Analytical solution for the polydisperse random close packing problem in 2D

Alessio Zaccone

REVIEW 3 major objections 3 minor 72 references

Analytical solution for the polydisperse random close packing problem in 2D

T0 review · 3 major / 3 minor · reviewed 2026-08-09 · deepseek-v4-flash

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Pith's one-line read The paper claims that the random close packing density of polydisperse hard disks is obtained by solving $C_0[Z(\phi_{\mathrm{RCP}})-1]=1$ with the polydisperse compressibility $Z$ from an effective-density equation of state, yielding…

desk verdict Useful first analytical 2D polydisperse RCP formula, but the central EOS transfer to jammed disks is unvalidated and one swap-MC point is too thin a benchmark. read the letter →

arxiv 2502.03354 v2 pith:SS7NCMHP submitted 2025-02-05 cond-mat.soft cond-mat.dis-nncond-mat.mtrl-scicond-mat.stat-mechphysics.chem-ph

Alessio Zaccone This is my paper · ORCID

classification cond-mat.softcond-mat.dis-nncond-mat.mtrl-scicond-mat.stat-mechphysics.chem-ph

keywords randomclosepackingpolydisperseharddiskseffectivefractionisostaticityequationofstatejammingsizepolydispersity2Ddiskpackings

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the densest disordered packing of hard disks with a spread of sizes can be computed analytically, without simulation, by solving a single algebraic equation. The equation, $C_0[Z(\phi_{\mathrm{RCP}})-1]=1$, combines the isostatic condition that each disk has on average four contacts with an equilibrium equation of state for polydisperse disks. The solution grows with the reduced standard deviation $s$ of the size distribution and saturates near $\phi_{\mathrm{RCP}}\approx 0.97$--$0.98$ at large $s$; at $s=0.25$ it gives $0.8924$, close to the recent numerical estimate $0.905$. If correct, the result turns a long-standing packing problem into a formula that depends only on the first three moments of the disk-size distribution, with consequences for granular monolayers, particle-stabilized interfaces, and two-dimensional materials.

What carries the argument

The load-bearing object is the effective-density equation of state for polydisperse hard disks, which maps a mixture at packing fraction $\phi$ onto an equivalent monodisperse fluid at $\phi_{\mathrm{eff}}=\phi/[\phi+\lambda(1-\phi)]$, with $\lambda=m_3/m_2^2$ determined by the dimensionless moments of the disk-size distribution. Its compressibility $Z(\phi)$, meaning the pressure divided by density in thermal units, enters the isostatic condition $C_0[Z(\phi_{\mathrm{RCP}})-1]=1$, obtained by writing the mean contact number as $z=4C_0[Z(\phi)-1]$ and setting $z=4$. The work this machinery does is to reduce the many-species contact problem to a one-species equation of state, so $\phi_{\mathrm{RCP}}$ becomes a function of the distribution's first three moments.

What would settle it

Simulate slow compressions or swap-based packings of log-normal polydisperse hard disks at several reduced standard deviations $s$ and compare the measured maximum disordered packing fraction with Eq. (13); a systematic gap that grows with $s$, or a non-monotone curve, would rule out the assumed mapping.

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Extended reading notes

Core claim

The central claim is that random close packing in two dimensions is the rigidity-onset density, where the mean contact number reaches the isostatic value $z=4$, and that this density is the solution of $C_0[Z(\phi_{\mathrm{RCP}})-1]=1$. Here $Z$ is the compressibility of the polydisperse disk fluid obtained from the effective packing fraction mapping $\phi_{\mathrm{eff}}=\phi/[\phi+\lambda(1-\phi)]$ with $\lambda=m_3/m_2^2$, and $C_0$ is fixed by the monodisperse triangular close packing at $\phi=\pi/\sqrt{12}\approx 0.9069$. The paper evaluates $C_0$ with three standard disk equations of state to show the prediction is robust, and compares the log-normal result at $s=0.25$ ($\phi_{\mathrm{RCP}}=0.8924$) with the recent numerical estimate $0.905$ from irreversible swap Monte Carlo. This is presented as the first analytical solution to the polydisperse random close packing problem for two-dimensional disks.

Load-bearing premise

The equilibrium pressure-density relation for polydisperse hard disks is assumed to remain valid for jammed, out-of-equilibrium packings near random close packing, a transfer that the paper does not test in two dimensions.

Editorial extensions

If this is right

The random close packing fraction of any polydisperse disk system can be estimated by solving one algebraic equation from the first three moments of its size distribution, bypassing expensive packing simulations.
At a fixed size spread $s=0.25$, the theory yields $\phi_{\mathrm{RCP}}=0.8924$, about 1.4 percent below the $0.905$ value from the latest irreversible swap Monte Carlo simulations.
Increasing polydispersity raises the predicted packing fraction monotonically toward a plateau near $\phi_{\mathrm{RCP}}\approx 0.97$--$0.98$, always below the geometric limit of 1.
The final formula is insensitive to which of the three standard disk equations of state is used at moderate polydispersity, so the prediction is not an artifact of a particular choice.
The same approach previously applied to three-dimensional polydisperse spheres now covers two-dimensional disk monolayers, connecting to interfacial, polymer, and biological packing problems.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the solution depends on the size distribution only through its first three moments, two very different distributions with the same mean, variance, and skewness would be predicted to have identical $\phi_{\mathrm{RCP}}$; this moment-collapse is a direct, testable consequence not highlighted in the paper.
The current numerical comparison is at a single value of $s$; a stronger test would be to simulate log-normal disks over a range of $s$ and compare the whole $\phi_{\mathrm{RCP}}(s)$ curve with Eq. (13).
If the equilibrium-to-jammed transfer holds, the same effective-density route could likely be applied to models with soft repulsive interactions at jamming, where isostaticity and the contact number are already known to control the mechanical response.
The predicted saturation near 0.97--0.98 suggests an upper bound for isostatic disordered disk packings; a natural extension is to ask whether any protocol that preserves strict disorder can exceed this plateau.

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Editorial analysis

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Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper gives the first analytical curve for phi_RCP as a function of polydispersity in 2D hard disks. It combines Zaccone's earlier crowding relation, z = 4C0[Z(phi)-1], with Santos's equilibrium polydisperse equation of state, then sets z=4 at isostaticity. The derivation is transparent and the algebra checks out. The scaling constant C0 is pinned at the exact triangular close-packing point rather than fitted to RCP data, which is a plus. The monodisperse limit lands near 0.886, inside the broad 0.81-0.89 range quoted in the literature, though that range is so wide it does not provide a sharp test.

The real soft spot is the transfer. Equation (8) assumes the jammed contact value is proportional to the equilibrium virial contact value. That proportionality is imported from the author's 3D work and is not independently checked for 2D jammed disks. The Santos EOS used for the polydisperse mixture is an equilibrium high-density mapping, and applying it to a jammed state with a singular contact peak is an extrapolation. The single comparison with swap Monte Carlo at s=0.25 is not a strong test: the authors themselves note those packings are not necessarily isostatic, so the agreement between 0.892 and 0.905 could easily be partly fortuitous. I would want at least one more polydispersity benchmark before believing the full curve, especially the saturation plateau near 0.97-0.98.

There are also smaller issues. The abstract calls the size distribution a power law, but the text uses a log-normal. Equation (12) as printed is ambiguous; the denominator structure is not clear. These are fixable but suggest the manuscript was not checked carefully enough.

Candidly, the central argument is coherent on its own terms. The paper does not oversell: it states the proportionality is an assumption and flags the non-isostatic nature of the swap packings. So this is a real theoretical proposal, not a hand-wave. I would send it to peer review rather than desk-reject. A good referee should push for either a direct 2D numerical check of Eq. (8) or a much more explicit caveat, and for correction of the abstract and Eq. (12).

Who benefits? People who need a quick analytical estimate for polydisperse monolayers, colloidal films, or granular packing, and who accept the equilibrium-to-jammed transfer as a working approximation. For my own work, I would not cite it as a validated result yet, but I would mention it as a testable prediction. It deserves a serious referee.

Referee Report

3 major / 3 minor

Summary. The manuscript proposes an analytical route to the random close packing density of polydisperse hard disks. It combines a jammed-state contact relation, z = 4 C0 [Z(phi) - 1], with equilibrium hard-disk equations of state, after calibrating C0 at the monodisperse triangular close-packing point. Polydispersity is introduced through the Santos effective-packing-fraction EOS, Eq. (12). Solving C0[Z(phi_RCP) - 1] = 1 yields phi_RCP as a function of the width of the size distribution; the result is compared with swap-Monte-Carlo data at s = 0.25, giving phi_RCP = 0.8924 versus 0.905.

Significance. If its assumptions held, this would be the first closed-form analytical formula for the 2D polydisperse random close packing fraction, with potential applications to granular monolayers, interfacial assemblies, and composite materials. The derivation is concise, and the s = 0.25 comparison is an external benchmark rather than a fit, because C0 is fixed at the monodisperse close-packing point. However, the two central transfers—equilibrium fluid equation of state to jammed packings, and monodisperse calibration to arbitrary polydispersity—are not validated in 2D, and Eq. (12) as printed does not reduce to the monodisperse equation of state. The paper therefore cannot be accepted without substantial revision.

major comments (3)

[Section 'To extend the EOS', Eq. (12)] Equation (12) as written is internally inconsistent: for a monodisperse distribution one has lambda = 1, alpha = 1, and phi_eff = phi, so the equation reduces to Z = (1 + phi) + phi^2 Z_s(phi), not to Z_s(phi). The Santos FMT mapping has the structure Z = 1/(1 - phi) + alpha [ Z_s(phi_eff) - 1/(1 - phi_eff) ], so Eq. (12) appears to contain an erroneous factor phi and to be missing the division by phi_eff (or an equivalent rearrangement). Since Eq. (13) and all polydisperse numerical values, including phi_RCP = 0.8924 and the curves in Fig. 1, are computed from Eq. (12), the polydisperse predictions must be recomputed with the correct EOS before the central claim can be assessed.
[Section 'EOS', Eq. (7)] The proportionality in Eq. (7) is the pivot of the theory: it takes the equilibrium virial contact value, (Z - 1)/(2 phi), and asserts that the jammed contact value g(sigma; phi) is proportional to it. The paper justifies this by Ref. [10] for 3D hard spheres, but provides no 2D numerical or theoretical check. In a jammed disk packing near RCP, the contact peak is singular and the pressure is not given by the equilibrium virial expression; C0 absorbs the proportionality factor at one density, but the assumed functional form z proportional to Z(phi) - 1 across all densities and polydispersities is unchecked. This makes the entire phi_RCP(s) curve in Fig. 1 conditional on an unvalidated analogy.
[Section 'Comparison with Ref. [69]'] The only quantitative validation of the polydisperse theory is a single point at s = 0.25, where the paper predicts phi_RCP = 0.8924 against the swap-MC value 0.905. The paper itself states that the packings in Ref. [69] are not necessarily isostatic, i.e. z may exceed 4, so the simulated phi_RCP may be set by a different criterion than z = 4. With one point and an acknowledged mismatch in the defining condition, the agreement cannot distinguish between the proposed mechanism and a systematic underestimate. The authors should compare with additional polydisperse data (e.g. bidisperse disk packings or other distribution widths) or directly measure z(phi) in simulated jammed packings to test Eq. (8).

minor comments (3)

[Section 'EOS', Eq. (3)] The constant g0 in the contact-peak ansatz is said to be determined by a boundary condition, but the boundary condition is never specified; please state it explicitly.
[Abstract and text] The abstract describes the comparison as a 'power-law size distribution' with s = 0.246, while the text and Fig. 1 use a log-normal distribution with sigma approximately 0.2462; please reconcile the nomenclature.
[Section 'Polydisperse contacts', Eq. (2)] For polydisperse disks, the average contact relation should specify how g(r) is averaged over ij pairs and how the single mean diameter sigma follows from that average; currently the reduction from the polydisperse contact condition sigma_ij = (sigma_i + sigma_j)/2 is implicit.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the phi_RCP(s) curve follows from the calibrated crowding relation plus independent equilibrium equations of state; the swap-MC comparison is an external benchmark, not a fitted input.

full rationale

The derivation chain is non-circular. Equation (13), C0[Z(phi_RCP)-1]=1, follows by combining the isostatic definition of RCP (z_c=4, from Ref. [32]) with the crowding relation z=4C0[Z-1], where C0 is calibrated at the monodisperse CP point (phi_CP=pi/sqrt(12), z=6), not at the predicted RCP. The monodisperse values in Table I are outputs of the same equation, not fitted inputs. The EOS choices (SPT, Henderson, CS) and the Santos polydisperse mapping are independent equilibrium inputs. The s=0.25 comparison with the swap-MC value phi=0.905 (Ref. [69]) is an external benchmark; the paper explicitly concedes that those packings may have z>4 and that its prediction may therefore underestimate them, which shows the agreement is not forced. The self-citations [9,10] provide prior numerical justification of the crowding ansatz in 3D; they do not supply the 2D polydisperse result, so the central claim does not collapse into those citations. The manuscript's own limitation, namely the unvalidated transfer of the equilibrium polydisperse hard-disk EOS to jammed 2D packings, is a correctness risk rather than a circular step. No reduction of Eq. (13) to a fitted parameter or to the comparison datum can be exhibited from the paper's equations.

Assumptions & free parameters 1 free parameters · 4 assumptions · 0 invented entities

The central claim depends on four assumptions, all inherited from the author's previous crowding framework or from equilibrium EOS theory. No new physical entities are introduced, but the calibration constant C0 and the EOS choice are modeling inputs that determine the predicted values.

free parameters (1)

C0 (dimensionless proportionality constant) = 0.0131 (SPT), 0.0119 (Henderson), 0.0218 (CS) from Table I
Introduced in Eq. (8) by combining the split-rdf assumption with the proportionality g(σ;φ) ∝ (Z-1)/(2φ). It is calibrated at monodisperse triangular close packing, z=6 at phi_CP=0.9069, and then used unchanged for all polydisperse predictions. It is a boundary-condition constant, not fitted to RCP data, but its value depends on the arbitrary choice of a monodisperse EOS.

assumptions (4)

ad hoc to paper The averaged contact value of the jammed packing rdf is proportional to the equilibrium virial contact value: g(σ;φ) ∝ (Z(phi)-1)/(2phi).
Introduced in Eq. (7) by analogy with the equilibrium virial theorem in Eq. (6), justified only by prior 3D numerical work [10]. It is the key step that connects jammed packings to equilibrium equations of state.
domain assumption RCP is identified with the isostatic rigidity onset with average contact number z=4.
Used in Eq. (13), citing Ref [32]. This defines RCP as the densest isostatic packing, one of several possible definitions.
domain assumption The Santos effective packing fraction mapping, phi_eff = phi/[phi+lambda(1-phi)], and the polydisperse compressibility Eq. (12) hold at jammed RCP densities.
Imported from Refs [66-68] for equilibrium polydisperse hard-sphere fluids; no derivation or numerical check is provided for its use in 2D jammed packings.
ad hoc to paper C0 calibrated at monodisperse CP remains valid for arbitrary polydispersity.
Used in Eq. (13) for all size distributions without further justification beyond the prior framework.

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Pith. "Pith review of Analytical solution for the polydisperse random close packing problem in 2D." pith.science (2026). https://pith.science/paper/SS7NCMHP

@misc{pith2026250203354,
  author       = {Pith},
  title        = {Pith review of: Analytical solution for the polydisperse random close packing problem in 2D},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SS7NCMHP}},
  note         = {Machine review of arXiv:2502.03354}
}

abstract

An analytical theory for the random close packing density, $\phi_\textrm{RCP}$, of polydisperse hard disks is provided using an equilibrium model of crowding [A. Zaccone, Phys. Rev. Lett. 128, 028002 (2022)] which has been justified on the basis of extensive numerical analysis of the maximally random jammed (MRJ) line in the phase diagram of hard spheres [Anzivino et al., J. Chem. Phys. 158, 044901 (2023)]. The solution relies on the equations of state for the hard disk fluid and provides predictions for $\phi_\textrm{RCP}$ as a function of the ratio, $s$, of the standard deviation of the distribution of disk diameters to its mean. For a power-law size distribution with $s=0.246$, the theory yields $\phi_\textrm{RCP} =0.892$, which compares well with the most recent numerical estimate $\phi_\textrm{RCP} =0.905$ based on the Monte-Carlo swap algorithms [Ghimenti, Berthier, van Wijland, Phys. Rev. Lett. 133, 028202 (2024)].

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