At the jamming point of frictionless spherical particles, the network of contact directions alone yields two interpretable particle-level structural descriptors: the contact-sum vector Ci\mathbf{C}_i (sum of unit contact vectors at particle ii), which predicts the preferred direction of motion, and the scalar SiS_i (sum of dot products of contact-vector pairs), which predicts single-particle mobility. This framework was introduced by Díaz Hernández Rojas, Parisi, and Ricci-Tersenghi (Soft Matter 17, 1056 (2021)), who verified it by molecular dynamics and Monte Carlo simulations near jammed configurations and measured a slowly decaying correlation between structure and dynamics. Their conclusions explicitly pose five linked open questions that define the current problem: (1) how far in time and in configurational distance from the original jammed minimum the descriptors remain quantitatively reliable, and what sets the rate and sharpness of the decorrelation as the system leaves the marginal (Gardner-like) regime toward stable glassy minima; (2) whether descriptors built from second-nearest neighbours or coarse-grained cluster averages extend the predictive horizon; (3) whether an operational procedure (e.g. Bayesian model selection) can infer full single-particle displacement distributions rather than only first moments; (4) whether the contact-direction descriptors exhibit genuine long-range structural correlations across jammed packings, and how periodic order or polydispersity (tunneled or slightly polydisperse crystals) modifies them; (5) whether the similarity of jammed minima within a free-energy basin can be quantified so as to predict the persistence of dynamical predictability from basin-level statistics.

Background

Jamming is the transition by which a disordered assembly of repulsive particles, such as grains, droplets, or colloids, acquires mechanical rigidity as its density (packing fraction) increases; at the jamming point of frictionless spheres the packing is isostatic, meaning the number of contacts just suffices to constrain all degrees of freedom, and a small number of particles (rattlers) have too few contacts to be locked in place. A jammed configuration is fully specified by the particle positions, which determine the contact network: the list of touching pairs and the unit contact vectors nij\mathbf{n}_{ij}. The physics of such marginally stable states is well developed statically: mean-field (infinite-dimensional) replica theory, extended to finite dimension, predicts the critical distributions of small contact forces and interparticle gaps, and the free-energy landscape below the glass transition is predicted to be fractal, with a Gardner phase in which a glass basin splits into a hierarchy of marginal sub-basins.

Whether and how this static structure controls the subsequent dynamics of individual particles is far less understood. A standard probe is the isoconfigurational ensemble: many independent dynamical trajectories are launched from one and the same initial configuration, so that averaging over trajectories isolates the dynamical propensity induced by structure rather than by thermal noise. Studies in this spirit established that local structural order correlates with mobility in a strongly system-dependent way, that soft vibrational modes partially predict irreversible rearrangements, and that machine learning can construct a linear combination of local structural features ('softness') that predicts rearrangements well but without transparent physical meaning; graph neural networks push this predictive power further at the cost of complete interpretability.

The source paper of the present problem (Díaz Hernández Rojas, Parisi, Ricci-Tersenghi, Soft Matter 17, 1056 (2021)) asked what the contact network alone, with no force magnitudes and no learning, says about the short-time, single-particle dynamics near a jammed configuration of monodisperse frictionless spheres in three dimensions. For each particle ii with neighbour set ∂i\partial i they defined the vectorial and scalar contact descriptors

Ci:=∑j∈∂inij,Si:=∑j<k∈∂inij⋅nik=12(∣Ci∣2−qi),\mathbf{C}_i := \sum_{j\in\partial i}\mathbf{n}_{ij}, \qquad S_i := \sum_{j<k\in\partial i}\mathbf{n}_{ij}\cdot\mathbf{n}_{ik} = \tfrac{1}{2}\big(|\mathbf{C}_i|^2 - q_i\big),

where qiq_i is the coordination number (number of contacts). Using isoconfigurational molecular dynamics and Monte Carlo simulations started from exactly generated isostatic jammed configurations, they found that Ci\mathbf{C}_i correlates with the mean displacement vector ⟨δri⟩\langle\delta\mathbf{r}_i\rangle (preferred direction of motion) and SiS_i correlates strongly with the mobility ⟨∣δri∣2⟩\langle|\delta\mathbf{r}_i|^2\rangle; here δri\delta\mathbf{r}_i is the displacement of particle ii from its position in the jammed configuration and ⟨⋅⟩\langle\cdot\rangle denotes the average over isoconfigurational trajectories. The strength of the association was measured by the Spearman rank correlation coefficient KK (a nonparametric measure of monotonic association), which is high at short times and comparable to state-of-the-art machine-learned predictors. Two further findings frame the open questions. First, the correlation coefficient KK decays only slowly as a function of the mean-square displacement Δ\Delta (the configurational distance from the jammed state), and curves at different distances from jamming collapse once Δ\Delta is rescaled, suggesting a universal, interaction-independent rate of information loss. Second, normal vibrational modes of the jammed minimum fail to capture the observed mean displacements, so the relevant regime is anharmonic. The authors argue, via the fractal free-energy landscape picture, that the Gardner marginal phase sets only a rough limit of validity: within it, nearby jammed states share a large fraction of contacts, so structure remains informative, but no criterion exists for how fast predictability is lost once the dynamics leaves the marginal regime toward more stable glassy minima.

Problem Statement

Determine the limits and extensions of contact-geometry-based inference of single-particle dynamics near the jamming point, as explicitly posed by the source paper's conclusions. The common objective uniting the requested results is to establish, quantitatively and physically, how long and how far the jammed contact geometry constrains the subsequent motion of individual particles, and to enlarge that predictive window while keeping the descriptors physically interpretable. Concretely, the following interlinked questions are open. (1) Predictive horizon: measure how far in time and in configurational distance (e.g. in mean-square displacement Δ\Delta from the jammed minimum) the contact-direction descriptors Ci\mathbf{C}_i (the sum of unit contact vectors at particle ii) and SiS_i (the scalar sum of dot products of contact-vector pairs at ii), both defined above, remain quantitatively reliable predictors of the mean displacement vector ⟨δri⟩\langle\delta\mathbf{r}_i\rangle and the mobility ⟨∣δri∣2⟩\langle|\delta\mathbf{r}_i|^2\rangle (both averaged over isoconfigurational trajectories launched from the same jammed configuration), and identify what sets the sharpness and rate of the decorrelation as the dynamics leaves the marginal, Gardner-like regime toward more stable glassy minima, for which no criterion currently exists; the threshold for ‘quantitatively reliable’ is left to the solver, who should state an operational criterion for it and report the predictive horizon as a function of that criterion. (2) Extended and coarse-grained descriptors: determine whether incorporating information beyond first-neighbour contacts (second-nearest neighbours, or cluster-averaged quantities such as the mean of SiS_i over a region) improves the predictive performance and extends the time window of validity. (3) Full-distribution inference: construct an operational procedure, e.g. combining contact-direction variables with Bayesian model selection or comparable statistical learning, that infers the full single-particle displacement distributions rather than only their first moments. (4) Long-range structural correlations: establish whether the contact-direction descriptors (or similar variables) exhibit genuine long-range structural correlations across jammed packings, at mesoscopic or system-size scales, and determine how the presence of periodic order or polydispersity, e.g. in tunneled crystals (low-density periodically ordered packings containing a regular array of void channels) or slightly polydisperse crystals, modifies these correlations and the role of second-nearest neighbours. (5) Basin-level control of predictability: quantify the relationship between the similarity of jammed minima within a free-energy basin (e.g. their overlap or contact-sharing fraction) and the persistence of dynamical predictability, so that the success and range of applicability of the inference can be predicted from basin-level statistics rather than case-by-case simulation. The five questions are independent targets, and the full open program comprises all five.

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