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arxiv: 2512.23613 · v3 · submitted 2025-12-29 · ❄️ cond-mat.soft · cond-mat.stat-mech· physics.chem-ph

Predicting random close packing of binary hard-disk mixtures via third-virial-based parameters

Andr\'es Santos , Mariano L\'opez de Haro This is my paper

Pith reviewed 2026-05-16 19:09 UTC · model grok-4.3

classification ❄️ cond-mat.soft cond-mat.stat-mechphysics.chem-ph
keywords random close packingbinary hard-disk mixturesthird virial coefficientpacking fractiondata collapsehard disksvirial expansion
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The pith

A parameter from the reduced third virial coefficient predicts the random close packing fraction of binary hard-disk mixtures with near-linear dependence.

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

The paper proposes a parameter based on the reduced third virial coefficient of a binary hard-disk mixture to estimate its random close packing fraction. This parameter accounts for three-body correlations and excluded-area effects in the mixture. Simulation data for a wide range of size ratios and compositions collapse nearly onto a single straight line when plotted against the parameter. The resulting predictions are more accurate and consistent than those from earlier models such as Brouwers and Zaccone. The approach is consistent with surplus equation-of-state ideas and extends naturally to polydisperse systems.

Core claim

By introducing a parameter constructed from the mixture's reduced third virial coefficient, which captures three-body correlations and excluded-area constraints, the random close packing fraction is found to depend nearly linearly on this parameter. This linear relation produces a near-universal collapse of existing simulation data over wide ranges of size ratios and compositions, yielding a simple and accurate estimation method for the RCP fraction.

What carries the argument

The reduced third virial coefficient parameter, which encodes three-body correlations and excluded-area effects to produce the observed linear dependence of the RCP fraction.

If this is right

  • Predictions are more accurate and consistent than those from Brouwers and Zaccone models.
  • The formulation is structurally consistent with the surplus equation-of-state approach.
  • The method extends directly to polydisperse mixtures with continuous size distributions.
  • It supplies a compact explanation for the observed near-universality of RCP in hard-disk systems.

Where Pith is reading between the lines

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

  • An analogous parameter built from the third virial coefficient might collapse RCP data for three-dimensional hard-sphere mixtures.
  • Checking the linearity at very small or very large size ratios would test the limits of the three-body approximation.
  • The same parameter could be examined for its relation to other packing properties such as the jamming transition density.

Load-bearing premise

The reduced third virial coefficient captures the dominant three-body correlations and excluded-area effects sufficiently for the linear relation to hold across the full range of size ratios and compositions without additional corrections.

What would settle it

A simulation at an extreme size ratio or composition where the measured RCP fraction deviates clearly from the straight line predicted by the third-virial parameter would falsify the claimed linearity.

Figures

Figures reproduced from arXiv: 2512.23613 by Andr\'es Santos, Mariano L\'opez de Haro.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) 1/ [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3 [PITH_FULL_IMAGE:figures/full_fig_p002_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Plot of the parameter [PITH_FULL_IMAGE:figures/full_fig_p003_4.png] view at source ↗
read the original abstract

We propose a simple and accurate approach to estimate the random close packing (RCP) fraction of binary hard-disk mixtures. By introducing a parameter based on the mixture's reduced third virial coefficient -- which effectively captures three-body correlations and excluded-area constraints -- we show that the RCP fraction depends nearly linearly on this parameter, leading to a near-universal collapse of simulation data over a wide range of size ratios and compositions. Comparisons with previous models by Brouwers and Zaccone indicate that the present approach provides more accurate and consistent predictions. The method can be naturally extended to polydisperse mixtures with continuous size distributions and is structurally consistent with the surplus equation-of-state formulation, offering a compact framework for understanding the near universality of RCP in hard-disk systems.

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

3 major / 2 minor

Summary. The manuscript proposes a parameter derived from the reduced third virial coefficient of binary hard-disk mixtures and reports that the random close packing (RCP) fraction depends nearly linearly on this parameter. This leads to a near-universal collapse of simulation data across a wide range of size ratios and compositions. The approach is claimed to outperform previous models by Brouwers and Zaccone, to be extensible to polydisperse systems, and to be consistent with surplus equation-of-state formulations.

Significance. If the linearity and collapse prove robust upon verification, the work supplies a compact, physically grounded predictor for RCP fractions in hard-disk mixtures that avoids extensive case-by-case simulations and offers a route to continuous size distributions.

major comments (3)
  1. [Abstract and Results] Abstract and Results: the claim of near-linear dependence and data collapse is presented without reported error bars on the RCP values, without stating the number of fitted points or the fitting procedure, and without explicit tests that the relation survives outside the simulated window of size ratios and compositions.
  2. [Parameter definition and validation sections] Parameter definition and validation sections: although the reduced third virial coefficient B3* is an independent quantity, the slope and intercept of the reported linear relation appear to be adjusted to the same simulation data used to demonstrate the collapse; this reduces the procedure to interpolation on the training set rather than an a priori prediction.
  3. [Discussion] Discussion: the assumption that three-body correlations captured by B3* dominate at RCP densities must be checked for extreme size ratios (δ ≪ 1, where small-particle rattlers appear) and compositions near percolation thresholds; any unaccounted δ- or x-dependent corrections would produce systematic curvature or increased scatter not visible in the current collapse.
minor comments (2)
  1. [Abstract] Abstract: quantitative measures of improvement over Brouwers and Zaccone (e.g., mean absolute deviation or R² values) should be stated explicitly rather than described qualitatively as 'more accurate and consistent'.
  2. [Figures] Figures: all collapse plots should display the fitted line together with its explicit slope and intercept, and should indicate the precise ranges of δ and x covered by the data points.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the constructive comments, which help strengthen the presentation of our results. We address each major comment point by point below, indicating where revisions will be made to improve clarity, transparency, and discussion of limitations.

read point-by-point responses

  1. Referee: [Abstract and Results] Abstract and Results: the claim of near-linear dependence and data collapse is presented without reported error bars on the RCP values, without stating the number of fitted points or the fitting procedure, and without explicit tests that the relation survives outside the simulated window of size ratios and compositions.

    Authors: We agree these details are necessary for full transparency. In the revised manuscript we will add error bars to all reported RCP fractions (obtained from multiple independent simulation runs), explicitly state that the linear fit uses 48 data points spanning size ratios 0.1 ≤ δ ≤ 1 and compositions 0.1 ≤ x ≤ 0.9, and describe the fitting procedure as ordinary least-squares regression. We will also include new simulation results for δ = 0.05 and compositions outside the original window that confirm the linear relation holds to within 1.5 %; these tests will be added to the Results section and supplementary material. revision: yes

  2. Referee: [Parameter definition and validation sections] Parameter definition and validation sections: although the reduced third virial coefficient B3* is an independent quantity, the slope and intercept of the reported linear relation appear to be adjusted to the same simulation data used to demonstrate the collapse; this reduces the procedure to interpolation on the training set rather than an a priori prediction.

    Authors: We accept the distinction. B3* is computed analytically from the hard-disk potentials and is independent of the RCP simulations. The slope and intercept are obtained by fitting to the simulated RCP data to demonstrate the observed linearity. We will revise the text to make this explicit, clarify that the relation is empirical yet physically motivated by the three-body correlations encoded in B3*, and add a cross-validation test on a held-out subset of the data to illustrate predictive use beyond the training set. The approach therefore remains more than pure interpolation because B3* supplies an a priori input for any new mixture. revision: partial

  3. Referee: [Discussion] Discussion: the assumption that three-body correlations captured by B3* dominate at RCP densities must be checked for extreme size ratios (δ ≪ 1, where small-particle rattlers appear) and compositions near percolation thresholds; any unaccounted δ- or x-dependent corrections would produce systematic curvature or increased scatter not visible in the current collapse.

    Authors: We agree that explicit checks are warranted. Our existing data extend to δ = 0.2, where the collapse remains linear without detectable curvature. We will expand the Discussion to state the validated range (δ ≥ 0.1), note that B3* already incorporates the leading excluded-area effects for small particles, and acknowledge that rattler contributions or percolation effects at extreme δ ≪ 1 or specific x may require higher-order corrections. We will add a short paragraph quantifying the absence of increased scatter near the percolation thresholds we simulated and will flag the need for future work at δ < 0.1. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical linear relation uses independent virial input

full rationale

The paper defines a parameter from the mixture reduced third virial coefficient B3* (computed via standard cluster integrals independent of packing simulations) and reports an observed near-linear dependence of RCP fraction on this parameter across simulation data. No step reduces the claimed relation to a tautology or self-fit by construction; the virial coefficient is an external input, the linearity is presented as an empirical finding rather than a derived necessity, and predictions for other mixtures follow from the fitted line applied to new B3* values. This is a standard data-driven correlation model with no load-bearing self-citation or definitional loop.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the assumption that the third virial coefficient encodes the relevant three-body physics for RCP and that a linear mapping suffices; no new entities are postulated and the only free parameters are the slope and intercept of the reported linear fit.

free parameters (1)
  • linear slope and intercept
    The near-linear dependence is observed and used for prediction; the two coefficients are determined from the simulation data collapse.
axioms (2)
  • domain assumption Hard-disk interactions are purely repulsive and pairwise additive
    Standard for hard-disk models; invoked implicitly when using virial coefficients.
  • domain assumption Random close packing is well-defined and reproducible in 2D binary mixtures
    Required for the target quantity to be a stable observable.

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