Finite-temperature and protocol-space boundaries of the universal microstructure scaling of jammed sphere packings

Jammed packings of frictionless spheres possess a universal zero-temperature microstructure: the cumulative pair correlation $Z(r)$ grows beyond the isostatic contact plateau as $Z(r)-z\propto(r-\sigma)^{1-\alpha}$, the cumulative force distribution has a weak-force tail $G(f)\propto f^{1+\theta}$, and the two regimes are matched by universal crossover scaling functions; the exponents are linked by the marginal-stability relation $\alpha=1/(2+\theta)$. Charbonneau, Corwin, Parisi and Zamponi (Phys. Rev. Lett. 109, 205501 (2012)) established this universality for spatial dimensions d=3-10, at zero temperature, for compactification-type preparation protocols, and posed the follow-up problems. Later literature resolved three of them: the full replica-symmetry-breaking theory of hard spheres in the limit d to infinity reproduces the nonzero exponents using non-Gaussian fractal cages; the stability bound is saturated rather than violated, with the underlying identity proven analytically in 2026; and photoelastic-disk experiments confirmed the weak-force and gap power laws quantitatively. The surviving open problem is to complete the universality phase diagram on the two axes the original study excluded: (a) how the microstructural scaling relations are modified, blurred, or preserved at finite temperature, and (b) whether they survive preparation protocols beyond the compaction of liquid configurations, in particular denser and partially crystallized packings.

Background

Jamming and isostaticity: a disordered packing of frictionless spheres becomes mechanically rigid ('jams') when its packing fraction exceeds a critical value. At the transition the packing is isostatic, meaning the number of force-bearing mechanical constraints exactly matches the degrees of freedom, so the average number of contacts per particle is $z=2d$ in spatial dimension $d$. The transition can be approached from below by compressing hard spheres (HS), which have infinite interaction energy on overlap, for instance with Lubachevsky-Stillinger dynamics in which particle diameters grow at a controlled rate, or from above by minimizing the energy of soft spheres (SS) with a harmonic overlap potential; $\Delta\varphi$ denotes the distance from the jamming packing fraction.

Microstructure observables: $g(r)$ is the pair correlation function (the density of particle centers at distance $r$) and $Z(r)$ is its cumulative version, giving the average number of neighbors within $r$ of a given particle. Charbonneau, Corwin, Parisi and Zamponi (2012) showed that near contact $Z(r)$ exhibits three universal regimes: (i) a contact regime in which $Z$ grows from 0 to $z$; (ii) a quasi-contact regime just beyond contact, $Z(r)-z\propto(r-\sigma)^{1-\alpha}$ with $\sigma$ the particle diameter, which means the density of near-touching gaps $h=r-\sigma$ grows as $h^{-\alpha}$; and (iii) an intermediate crossover regime with scaling functions $H_{\pm}$ and matching relations $\nu=\alpha\mu$, $\mu=(1+\theta)/(2+\theta-\alpha)$. For the inter-particle forces, whose cumulative distribution is $G(f)$, the weak-force tail behaves as $G(f)\propto f^{1+\theta}$. The exponents were found universal within the studied class, i.e. independent of dimension d=3-10, density, and the compactification protocols used: $\alpha\approx0.4$ throughout, with $\theta=0.42(2)$ for SS and $\theta=0.28(3)$ for HS. Rattlers (particles with no mechanical contacts) were excluded from the analysis.

Stability bound: Wyart (2012) argued that jammed packings are marginally stable, that is, stable but on the verge of instability: opening a contact carrying a weak force $f$ destabilizes the packing unless quasi-contacts are abundant enough, which yields the bound $\alpha\geq1/(2+\theta)$ between the gap and force exponents, and a generalized form $\alpha\geq(1-\delta/2)/(2+\theta-\delta/2)$ when the displacement response on a weak contact scales as $f^{\delta}$. The 2012 measurements slightly violate the simple bound, and a direct test of the extended relation was explicitly left open.

Replica theory: the Gaussian replica theory (G-RT) was the then-available first-principles mean-field theory of the glass transition and of jamming; it describes glass states as long-lived metastable minima of the free energy treated with the replica method (coupled copies of the system) and approximates the cage of each particle (the region it can explore while its neighbors stay fixed) by a Gaussian. G-RT correctly predicts several features, including the contact-regime scaling functions, but yields $\alpha=\theta=0$ and so misses the observed power laws; the 2012 paper attributed this failure to the Gaussian cage assumption. The full replica-symmetry-breaking (fullRSB) solution of hard spheres in the limit $d\to\infty$ replaces the single Gaussian basin by a hierarchical, fractal structure of nested sub-basins — the structure that appears below the Gardner transition, the transition inside the glass phase at which each glass basin splits into such a hierarchy; it predicts isostaticity and yields the critical exponents analytically, $\gamma=0.41269\ldots$ for the gaps and $\theta_f=0.42311\ldots$ for the weak forces (in this mean-field notation $\gamma$ and $\theta_f$ are the exponents denoted $\alpha$ and $\theta$ in the scaling relations above), in agreement with the numerics. Bucklers are particles with barely enough contacts to be constrained, hosting localized low-frequency excitations; separating their contribution (Charbonneau, Corwin, Parisi and Zamponi, 2015) resolved the long-standing ambiguity in the weak-force exponent.

Status after the source paper: the fullRSB program (Charbonneau, Kurchan, Parisi, Urbani and Zamponi, 2014) resolved the theory-refinement thread, and the stability bound is now understood to be saturated, $\alpha=1/(2+\theta)$, with the identity underlying this relation proven analytically by Parisi and Zamponi (2026). Experiments on jammed photoelastic disks (Wang, Shang, Jin and Zhang, 2022) measured the weak-force and gap power laws and found excellent agreement with the theory. What remains open are the two axes the 2012 study explicitly excluded: finite temperature, and the broader preparation-protocol space.

Problem Statement

Complete the universality phase diagram of the microstructure and stability scaling relations of jammed frictionless-sphere packings on the two axes excluded by the 2012 study. Both strands serve one objective: establishing the full domain of validity of the universal zero-temperature scalings, namely the quasi-contact law $Z(r)-z\propto(r-\sigma)^{1-\alpha}$, the weak-force tail $G(f)\propto f^{1+\theta}$, the intermediate crossover scaling functions that match the two regimes, and the saturated marginal-stability relation $\alpha=1/(2+\theta)$, beyond the regime in which they were established ($T=0$, compactification-type preparation protocols, $d=3$-$10$).

(a) Finite temperature: for soft-sphere and related thermal systems at $T>0$, determine how thermal fluctuations and anharmonicity modify, blur, or preserve these scaling relations and the saturation of the stability bound. This is the future work announced by the source paper: detailing how temperature and its associated anharmonicities affect the $T=0$ scaling relations identified there.

(b) Broader protocol space: for jammed packings generated by preparation protocols beyond the compaction of liquid configurations, in particular denser packings and partially crystallized packings, determine whether the same exponents and scaling functions are obtained; where they are not, characterize what replaces them, thereby completing the delineation of the jamming universality class in preparation-protocol space.

References

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