing self-citations or uniqueness theorems from the authors' prior work.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on extending the standard coherent potential approximation; the abstract implies standard assumptions of effective-medium theory plus a new modeling choice for incorporating spatial correlations into the self-consistency condition.

axioms (1)

domain assumption The effective-medium approximation remains valid when spatial correlations are introduced via a modified self-consistency condition.
Invoked implicitly when the authors state that the generalization describes mechanical behavior of networks with correlated disorder.

pith-pipeline@v0.9.0 · 5720 in / 1125 out tokens · 31663 ms · 2026-05-18T10:21:28.306013+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

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

the generalized self-consistent condition ⟨T⟩=0 yields ⟨F⟩=0 ... Hii = d/zNN
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

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

rigidity transition occurs at the Maxwell threshold zc=4

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

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[1] [1]

C. P. Broedersz and F. C. MacKintosh,Modeling semi- flexible polymer networks, Rev. Mod. Phys.86, 995 (2014)

work page 2014

[2] [2]

A. J. Liu and S. R. Nagel,Jamming is not just cool any more, Nature396, 21 (1998)

work page 1998

[3] [3]

A. J. Liu and S. R. Nagel,The jamming transition and the marginally jammed solid, Annu. Rev. Condens. Mat- ter Phys.1, 347 (2010)

work page 2010

[4] [4]

Wyart,On the rigidity of amorphous solids, Ann

M. Wyart,On the rigidity of amorphous solids, Ann. Phys. Fr.30, 1 (2005)

work page 2005

[5] [5]

D. Bi, J. Lopez, J. M. Schwarz, and M. L. Manning,A density-independent rigidity transition in biological tis- sues, Nature Physics11, 1074 (2015)

work page 2015

[6] [6]

A. V. Tkachenko and T. A. Witten,Stress propagation through frictionless granular material, Phys. Rev. E60, 687 (1999)

work page 1999

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S. F. Edwards and D. V. Grinev,Statistical mechanics of stress transmission in disordered granular arrays, Phys. Rev. Lett.82, 5397 (1999)

work page 1999

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Miguel, A

M.-C. Miguel, A. Vespignani, M. Zaiser, and S. Zapperi, Dislocation jamming and andrade creep, Phys. Rev. Lett. 89, 165501 (2002)

work page 2002

[9] [9]

Tsekenis, N

G. Tsekenis, N. Goldenfeld, and K. A. Dahmen,Disloca- tions jam at any density, Phys. Rev. Lett.106, 105501 (2011)

work page 2011

[10] [10]

D. B. Liarte, S. J. Thornton, E. Schwen, I. Cohen, D. Chowdhury, and J. P. Sethna,Universal scaling for disordered viscoelastic matter near the onset of rigidity, Phys. Rev. E106, L052601 (2022)

work page 2022

[11] [11]

S. J. Thornton, D. B. Liarte, P. Abbamonte, J. P. Sethna, and D. Chowdhury,Jamming and unusual charge density fluctuations of strange metals, Nature Communciations 14, 3919 (2023)

work page 2023

[12] [12]

V. K. de Souza and P. Harrowell,Rigidity Percolation and the spatial heterogeneity of soft modes in disordered ma- terials, Proc. Natl. Acad. Sci. U.S.A.106, 15136 (2009)

work page 2009

[13] [13]

Feng and P

S. Feng and P. N. Sen,Percolation on elastic networks: New exponent and threshold, Phys. Rev. Lett.52, 216 (1984)

work page 1984

[14] [14]

Souslov, A

A. Souslov, A. J. Liu, and T. C. Lubensky,Elasticity and response in nearly isostatic periodic lattices, Phys. Rev. Lett.103, 205503 (2009)

work page 2009

[15] [15]

Mao and T

X. Mao and T. C. Lubensky,Coherent potential approx- imation of random nearly isostatic kagome lattice, Phys. Rev. E83, 011111 (2011)

work page 2011

[16] [16]

C. P. Goodrich, A. J. Liu, and S. R. Nagel,The principle of independent bond-level response: Tuning by pruning to exploit disorder for global behavior, Phys. Rev. Lett.114, 225501 (2015)

work page 2015

[17] [17]

D. B. Liarte, X. Mao, O. Stenull, and T. C. Lubensky, Jamming as a multicritical point, Phys. Rev. Lett.122, 128006 (2019)

work page 2019

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S. Feng, M. F. Thorpe, and E. Garboczi,Effective- medium theory of percolation on central-force elastic net- works, Phys. Rev. B31, 276 (1985)

work page 1985

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M´ ezard, G

M. M´ ezard, G. Parisi, and M. A. Virasoro,Spin glass the- ory and beyond: An Introduction to the Replica Method and Its Applications, Vol. 9 (World Scientific Publishing Company, 1987)

work page 1987

[20] [20]

Zhang, L

S. Zhang, L. Zhang, M. Bouzid, D. Z. Rocklin, E. Del Gado, and X. Mao,Correlated rigidity percolation and colloidal gels, Phys. Rev. Lett.123, 058001 (2019)

work page 2019

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P. L. Leath,Self-consistent-field approximations in dis- ordered alloys, Phys. Rev.171, 725 (1968)

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D. J. Jacobs and M. F. Thorpe,Generic rigidity percola- tion: The pebble game, Phys. Rev. Lett.75, 4051 (1995)

work page 1995

[23] [23]

D. J. Jacobs and M. F. Thorpe,Generic rigidity percola- tion in two dimensions, Phys. Rev. E53, 3682 (1996)

work page 1996

[24] [24]

D. B. Liarte, O. Stenull, and T. C. Lubensky,Multi- functional twisted kagome lattices: Tuning by pruning mechanical metamaterials, Phys. Rev. E101, 063001 (2020)

work page 2020

[25] [25]

D¨ uring, E

G. D¨ uring, E. Lerner, and M. Wyart,Phonon gap and localization lengths in floppy materials, Soft Matter9, 146 (2013)

work page 2013

[26] [26]

M. G. Yucht, M. Sheinman, and C. P. Broedersz,Dynam- ical behavior of disordered spring networks, Soft Matter 9, 7000 (2013)

work page 2013

[27] [27]

M. Das, F. C. MacKintosh, and A. J. Levine,Effec- tive medium theory of semiflexible filamentous networks, Phys. Rev. Lett.99, 038101 (2007)

work page 2007

[28] [28]

X. Mao, O. Stenull, and T. C. Lubensky,Effective- 9 medium theory of a filamentous triangular lattice, Phys. Rev. E87, 042601 (2013)

work page 2013

[29] [29]

X. Mao, O. Stenull, and T. C. Lubensky,Elasticity of a filamentous kagome lattice, Phys. Rev. E87, 042602 (2013)

work page 2013

[30] [30]

D. B. Liarte, O. Stenull, X. Mao, and T. C. Lubensky, Elasticity of randomly diluted honeycomb and diamond lattices with bending forces, Journal of Physics: Con- densed Matter28, 165402 (2016)

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