Analytic origin and universality of the two-component oscillatory structure and anomalous long-range decay of correlations in jammed hard-sphere packings

At the jamming point of three-dimensional monodisperse frictionless hard-sphere packings, the pair correlation function of the contact-point representation ('contacts RDF') was found numerically to decompose into two exponentially damped oscillatory components with commensurate wave numbers and damping rates ($p_m \approx 2p_l$, $a_m \approx 2a_l$), while correlations weighted on weak contact forces or on deviations from isostatic coordination decay as power laws with exponent $\gamma \approx 4$ (Rissone, Corwin, and Parisi, Phys. Rev. Lett. 127, 038001 (2021)). These findings are purely phenomenological: no analytic theory derives them, and their robustness and their evolution below jamming are unmeasured. The problem is (i) to derive the two-component damped-oscillatory form and the anomalous-decay exponents from a microscopic theory of the jammed ensemble, and (ii) to establish numerically the robustness of the exponents in larger systems and their evolution as the packing fraction approaches the jamming point from below.

Background

When a disordered assembly of frictionless hard spheres in three dimensions is compressed until it just acquires mechanical rigidity, it reaches the jamming point at packing fraction $\phi_J \approx 0.64$ (the fraction of volume covered by spheres). At jamming the contact network is isostatic: the average number of contacts per particle equals the minimum required for mechanical stability, $Z \approx Z_{iso} = 2d = 6$ in $d = 3$; particles that carry no contacts ('rattlers') are excluded from the count. The standard structural probe is the radial distribution function (RDF) $g(r)$, proportional to the probability of finding a pair of particles at separation $r$. A packing is called hyperuniform if its density fluctuations are anomalously suppressed at large length scales, equivalently if its static structure factor $S(k)$ (the Fourier-space density-correlation function) vanishes as a positive power of the wave number as $k \to 0$; in the marginal case of a linearly vanishing structure factor, $S(k) \propto k$, hyperuniformity implies a long-range power-law tail $g(r)-1 \propto -r^{-(d+1)}$ in $d$ dimensions instead of a rapid approach to 1.

Whether randomly jammed sphere packings are hyperuniform has been contested. Donev, Stillinger, and Torquato (2005) found $S(k) \propto |k|$ and hence a weak tail $h(r) \equiv g(r)-1 \approx -r^{-4}$ in large jammed disordered packings. Ikeda, Berthier, and Parisi (2017) reanalyzed the large-scale density field of randomly jammed packings and found no genuine hyperuniform signatures in the raw density, a short-ranged direct correlation function, and a 'surprisingly complex' large-distance structure of $g(r)$.

Rissone, Corwin, and Parisi (2021) attacked the large-scale structure with two constructions. First, they introduced observable-weighted correlation functions: for an observable $O$ attached to particles or to contacts (e.g. the contact force $f$ carried by a particle, or the coordination deficit $\Delta Z' = Z'-2d$ attached to a contact point, where $Z' = (Z_i+Z_j)/2$ is the mean contact number of the two spheres $i, j$ meeting at that contact), they define $C_O^s(r) = g_O^s(r)/g(r)$, where $g_O^s(r)$ is a pair-distance histogram weighted by $O_i^s O_j^s$ with a real control exponent $s$; $s = 0$ recovers the ordinary RDF, and $s < 0$ emphasizes weak forces (the force weighting is restricted to $s > -1-\theta \approx -1.4$ by the weak-force distribution $P(f) \propto f^\theta$, $\theta \approx 0.4$). Second, they used a contact-point representation: each contact is replaced by a fictive particle of diameter $\sigma' = \sigma/2$ centered at the contact point, giving up to $2dN$ fictive particles whose RDF (the 'contacts RDF') resolves structural features of the contact network that are smeared out in the particle-center representation. In packings of $N = 16384$ spheres at jamming they found: (i) the weak-force-weighted correlation for $s = -1/2$ and the $\Delta Z'$-weighted correlation for $s = 1$ decay as power laws with exponents $\gamma_f^{s=-1/2} = 4.1 \pm 0.3$ and $\gamma_{\Delta Z'} = 3.9 \pm 0.2$, consistent with the $r^{-(d+1)}$ decay expected for hyperuniform systems in $d = 3$, while $s = -1$ gives $\gamma_f^{s=-1} = 0.7 \pm 0.3$; (ii) after multiplying the contacts RDF by $e^{ar}$ ($a \approx 0.3$) to undo an overall exponential damping, the remaining mid- and long-range oscillations decompose into a sum of two exponentially damped cosines $g(r) = f_m(r) + f_l(r)$ with $f_i(r) = c_i e^{a_i r}\cos(p_i r + \psi_i)$, an eight-parameter fit yielding commensurate wave numbers and damping rates, $p_m \approx 2p_l = 7.514 \pm 0.003$ and $a_m \approx 2a_l = 0.70 \pm 0.01$ (distances in units of $\sigma'$), the long-range term $f_l$ being only a small correction to the mid-range term $f_m$.

All of these findings are numerical. The authors state that a theoretical explanation of the two-term form 'could be recovered by writing integral equations of the hypernetted chain kind ... for the correlation function. However this problem has not been explored yet.' The hypernetted-chain (HNC) approximation is a standard closure of equilibrium liquid-state theory: the Ornstein-Zernike relation $S(k) = 1/(1+\rho\tilde{c}(k))$ ($\rho$ the number density, $\tilde{c}$ the Fourier transform of the direct correlation function, in terms of which $g(r)-1$ splits into a direct part plus an indirect chain-dressed part) is closed by an approximation for the bridge function. Whether such an equilibrium-style integral-equation theory can be built for the athermal, mechanically constrained, history-dependent jammed ensemble is itself open. The authors further state that bigger systems would ease the isolation of the power-law decay from residual oscillations, and that characterizing these correlation functions at packing fractions $\phi < \phi_J$ approaching the jamming point 'would be of great interest'.

Problem Statement

The common objective is to turn the phenomenological findings of Rissone, Corwin, and Parisi (2021) on the large-scale correlations of jammed hard-sphere packings into an established physical picture, by explaining the observed structures analytically and establishing their universality numerically. For three-dimensional monodisperse frictionless hard-sphere packings at jamming:

(1) Analytic origin. Provide a microscopic statistical-mechanical derivation of (a) the decomposition of the contacts RDF into two exponentially damped oscillatory components $f_m(r) + f_l(r)$ with commensurate wave numbers and damping rates ($p_m \approx 2p_l$, $a_m \approx 2a_l$), and (b) the anomalous power-law decay of the weak-force-weighted and $\Delta Z'$-weighted correlations, with predictions compatible with the measured exponents $\gamma_f^{s=-1/2} = 4.1 \pm 0.3$, $\gamma_{\Delta Z'} = 3.9 \pm 0.2$ (and $\gamma_f^{s=-1} = 0.7 \pm 0.3$) and accounting for their dependence on the weighting exponent $s$.

(2) Approach from below jamming. Characterize these observable-weighted correlation functions at packing fractions $\phi < \phi_J$ approaching the jamming point, making explicit how the contact-network observables (weak forces, $\Delta Z'$) are defined off jamming, and determine how the anomalous long-range decay and the two-component oscillatory structure emerge as $\phi \to \phi_J$.

(3) Universality and robustness. Establish the robustness of the measured exponents and of the two-component structure with larger systems ($N$ well above 16384), resolving whether the fitted exponents are universal numbers — independent of system size at fixed weighting exponent $s$ — and what physical mechanism they encode.

These three components — (1) the analytic derivation, and (2) and (3) the numerical studies — together establish whether the anomalous decay is a genuine, universal signature of the jammed state and where it comes from.

References

1. Paolo Rissone, Eric I. Corwin, Giorgio Parisi, 'Long-Range Anomalous Decay of the Correlation in Jammed Packings', Phys. Rev. Lett. 127, 038001 (2021). DOI: 10.1103/PhysRevLett.127.038001, https://doi.org/10.1103/PhysRevLett.127.038001 (preprint arXiv:2012.10181, https://arxiv.org/abs/2012.10181)
2. Aleksandar Donev, Frank H. Stillinger, Salvatore Torquato, 'Unexpected Density Fluctuations in Jammed Disordered Sphere Packings', Phys. Rev. Lett. 95, 090604 (2005). DOI: 10.1103/PhysRevLett.95.090604, https://doi.org/10.1103/PhysRevLett.95.090604 (preprint arXiv:cond-mat/0506406, https://arxiv.org/abs/cond-mat/0506406)
3. Atsushi Ikeda, Ludovic Berthier, Giorgio Parisi, 'Large-scale structure of randomly jammed spheres', Phys. Rev. E 95, 052125 (2017). DOI: 10.1103/PhysRevE.95.052125, https://doi.org/10.1103/PhysRevE.95.052125 (preprint arXiv:1701.00936, https://arxiv.org/abs/1701.00936, titled 'Large-scale structure of randomly jammed particles')
4. Charles Emmett Maher, Yang Jiao, Salvatore Torquato, 'Hyperuniformity of maximally random jammed packings of hyperspheres across spatial dimensions', Phys. Rev. E 108, 064602 (2023). DOI: 10.1103/PhysRevE.108.064602, https://doi.org/10.1103/PhysRevE.108.064602 (preprint arXiv:2311.06970, https://arxiv.org/abs/2311.06970)
5. Duc T. Dam, Takeshi Kawasaki, Atsushi Ikeda, Kunimasa Miyazaki, 'Hyperuniformity near jamming transition over a wide range of bidispersity', Phys. Rev. E 114, 015407 (2026). DOI: 10.1103/vclw-6h8s, https://doi.org/10.1103/vclw-6h8s
6. Salvatore Torquato, 'Physics of sphere packings', Nat. Rev. Phys. 8, 261-275 (2026). DOI: 10.1038/s42254-026-00925-w, https://doi.org/10.1038/s42254-026-00925-w
7. Junchao Huang, Jianhua Zhang, Ding Xu, Shiyun Zhang, Hua Tong, Ning Xu, 'Revealing bond level origin of stability in disordered solids from marginal stability to ultrastability', Nat. Commun. 17, 7027 (2026). DOI: 10.1038/s41467-026-73939-8, https://doi.org/10.1038/s41467-026-73939-8
8. Claudia Artiaco, Rafael Diaz Hernandez Rojas, Giorgio Parisi, Federico Ricci-Tersenghi, 'Hard-sphere jamming through the lens of linear optimization', Phys. Rev. E 106, 055310 (2022). DOI: 10.1103/PhysRevE.106.055310, https://doi.org/10.1103/PhysRevE.106.055310 (preprint arXiv:2203.05654, https://arxiv.org/abs/2203.05654)
9. Yinqiao Wang, Jin Shang, Yuliang Jin, Jie Zhang, 'Experimental observations of marginal criticality in granular materials', Proc. Natl. Acad. Sci. USA 119, e2204879119 (2022). DOI: 10.1073/pnas.2204879119, https://doi.org/10.1073/pnas.2204879119 (preprint arXiv:2201.10755, https://arxiv.org/abs/2201.10755)
10. Hiroshi Frusawa, 'Non-hyperuniform metastable states around a disordered hyperuniform state of densely packed spheres: stochastic density functional theory at strong coupling', Soft Matter 17, 8810-8831 (2021). DOI: 10.1039/D1SM01052B, https://doi.org/10.1039/D1SM01052B