Can a Large Packing Be Assembled from Smaller Ones?

Can a large packing be assembled from smaller ones?

Daniel Hexner,1, 2, ∗ Pierfrancesco Urbani,3 and Francesco Zamponi4, 5 1The James Franck Institute and Department of Physics, The University of Chicago, Chicago, IL 60637, USA 2Department of Physics and Astronomy, The University of Pennsylvania, Philadelphia, PA, 19104, USA 3Institut de Physique Théorique, Université Paris Saclay, CNRS, CEA, F-91191, Gif-sur-Yvette 4Laboratoire de Physique de l’Ecole Normale Supérieure, CNRS, Paris, France 5Université PSL, Sorbonne Université, Université Paris-Diderot, Sorbonne Paris Cité, Paris, France We consider zero temperature packings of soft spheres, that undergo a jamming to unjamming transition as a function of packing fraction. We compare differences in the structure, as measured from the contact statistics, of a finite subsystem of a large packing to a whole packing with periodic boundaries of an equivalent size and pressure. We find that the fluctuations of the ensemble of whole packings are smaller than those of the ensemble of subsystems. Convergence of these two quantities appears to occur at very large systems, which are usually not attainable in numerical simulations. Finding differences between packings in two dimensions and three dimensions, we also consider four dimensions and mean-field models, and find that they show similar system size dependence. Mean- field critical exponents appear to be consistent with the 3d and 4d packings, suggesting they are above the upper critical dimension. We also find that the convergence as a function of system size to the thermodynamic limit is characterized by two different length scales. We argue that this is the result of the system being above the upper critical dimension.

A starting point for characterizing the structure of or- to the thermodynamic limit we measure the contact fluc- dered materials are their microscopic subunits, or build- tuations in whole systems as a function of the distance to ing blocks. Crystalline materials, in their ground state, the jamming transition. We perform finite size scaling to are defined by a single unit cell repeating throughout the identify a length scale, and find that it differs from those system. Quasicrystalline materials, while aperiodic, still previously measured in the contact statistics. Finding have a rather small number of building blocks. In the differences between 2d and 3d, we also consider 4d and case of disordered materials, each subsystem is different mean-field variants of the jamming model. These appear because of geometrical frustration, and the multiplicity to be consistent with results from 3d, suggesting that the of different subsystems is huge [1]. Nonetheless, it is in- the upper critical dimension is below three [7]. teresting to ask, how different is a subsystem from the To summarize, in this paper (i) we introduce an en- whole packing it composes? This question addresses, in tirely new procedure (in the context of jamming) for fi- part, the effect of boundaries, correlations in the struc- nite size scaling analysis, by comparing subsystems of ture, and multiplicity of ground states. a large packing with periodic packings of the same size; In this paper, we ask this aforementioned question in (ii) we identify a new length scale, in addition to the ones a commonly studied model for amorphous solids: disor- reported in [6]; (iii) we provide accurate measurements dered packings of soft spheres at zero temperature [2–4]. of this new length scale in dimension d = 2, 3, 4 (note This model undergos a rigidity transition, as a function that previous calculations of jamming length scales could of the packing fraction [5]. We compare the ensemble of only obtain accurate results for 2d systems); (iv) thanks subsystems cut out from large packings, to the ensem- to this improved finite size scaling analysis, we obtain ble of whole systems of the same volume with periodic strong quantitative evidence that du ∼ 2 is the upper boundary conditions (Fig. 1). Recently it has been found critical dimension; (v) we analyze models with hypostatic that the contact statistics possess unusual long range cor- jamming and find that the corresponding suppression of relations near the transition [6]. We therefore focus on fluctuations does not take place. Our analysis thus shows contact fluctuations to compare the two ensembles. that anomalous contact fluctuations survive in mean-field While we expect convergence of the two ensembles for like models, and are crucially related to isostaticity. arXiv:1902.00630v2 [cond-mat.soft] 5 Jul 2019 large enough systems, the system size V ∗ for which these We begin by defining the jamming model, in which converge appears to be in many cases well beyond that overlapping particles of radius Ri interact via a harmonic accessible via simulations. For system sizes smaller than potential: V ∗, fluctuations in contacts are significantly smaller in 2 systems with periodic boundary conditions. When ap- 1 rij Uij = k 1 − Θ(Ri + Rj − rij) . (1) proaching the jamming transition this disparity grows, 2 Ri + Rj and V ∗ appears to diverge, suggesting that it is associated with a diverging length scale. To study convergence Here, rij is the distance between the centers of the particles and the Heaviside step function, Θ(x), insures that only overlapping particles interact. In 2d the radii are chosen to be polydisperse [8] to avoid crystallization, ∗ [email protected] while in higher dimension monodisperse particles lead to 2

2 were uncorrelated random variables, then σZ (`) would not depend on `, since the number of particles scales as the volume. At the jamming transition, ∆Z = 0, the fluc- 2 −1 tuations scale in the smallest possible way, σZ (`) ∝ ` , implying that the sub-extensive fluctuations are dominated by the surface of the enclosure [6]. Such kind of fluctuations are called “hyperuniform” [13] and have been observed in several strongly constrained physical systems [14–22]. At a finite distance from jamming, ∆Z > 0, the fluctuations are only suppressed up to a length scale −ν ξf ∝ ∆Z f ; above ξf , the lack of correlations imply 2 that σZ (`) is independent of `. We also remark that unlike typical critical systems, here there are two different diverging length scales. A second length scale, −νz ξZ ∝ ∆Z , can be measured from hδZ (r) δZ (0)i and is different than ξf , having different exponents νf > νz [6]. 2 In this paper, in addition to σZ (`), we characterize sample-to-sample fluctuations of many jamming config- Figure 1. Illustration of the two ensembles: a subsystem of a large packing is on the left, while a whole system of the same urations with periodic boundaries, at the same value of size is on the right. pressure. We define h 2i δ2Z (N) ≡ N Z2 − Z , (3) amorphous packings. We begin with particles distributed where the average • is over distinct packings at constant randomly and uniformly throughout space, and minimize pressure, and Z = 1 PN Z , as before. The factor of the energy at a constant pressure via FIRE algorithm [9] N i=1 i until the system reaches force balance. N on the right hand side of Eq. (3) ensures convergence to a finite value in the limit of N → ∞. We also note An important quantity in understanding the geome- that in the infinite size limit, where boundary condition try of the jamming transition is the average coordina- are unimportant, ρδ2Z(N = ∞) = σ2 (` = ∞), where ρ tion number, Z. At the marginally rigid state at zero Z is the number density. To see how this is related to ξ , pressure, the average coordination number Z attains a f we note that a sufficiently large system can decomposed universal value that approaches Z = 2d for infinite sys- c into uncorrelated sub-regions of volume ξd. In each such tem [7, 10–12]. This amounts to the smallest number of f d−1 contacts to maintain rigidity. The excess coordination region the fluctuations scale as the surface, ξf . Because d number ∆Z = Z − Zc, also characterizes the distance the number of uncorrelated regions is (L/ξf ) , we obtain from the jamming transition. Recently, it has been real- 1 ized that the coordination number possess subtle spatial 2 d −1 νf δ Z N ξf ∝ ξf ∝ ∆Z . (4) correlations [2, 6]. Unlike equilibrium critical systems ρ that have diverging fluctuations (associated with a di- 2 2 We now turn to show how σZ (`) and δ Z(N) compare verging susceptibility), packings have anomalously small on a finite length scale. To make this comparison, we plot fluctuations. At the jamming transition the bulk contact 2 d σZ (`) /ρ as a function of N = ρ` . Results in 2d and in fluctuations vanish, and the fluctuations inside a subsys- 3d (Fig. 2) show that for finite N sample-to-sample fluc- tem scale as its surface. 2 2 tuations, δ Z, are smaller than σZ /ρ and have a fairly We now briefly review the metrics and the results of 2 weak dependence on system size. Because σZ (`) /ρ and Ref. [6] for characterizing the fluctuations. For a pack- δ2Z(N) converge in the thermodynamic limit, one can ing of N particles excluding the rattlers, we define Zi define a length scale at which the two ensembles con- to be the number of particles in contact with particle i, verge. This length scale is surprisingly large, especially 1 PN Z = N i=1 Zi the average contact number in a given in 3d, and the difference between the two ensembles grows packing, and δZi = Zi−Z the deviation from the average. dramatically upon approaching the jamming transition, The fluctuations are then characterized by measuring the ∆Z → 0. Even at ∆Z ≈ 1.22, which is usually consid- d variance in a hyper-cube of volume ` , ered to be far from the jamming transition, convergence occurs for N > 105. For ∆Z ≈ 0.12 convergence can be 1 X 2 8 σ2 (`) = δZ . (2) extrapolated to occur for system sizes of N > 10 which Z `d i i∈`d are currently not attainable numerically. The fact that convergence in 2d appears to occur at smaller N suggests Here the average h•i is over many subsystems of a large that it is dominated by a correlation length. packing, and the average • is over many large packings It is interesting to speculate on why the sample-to- all at the same pressure, realized by different initial par- sample fluctuations are much smaller than the fluctua- ticle positions prior to the energy minimization. If δZi tions in a subsystem. Subsystems, by definition, have 3

(a) (b) 2d 3d (a) 2d (b) N = 200 1.0 100 ∆Z ∆Z 1.0 N = 1000 1 2 2 1 2 ρ σZ 1 10 N = 4000 ρ σZ 10− 1 N = 16000 7 . 2 0 N = 64000 1 2 δ Z Z 2 10− δ Z N 1 δ

1 Z 10 10− ∆Z = 1.22 2 ∆Z = 0.53 δ ∆Z = 0.16 ∆Z = 0.39 0 ∆Z = 0.12 2 10 ∆Z = 0.05 10−

1 2 3 4 5 1 2 3 4 5 2 1 0 1 2 3 10 10 10 10 10 10 10 10 10 10 10− 10− 10 10 10 10 N N ∆Z ∆ZN 0.71

3 (c) 3d (d) 10 N = 200 N = 1000 Figure 2. Comparison of the ﬂuctuations of contacts of entire N = 4000 2 5 2 N = 16000 packings (full line), δ Z, to the contact ﬂuctuations of a sub- 7 10 1.25 1 .

0 ∆Z 2 10− N = 64000 Z 2 system (dashed line) σZ /ρ with the same number of particles N δ 2 1.25 Z ∆Z 2 in 2d (a) and 3d (b). For σ /ρ, error bars are shown only for 1 Z δ 10 some points to ease visualization.

100 1 0 0 1 2 3 10− 10 10 10 10 10 boundaries, and these always entail a surface contribu- ∆Z ∆ZN 0.6 tion that scales as `−1. This suggests that δ2Z, which (e) 4d (f) 103 N = 200 lacks these surface ﬂuctuations, measures the bulk con- N = 1000 tribution to the ﬂuctuations, scaling as ∆Zνf , as noted in N = 4000 5 N = 16000 1.25 7 ∆Z . 2 0 10 N = 64000 Eq. (4). We will demonstrate that this scaling holds for Z 1 2 10− N

−ν δ

d Z both ensembles on length scales larger than ξd = ∆Z , 1.25 2 ∆Z δ and below this length scale finite size effects are present. 101 The length ξd is smaller than both ξZ and ξf in d > 2, so that ν < ν < ν . d z f 10 1 100 100 101 102 103 2 − 0.6 In order to measure νd and νf , in Fig. 3 we plot δ Z as ∆Z ∆ZN a function of ∆Z, for different system sizes, in d = 2, 3, 4. In all cases there is a dependence on system size, mostly Figure 3. δ2Z as a function of ∆Z in two (a), three (c) and observed at small values of ∆Z, where δ2Z decreases with four dimensions (e). The curves are collapsed by rescaling the system size. This effect seems to be smaller in 2d than in axis with different power of N in two (b), three (d) and four 3d and 4d. For large enough systems the curves appear to dimensions (f). converge, and for these we measure νf , from Eq. (4). The 2d values of νf are consistent with Ref. [6], where νf ≈ 1.0 3d and νf ≈ 1.25 (see Appendix for a comparison between 0 2d and 3d). The variation between 2d and 3d is also (a) 10 MK2d (b) 103 N = 200 N = 1000 N = 4000 manifest in the qualitative shape of the curves: unlike 102 5

7 N = 8000 1.25 1.25 . the 3d curves, the 2d results taper oﬀ at large values of 1 ∆Z 0 ∆Z

Z 10− 2 N 1 δ ∆Z. The behavior in 4d is consistent with the 3d case, Z 10 2

4d δ νf ≈ 1.25. To characterize the system size dependence, 100 2 we collapse the different curves by assuming a scaling 10− form, 2 1 0 1 0 1 2 3 10− 10− 10 10− 10 10 10 10 ∆Z ∆ZN 0.6 δ2Z = N −βf (∆ZN α) . (5) (c) (d) N = 200 0 MK3d 103 Requiring that in the limit of N → ∞, δ2Z ∝ ∆Zνf is 10 N = 1000 N = 4000 5 independent of the system size, yields β = ανf . Hence, 7 2 N = 16000 . 10 0 ∆Z 1.25 Z 2 given ν , the curves can be collapsed by varying a single 1.25 N f 1 ∆Z δ 10− Z 2 1 exponent. Because the number of particles is propor- δ 10 tional to the volume, Ld, the argument of Eq. (5) can αd 100 2 α L −νd 10− be rewritten as ∆ZN ∝ , with ξd ≡ ∆Z and 2 1 0 0 1 2 3 ξd 10− 10− 10 10 10 10 10 ∆Z ∆ZN 0.6 νd ≡ 1/(αd). The collapse shown in Fig.3 yields approx- imately α ≈ 0.6 both in 3d and 4d, implying that νd depends on dimension. In 3d the exponent νd is smaller Figure 4. δ2Z as a function of ∆Z for the MK mean-field than νz and νf , measured in Ref. [6]. In 2d it is difficult model in (a) 2d and (c) 3d. The curves are collapsed by 2d to determine α; because νf ≈ 1.0 there is a range of rescaling the axis with different powers of N in (b) and (d), exponents α=β which collapse the data reasonably well. respectively. We also consider the mean-field limit of the jamming 4

(a) 0 (b) model, by simulating the Mari-Kurchan (MK) model [23]. 10 This model retains most of the details of the jamming 0.2 model, including the interaction potential in Eq. (1), but 0.15 aims at disrupting the spatial correlations by varying the spatial metric. A given particle i sees particle j at a lo- 10-1 0.1 cation shifted by a random value, dij, which is uniformly distributed through space. The potential between par- 10-5 10-4 10-3 10-2 10-5 10-4 10-3 10-2 ticle i and j is thus given by Uij = U (|ri − rj − dij|), and the interaction between particles does not depend on the actual Euclidean distance between them. As a Figure 5. The average ∆Z (a) and fluctuations δ2Z (b) as a consequence, spatial dimension d is not expected to play function of pressure for the breathing particle model. Note any role in the criticality, and the model is mean-field. that unlike the usual jamming transition, the contact fluc- Fig. 4 shows the results of the simulations of the MK tuations are non-zero at the transition. The system is two dimensional with N = 4000 particles. model in 2d and 3d, which overall appear very similar to the 3d and 4d non-mean field variant. In the large system νMK size limit, the MK curves appear to converge to ∆Z f with our findings that 2d appears somewhat different. In MK and νf ≈ 1.25 independently of dimension. The expo- the latter case, δ2Z would collapse with the 2d correla- nent is also very similar to the 3d and 4d result, suggest- 2d tion length, implying that α2d = 1/(2νz ). The collapse ing that the upper critical dimension is less than three [7]. in Fig. 3 shows reasonable agreement, but the finite range The MK curves can be collapsed using the scaling form of our data allows other values as well. Note that it has in Eq. (5), with exponents agreeing within errors with long been thought that du = 2 [4, 26]. This is mostly those of the 3d and 4d jamming model. Hence, the col- based on observations that exponents appear to be inde- 2 lapse of δ Z does not depend on the length of the system pendent of dimension, for d ≥ 2 [27–30]. but rather its volume, or number of particles. Because We also consider a different class of jamming models, α is independent of dimension (for 3d and above), the which are not isostatic at the jamming transition [31], as −ν scaling of the length scale ξd ∝ ∆Z d , with νd = 1/αd, it is the case for packings of non-spherical particles [32– does depends on dimension. This is contrasted by the 34]. The inter-particle interactions are given by Eq. (1), collapse of the fluctuations in a subsystem with `/ξf , the but the particle radii are also considered as degrees of length ξf scaling independently of dimension. freedom of “breathing particles”. To insure that particles We now discuss the finite size scaling of δ2Z. The do not shrink to zero, a confining potential is assigned to theory of finite size scaling above the upper critical di- the radii: mension du that has emerged from work on the Ising X kr 2 model [24, 25] predicts two types of scalings, depending U = R − R0 . (6) r 2 i i on the quantities considered. Finite size scaling of fluc- i tuations of whole systems collapse as a function of the system size (number of particles). The intuitive reason is To avoid crystallization, in 2d we consider a bidisperse 0 that some mean-field models cannot be embedded in Eu- particle distribution, where the diameter Ri of half of the clidean space, and are thus defined by system size alone. particles is larger by a factor 1.4 than that of the others. Nonetheless, the scaling of quantities associated to a fi- An important feature of this model is the scaling of the nite wavelength depend on the ratio of a length and the stiffness kr. To achieve a radii distribution with finite correlation length, whose exponent is given by its mean- width at jamming, kr = P k0, where P is the pressure field value. Hence, quite generally one can define two and k0 sets the overall magnitude of the stiffness. In diverging length scales for systems above du, in contrast the limit of k0 → ∞, the behavior of the usual jamming to dimension below du where only one correlation scale is transition is recovered. relevant. In the jamming model this scenario is realized Unlike the usual jamming transition, in the limit of 2 2 by a different collapse of σZ and δ Z. One can exploit P → 0, the system is not isostatic, ∆Z > 0. In Fig. 5(a) these different collapses to measure du, by estimating the we show that our simulations are consistent with the re- dimension where the two length scales coincide. Extrap- sults of Ref. [31]. In Fig. 5(b) we plot the contact fluctu- olating νd = 1/(αd) to the dimension where it is equal ations as a function of pressure. In the limit of P → 0, 2 to the mean field exponent yields du. However, as men- the contact fluctuations δ Z remain finite, in contrast to tioned above, in the jamming transition there are two the vanishing fluctuations of the conventional jamming length scales that characterize contact statistics [6]: ξZ , transition. This suggests that isostaticity is crucial for which characterizes the two point correlation function, the suppression of fluctuations and the divergence of cor- and ξf , which characterizes the cross-over of the hyper- relation lengths. uniform fluctuations to the normal fluctuations. We ar- We conclude by reiterating our main results. Fluc- gue that the first is a more fundamental quantity, and tuations in whole periodic systems are smaller than in 3d using the exponent νz ≈ 0.85 measured in [6] we obtain subsystems of the same size. This is most significant 2 that νd = νz when du ≈ 2. This estimate is consistent at the jamming transition where δ Z → 0, while the 5

−1 ∂U fluctuations in a subsystem scale as ` , which is the implying that P = − ∂V = P0. The pressure of the sys- 2 2 fastest possible decay. Moreover, δ Z and σZ converge tem, P , is given by the diagonal of the virial stress tensor: to their thermodynamic value with different exponents 1 X for 3d and above. The sample-to-sample fluctuations, τ = r f . (9) δ2Z, approach their asymptotic value at system sizes ij V b,i b,j d −1/α b N ∼ (ξd) ∼ ∆Z , where α ≈ 0.6 is independent of dimension. Fluctuations in a subsystem only reach their The sum is over all bonds, V is the volume, rb is a vector −ν asymptotic value at a system length ` ∼ ξf ∼ ∆Z f , that connects the center of two interacting particles and where ν ≈ 1.25 is also independent of dimension. Un- f fb is inter-particle force along and it is pointed in the der typical system sizes and values of ∆Z, usually con- same direction as fb. sidered in simulations, the system is well below ξf . It is therefore interesting to explore if new behaviors arise for large systems and how ξ and ξ affect the behav- f Z Definition of σ2 (`). ior of the packings. As a byproduct of this analysis, Z we obtain an estimate of the upper critical dimension 2 du ≈ 2, as the dimension where the distinction between The definition of σZ (`) given in the main text is ξd and ξZ disappears. Our results also show that a signa- ture of suppressed fluctuations survive in mean-field vari- 2 1 X 2 σ (`) = δZi (10) ants of the jamming model. The exact solution of these Z `d i∈`d models [35] enables, in principle, the analytic calculation of νf , although the calculation is technically involved. where The exponent νz that characterizes the spatial decay of hδZ (r) δZ (0)i still remain inaccessible to present theory. δZi = Zi − Z (11) Acknowledgments – We warmly thank Andrea Liu and Sid Nagel for important discussions. This work was sup- and ported by a grant from the Simons Foundation (#348125, Sid Nagel and Daniel Hexner, and #454955, Francesco N 1 X Zamponi) and from "Investissements d’Avenir" LabEx Z = Zi (12) N PALM (ANR-10-LABX-0039-PALM). i=1

A small variant of Eq. (10) is to replace Z with Z which APPENDIX is the sample-to-sample average of Z. We denote the 2 2 corresponding σZ as σZ (`). For ` → ∞ we have that

Minimization at constant pressure 2 2 σZ (` → ∞)/ρ ≡ δ Z(N → ∞) (13)

In this section we present further details of how pack- However, since for N → ∞, Z → Z > 0 we get that the ings are prepared at a constant pressure. As discussed in 2 2 difference between σZ and σZ is a subleading term that the main text the energy of the system depends on the vanishes for N → ∞ and ` → ∞, so that inter-particle distance: 2 2 2 σZ (` → ∞)/ρ ≡ δ Z(N → ∞) (14) 1 rij Uij = k 1 − Θ(Ri + Rj − rij) . (7) 2 Ri + Rj Finally the same argument holds if we replace Z by its local average Z(i), meaning its average inside the box in Working at a finite pressure provides a tighter control which Zi is computed in Eq. (11). Since for ` → ∞, over the distance from the jamming transition, than at Z(i) → Z > 0 up to subleading corrections, one can in- constant packing fraction. In the latter, near the jam- terchange the definitions without affecting the large ` be- ming transitions some of the packings may be under con- havior. strained and some packings could be over constrained. To maintain a constant pressure we minimize the enthalpy

Comparison of νf in 2d and 3d H = U + P0V. (8)

Here the target pressure is P0. We employ the FIRE In the main text we showed that in the large system 2 νf 2d minimization algorithm, which evolves based in the gra- limit δ Z ∝ ∆Z , where our data suggested that νf ≈ dients of energy (or enthalpy) [9]. The volume of the box 1.0, while in higher dimension νf ≈ 1.25. To visualize is also a coordinate that varies during the minimization these two possible scalings we plot these two power-laws. and its dynamics depend on the gradient with respect to We note that the exponents are deduced based on the ∂U ∂H 2 2 the volume, P = − ∂V . When H is a minimum ∂V = 0, collapse of δ Z, as well as the collapse of σf in Ref. [6]. 6

(a) N = 200 ∆Z 1.0 N = 1000 102 N = 4000

1 N = 16000 7 . 0 N = 64000

N 1.25 1 ∆Z

Z 10 2 δ

100 2d

100 101 102 103 ∆ZN 0.71

(b) 103 N = 200 N = 1000 1.0 N = 4000 ∆Z

5 2 N = 16000 7

. 10 1.25 0 N = 64000 ∆Z N Z 2

δ 101

100 3d 100 101 102 103 ∆ZN 0.6

5 N = 16000 1.25 7 ∆Z . 2 0 10 N = 64000 N Z 2 δ 101

4d 100 101 102 103 ∆ZN 0.6

Figure 6. A comparison of the two slopes, ∆Z1.0 and ∆Z1.25. 2d In two dimension the collapse suggests νf ≈ 1.0, while in three and four dimensions νf ≈ 1.25.

[1] J. Kurchan and D. Levine, Journal of Physics A: Math- [13] S. Torquato and F. H. Stillinger, Phys. Rev. E 68, 041113 ematical and Theoretical 44, 035001 (2010). (2003). [2] Wyart, M., Ann. Phys. Fr. 30, 1 (2005). [14] A. Donev, F. H. Stillinger, and S. Torquato, Phys. Rev. [3] M. van Hecke, Journal of Physics: Condensed Matter 22, Lett. 95, 090604 (2005). 033101 (2009). [15] A. Ikeda and L. Berthier, Phys. Rev. E 92, 012309 (2015). [4] A. J. Liu and S. R. Nagel, Annu. Rev. Condens. Matter [16] B. Jancovici, Phys. Rev. Lett. 46, 386 (1981). Phys. 1, 347 (2010). [17] D. Hexner and D. Levine, Phys. Rev. Lett. 114, 110602 [5] C. S. O’Hern, S. A. Langer, A. J. Liu, and S. R. Nagel, (2015). Phys. Rev. Lett. 88, 075507 (2002). [18] E. Tjhung and L. Berthier, Phys. Rev. Lett. 114, 148301 [6] D. Hexner, A. J. Liu, and S. R. Nagel, Physical review (2015). letters 121, 115501 (2018). [19] J. H. Weijs, R. Jeanneret, R. Dreyfus, and D. Bartolo, [7] C. P. Goodrich, A. J. Liu, and S. R. Nagel, Phys. Rev. Physical review letters 115, 108301 (2015). Lett. 109, 095704 (2012). [20] T. Goldfriend, H. Diamant, and T. A. Witten, Phys. [8] The radii are uniformly distributed in the range Rev. Lett. 118, 158005 (2017). [Rmin,Rmax], with Rmax/Rmin = 1.4. [21] Q.-L. Lei, M. P. Ciamarra, and R. Ni, Science advances [9] E. Bitzek, P. Koskinen, F. Gähler, M. Moseler, and 5, eaau7423 (2019). P. Gumbsch, Phys. Rev. Lett. 97, 170201 (2006). [22] S. Torquato, Physics Reports (2018), [10] S. Alexander, Physics reports 296, 65 (1998). https://doi.org/10.1016/j.physrep.2018.03.001. [11] S. Edwards, Physica A: Statistical Mechanics and its Ap- [23] R. Mari and J. Kurchan, The Journal of chemical physics plications 249, 226 (1998). 135, 124504 (2011). [12] C. F. Moukarzel, Physical review letters 81, 1634 (1998). 7

[24] K. Binder, M. Nauenberg, V. Privman, and A. P. Young, Physics 8, 265 (2017). Phys. Rev. B 31, 1498 (1985). [31] C. Brito, E. Lerner, and M. Wyart, Phys. Rev. X 8, [25] M. Wittmann and A. P. Young, Phys. Rev. E 90, 062137 031050 (2018). (2014). [32] C. Brito, H. Ikeda, P. Urbani, M. Wyart, and F. Zam- [26] M. Wyart, in Annales de Physique, Vol. 30 (EDP Sci- poni, Proceedings of the National Academy of Sciences ences, 2005) pp. 1–96. 115, 11736 (2018). [27] C. S. O’Hern, L. E. Silbert, A. J. Liu, and S. R. Nagel, [33] A. Donev, R. Connelly, F. H. Stillinger, and S. Torquato, Phys. Rev. E 68, 011306 (2003). Phys. Rev. E 75, 051304 (2007). [28] P. Charbonneau, E. I. Corwin, G. Parisi, and F. Zam- [34] A. Donev, I. Cisse, D. Sachs, E. A. Variano, F. H. Still- poni, Physical review letters 114, 125504 (2015). inger, R. Connelly, S. Torquato, and P. M. Chaikin, [29] P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and Science 303, 990 (2004). F. Zamponi, Nature communications 5, 3725 (2014). [35] P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, and [30] P. Charbonneau, J. Kurchan, G. Parisi, P. Urbani, F. Zamponi, Journal of Statistical Mechanics: Theory and F. Zamponi, Annual Review of Condensed Matter and Experiment 2014, P10009 (2014).