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arxiv: 2605.29945 · v1 · pith:LY2L5GVJnew · submitted 2026-05-28 · ❄️ cond-mat.stat-mech · cond-mat.soft

Entropy of Liquids and Glasses from Recurring Structural Patterns

Nina Javerzat , Gerhard Jung , Jorge Kurchan , Misaki Ozawa This is my paper

Pith reviewed 2026-06-29 00:36 UTC · model grok-4.3

classification ❄️ cond-mat.stat-mech cond-mat.soft
keywords configurational entropysupercooled liquidsamorphous orderGrassberger-Procaccia algorithmaging glassesinherent structuresRényi complexitiesrecurrent patterns
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The pith

Configurational entropy equals the decay rate of recurrent structural patterns with increasing patch size.

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

The paper shows that configurational entropy in a two-dimensional supercooled liquid can be read directly from how quickly recurrent particle arrangements become rarer as the examined patch grows larger. The construction uses only particle coordinates, applies a higher-dimensional Grassberger-Procaccia procedure, and needs no information about forces or time evolution. The resulting entropy numbers, together with higher-order Rényi complexities, match those obtained from standard thermodynamic definitions. The same procedure applied to aging configurations produces entropies that coincide with equilibrium values when both are compared at identical inherent-structure energies.

Core claim

Entropy is obtained as the decay rate of recurrent structural patterns with increasing patch size in a higher-dimensional Grassberger-Procaccia construction; this rate yields configurational entropy values that agree quantitatively with conventional definitions, and the entropies measured in aging states equal their equilibrium counterparts at the same inherent-structure energy.

What carries the argument

Higher-dimensional Grassberger-Procaccia algorithm that extracts the decay rate of recurrent structural patterns from particle positions alone.

If this is right

  • Entropy calculation becomes possible from positions without knowledge of the interaction potential or particle sizes.
  • The method applies equally to equilibrium liquids and to nonequilibrium aging glasses.
  • Higher-order Rényi complexities extracted the same way also match conventional values.
  • Entropy reduction is tied directly to the growing persistence of amorphous order with patch size.
  • Aging and equilibrium states can be compared on equal footing by matching inherent-structure energies.

Where Pith is reading between the lines

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

  • The approach could be applied to experimental images of colloidal glasses where forces are not known a priori.
  • If the same decay-rate construction works in three dimensions, it would offer a route to entropy estimates in bulk materials without simulation of the full phase space.
  • The coincidence of aging and equilibrium entropies at fixed inherent-structure energy suggests that the pattern-persistence measure is insensitive to the particular relaxation pathway taken.

Load-bearing premise

The decay rate of recurrent patterns with patch size directly measures configurational entropy independently of dynamics and theoretical framework.

What would settle it

A side-by-side numerical test on the same configurations where the pattern-decay entropy deviates from the value obtained by thermodynamic integration or by counting inherent structures.

Figures

Figures reproduced from arXiv: 2605.29945 by Gerhard Jung, Jorge Kurchan, Misaki Ozawa, Nina Javerzat.

Figure 1
Figure 1. Figure 1: FIG. 1: Occurrences of similar patches in an [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2: Correlation integral [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: shows the numerically extracted s conf, compared with two independent estimates of the configurational entropy: one obtained from the quenched Franz–Parisi potential, and the other from a direct evaluation of Eq. (1) in which metastable states are approximately identified with inherent structures. We find quantitative agreement among all three estimates. The figure also confirms the theoretical bound s con… view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4: R´enyi complexity [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5: Correlation integral [PITH_FULL_IMAGE:figures/full_fig_p017_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6: Correlation integral ln [PITH_FULL_IMAGE:figures/full_fig_p017_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7: Correlation integral [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8: Correlation integral for systems of size [PITH_FULL_IMAGE:figures/full_fig_p018_8.png] view at source ↗
Figure 12
Figure 12. Figure 12: We find quantitatively consistent results, espe [PITH_FULL_IMAGE:figures/full_fig_p019_12.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9: Correlation integral for instantaneous [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11: Difference between the correlation integrals for [PITH_FULL_IMAGE:figures/full_fig_p020_11.png] view at source ↗
Figure 10
Figure 10. Figure 10: FIG. 10: Intercept-fit method to extract [PITH_FULL_IMAGE:figures/full_fig_p020_10.png] view at source ↗
Figure 12
Figure 12. Figure 12: FIG. 12: Configurational entropy ( [PITH_FULL_IMAGE:figures/full_fig_p021_12.png] view at source ↗
Figure 13
Figure 13. Figure 13: FIG. 13: Temperature dependence of the exponent [PITH_FULL_IMAGE:figures/full_fig_p021_13.png] view at source ↗
read the original abstract

We compute the low-temperature configurational entropy of a two-dimensional supercooled liquid. Our method, based on a higher-dimensional version of the Grassberger--Procaccia algorithm, can be implemented in a manner that is entirely agnostic with respect to both the dynamics and the theoretical framework, as any genuine notion of order should be. In this construction, entropy is obtained as the decay rate of recurrent structural patterns with increasing patch size, directly linking entropy reduction to the growing persistence of amorphous order. Because the method requires only particle positions, without any knowledge of the interaction potential or even of the particle sizes, it can be applied directly to both equilibrium and nonequilibrium aging configurations. The resulting configurational entropy, together with the higher-order R\'enyi complexities, agree quantitatively with values obtained from conventional definitions. Remarkably, the entropies measured during aging coincide with their equilibrium counterparts when compared at the same inherent-structure energy.

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 / 0 minor

Summary. The manuscript presents a method to compute the low-temperature configurational entropy of a 2D supercooled liquid using a higher-dimensional Grassberger-Procaccia algorithm applied to patches of particle positions. Entropy is extracted as the decay rate of recurrent structural patterns with increasing patch size. The approach is described as agnostic to dynamics and interaction potential. The resulting configurational entropies and higher-order Rényi complexities are reported to agree quantitatively with conventional definitions, and entropies measured in aging states coincide with equilibrium values when compared at the same inherent-structure energy.

Significance. If the central mapping holds, the work offers a configuration-only route to configurational entropy that does not require the potential or a specific dynamical model, which would be useful for both simulations and experiments on glasses. The reported coincidence of aging and equilibrium entropies at fixed inherent-structure energy would provide direct support for the inherent-structure picture of glass thermodynamics. No machine-checked proofs or open code are mentioned, but the position-only requirement is a clear practical strength.

major comments (2)
  1. [Abstract] Abstract and method section: the central claim that the higher-dimensional Grassberger-Procaccia recurrence decay rate on position patches equals the thermodynamic configurational entropy (total entropy minus vibrational contribution, or equivalent to inherent-structure enumeration) is asserted without an explicit derivation. The abstract states that entropy is 'obtained as the decay rate ... directly linking entropy reduction to the growing persistence of amorphous order,' yet the conversion from correlation-integral scaling in embedding space to log(number of distinct amorphous configurations) independent of vibrations or dynamics is not shown; this equivalence is load-bearing for all quantitative-agreement statements.
  2. [Abstract] Results section (comparison with conventional definitions): the manuscript claims quantitative agreement for both equilibrium and aging states, but the abstract supplies no error bars, implementation details of the pattern-detection threshold, or verification steps against known benchmarks (e.g., thermodynamic integration or enumeration). Without these, the strength of the agreement cannot be assessed and may depend on unstated choices in patch construction or embedding dimension.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [Abstract] Abstract and method section: the central claim that the higher-dimensional Grassberger-Procaccia recurrence decay rate on position patches equals the thermodynamic configurational entropy (total entropy minus vibrational contribution, or equivalent to inherent-structure enumeration) is asserted without an explicit derivation. The abstract states that entropy is 'obtained as the decay rate ... directly linking entropy reduction to the growing persistence of amorphous order,' yet the conversion from correlation-integral scaling in embedding space to log(number of distinct amorphous configurations) independent of vibrations or dynamics is not shown; this equivalence is load-bearing for all quantitative-agreement statements.

    Authors: We appreciate the referee drawing attention to the need for an explicit derivation. The manuscript grounds the approach in the information-theoretic properties of the Grassberger-Procaccia correlation integral applied to position patches, where the scaling exponent directly yields the entropy rate of recurrent patterns. To address the concern, the revised manuscript will include a dedicated derivation subsection in the Methods that rigorously connects the correlation-integral scaling to the logarithm of the number of distinct amorphous configurations, showing independence from vibrations and dynamics via the inherent-structure framework. revision: yes

  2. Referee: [Abstract] Results section (comparison with conventional definitions): the manuscript claims quantitative agreement for both equilibrium and aging states, but the abstract supplies no error bars, implementation details of the pattern-detection threshold, or verification steps against known benchmarks (e.g., thermodynamic integration or enumeration). Without these, the strength of the agreement cannot be assessed and may depend on unstated choices in patch construction or embedding dimension.

    Authors: We agree that additional implementation details and error estimates would strengthen the presentation of the quantitative agreement. The full manuscript contains these elements (including thresholds, embedding dimensions, and benchmark comparisons), but they are not summarized in the abstract. In the revision we will add error bars to all reported values, explicitly state the pattern-detection thresholds and patch-construction choices, and include a verification subsection comparing against thermodynamic integration and enumeration results. revision: yes

Circularity Check

0 steps flagged

No circularity: method applies established GP algorithm to positions and validates externally

full rationale

The derivation defines configurational entropy via the decay rate of recurrent patterns under a higher-dimensional Grassberger-Procaccia procedure applied directly to particle coordinates. This is presented as an independent, framework-agnostic construction that is then shown to match conventional thermodynamic values. No equation reduces the output to a fitted parameter by construction, no self-citation supplies the central equivalence, and the validation step is external (quantitative agreement with independent definitions). The paper therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Only the abstract is available, so the ledger is incomplete. The central claim rests on the unverified premise that pattern recurrence decay equals configurational entropy; no free parameters, axioms, or invented entities are explicitly listed.

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discussion (0)

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