Abstract
In the late 1980s, Sam Edwards proposed a possible statisticalmechanical framework to describe the properties of disordered granular materials^{1}. A key assumption underlying the theory was that all jammed packings are equally likely. In the intervening years it has never been possible to test this bold hypothesis directly. Here we present simulations that provide direct evidence that at the unjamming point, all packings of soft repulsive particles are equally likely, even though generically, jammed packings are not. Typically, jammed granular systems are observed precisely at the unjamming point since grains are not very compressible. Our results therefore support Edwards’ original conjecture. We also present evidence that at unjamming the configurational entropy of the system is maximal.
Main
In science, most breakthroughs cannot be derived from known physical laws: they are based on inspired conjectures^{2}. Comparison with experiment of the predictions based on such a hypothesis allows us to eliminate conjectures that are clearly wrong. However, there is a distinction between testing the consequences of a conjecture and testing the conjecture itself. A case in point is Edwards’ theory of granular media. In the late 1980s, Edwards and Oakeshott^{1} proposed that many of the physical properties of granular materials (‘powders’) could be predicted using a theoretical framework that was based on the assumption that all distinct packings of such a material are equally likely to be observed. The logarithm of the number of such packings was postulated to play the same role as entropy does in Gibbs’ statisticalmechanical description of the thermodynamic properties of equilibrium systems. However, statisticalmechanical entropy and granular entropy are very different objects. Until now, the validity of Edwards’ hypothesis could not be tested directly—mainly because the number of packings involved is so large that direct enumeration is utterly infeasible—and, as a consequence, the debate about the Edwards hypothesis has focused on its consequences, rather than on its assumptions. Here we present results that show that now, at last, it is possible to test Edwards’ hypothesis directly by numerical simulation. To our own surprise, we find that the hypothesis appears to be correct precisely at the point where a powder is just at the (un)jamming threshold. However, at higher densities, the hypothesis fails. At the unjamming transition, the configurational entropy of jammed states appears to be at a maximum.
The concept of ‘ensembles’ plays a key role in equilibrium statistical mechanics, as developed by J. Willard Gibbs, well over a century ago^{3}. The crucial assumption that Gibbs made to arrive at a tractable theoretical framework to describe the equilibrium properties of gases, liquid and solids was that, at a fixed total energy, every state of the system is equally likely to be observed. The distinction between, say, a liquid at thermal equilibrium and a granular material is that in a liquid, atoms undergo thermal motion whereas in a granular medium (in the absence of outside perturbations) the system is trapped in one of (very) many local potential energy minima. Gibbsian statistical mechanics cannot be used to describe such a system. The great insight of Edwards was to propose that the collection of all stable packings of a fixed number of particles in a fixed volume might also play the role of an ‘ensemble’ and that a statistical mechanicslike formalism would result if one assumed that all such packings were equally likely to be observed, once the system had settled into a mechanically stable ‘jammed’ state. The nature of this ensemble has been the focus of many studies^{1,4,5,6}.
Jamming is ubiquitous and occurs in materials of practical importance, such as foams, colloids and grains when they solidify in the absence of thermal fluctuations. Decompressing such a solid to the point where it can no longer achieve mechanical equilibrium leads to unjamming. Studies of the unjamming transition in systems of particles interacting via soft, repulsive potentials have shown that this transition is characterized by powerlaw scaling of many physical properties^{7,8,9,10,11,12}. However, the exact nature of both the ensemble of jammed states and the unjamming transition remains unclear.
Here, we report a direct test of the Edwards conjecture, using a numerical scheme for computing basin volumes of distinct jammed states (energy minima) of N polydisperse, frictionless discs held at a constant packing fraction φ. Uniquely, our numerical scheme allows us to compute Ω, the number of distinct jammed states, and the individual probabilities p_{i∈{1, …, Ω}} of each observed packing to occur. Figure 1a shows a snapshot of a section of the system, consisting of particles with a hard core and a soft shell. We obtain jammed packings by equilibrating a harddisc fluid and inflating the particles instantaneously to obtain the desired softdisc volume fractions (φ), followed by energy minimization (see Methods). The minimization procedure finds individual stable packings with a probability p_{i} proportional to the volume v_{i} of their basin of attraction. Averages computed using this procedure, represented by , would then lead to a bias originating from the different v_{i} values. Recent advances in numerical methods^{13,14,15,16} have now enabled direct computation of v_{i}, and therefore, an unbiased characterization of the phase space. A summary of the technique is provided in the Methods.
We report a detailed analysis of the distribution of v_{i} for a fixed number of discs N = 64 (all maximum system sizes in our study were set by the current limits on computing power). We compute v_{i} using a thermodynamic integration scheme^{13,14,15,16}, and compute the average basin volume 〈v〉(φ). The number of jammed states is, explicitly, Ω(φ) = V_{J}(φ)/〈v〉(φ), where V_{J}(φ) is the total available phasespace volume at a given φ. A convenient way to check equiprobability is to compare the Boltzmann entropy S_{B} = lnΩ − lnN!, which counts all packings with the same weight, and the Gibbs entropy S_{G} = −∑ _{i}^{Ω}p_{i} lnp_{i} − lnN! (refs 17,18,19). The Gibbs entropy satisfies S_{G} ≤ S_{B}, saturating the bound when all p_{i} are equal: p_{i∈{1, …, Ω}} = 1/Ω. As shown in Fig. 2a, S_{G} approaches S_{B} from below as φ → φ_{N=64}^{∗(S)} ≈ 0.82. Fig. 1b, c schematically illustrates the evolution of the basin volumes as the packing fraction is reduced.
To characterize the distribution of basin volumes, we analyse the statistics of v_{i} along with the pressure P_{i} of each packing. It is convenient to study F_{i} ≡ − lnv_{i} as a function of Λ_{i} ≡ lnP_{i}. As shown in Fig. 2b, we observe a strong correlation between F_{i} and Λ_{i}, which we quantify by fitting the data to a bivariate Gaussian distribution. The conditional expectation of F given Λ then yields a linear relationship (denoted by solid lines in Fig. 2b) such that , where represents the average over all basins at a given Λ. Previous studies at higher packing fractions^{13} indicate that this relationship is preserved in the thermodynamic limit. Defining f = F/N, we have (see Methods for details)
where . For Edwards’ hypothesis to be valid, we require that in the thermodynamic limit: the distribution of volumes approaches a Dirac delta, which follows immediately from the fact that the variance σ_{f}^{2} ∼ 1/N (ref. 15) (see Supplementary Information); and F_{i} needs to be independent of Λ_{i}, as well as of all other structural observables (see Supplementary Information), and therefore λ(φ) must necessarily vanish. As can be seen from Fig. 2c, d, within the range of volume fractions studied, λ(φ)decreases but saturates to a minimum as φ → φ_{N=64}^{∗(λ)}. We argue below that the saturation is a finitesize effect. An extrapolation using the linear regime in Fig. 2c indicates that λ → 0 at packing fraction φ_{N=64}^{∗(λ)} = 0.82 ± 0.07, remarkably close to where our extrapolation yields S_{G} = S_{B}. The analysis of basin volumes, therefore, strongly suggests that equiprobability is approached only at a characteristic packing fraction and that the vanishing of λ(φ) can be used to estimate the point of equiprobability.
We next show that λ(φ) does indeed tend to zero in the thermodynamic limit. We use the fluctuations σ_{f}^{2}, σ_{Λ}^{2}, and the covariance σ_{fΛ}^{2}, obtained from the elements of the covariance matrix of the joint distribution of f and Λ (see Methods for details), to define λ and c as:
From Fig. 2b we observe that the decrease of λ is driven by σ_{Λ}^{2} increasing to a maximum, while σ_{f}^{2} and σ_{fΛ}^{2} decrease (see Supplementary Fig. 2). We expect the main features of these distributions to be preserved as the system size N is increased^{13}, which suggests that for larger N, where basin volume calculations are still intractable for multiple densities, the maximum in σ_{Λ}^{2} can be used to identify φ_{N}^{∗}. We have directly measured χ_{Λ} = Nσ_{Λ}^{2} using our sampling scheme—which samples packings with probability proportional to the volume of their basin of attraction—for systems of up to N = 128 discs (see inset of Fig. 3a) and finitesize scaling indicates that χ_{Λ} diverges as φ → φ_{N→∞}^{∗(Λ)} = 0.841(3) (see Supplementary Information). The saturation of λ to a minimum as φ → φ_{N}^{∗}, for small N, is determined by the fact that χ_{Λ} diverges only in the thermodynamic limit; a detailed discussion is given in the Methods.
Interestingly, we find evidence that in the thermodynamic limit, the point of equiprobability, φ_{N→∞}^{∗}, coincides with the point at which the system unjams, φ_{N→∞}^{J}. We use two characteristics of the unjamming transition to locate φ_{N→∞}^{J}: the average pressure of the packings goes to zero, and therefore 〈Λ〉 → −∞ (see Fig. 3b); and the probability of finding jammed packings, p_{J}, goes to zero (see inset of Fig. 3b). A scaling analysis indicates that 〈Λ〉 → −∞ as φ_{N→∞}^{J(Λ)} = 0.841(3), and p_{J} → 0 as φ_{N→∞}^{J(pJ)} = 0.844(2) (see Supplementary Information). We thus find that φ_{N→∞}^{∗} = φ_{N→∞}^{J} within numerical error and up to corrections to finitesize scaling^{20}. Our simulations therefore lead to the surprising conclusion that the Edwards conjecture appears to hold precisely at the (un)jamming transition. We note that our earlier simulations, which were performed at densities much above jamming^{13,15}, did not support the equiprobability hypothesis. The earlier simulations were in fact far enough away from unjamming that the emergence of equiprobability at this point could not be anticipated. Our earlier findings, therefore, do not contradict the more recent ones and are completely consistent with these.
Why is χ_{Λ} related to the unjamming transition? As the particles interact via purely repulsive potentials, the pressure P is strictly positive, which implies that the fluctuations of P have a floor and go to zero at unjamming. The relative fluctuations can be nonzero, and a diverging χ_{P} would then imply a diverging χ_{Λ}. Because of the bounded nature of P (refs 21,22,23), however, χ_{P} can diverge only at the unjamming transition where (see Methods). We find that χ_{P} does diverge (Fig. 3a) and finitesize scaling yields φ_{N→∞}^{∗(P)} = 0.841(3), in agreement with what has been found for χ_{Λ}. Returning to the N = 64 case that we have analysed using the basin volume statistics, we find that both χ_{P} and χ_{Λ} saturate to their maximum values over similar ranges of φ and our estimate φ_{N=64}^{∗} ≈ 0.82, where S_{G} = S_{B} and λ → 0, falls in this region. In addition, the average number of contacts is close to the isostatic value z_{N=64}^{(iso)} ≡ 2d − 2/64 ≈ 3.97 (ref. 10) (see Supplementary Information).
Finally, we note that the states in the generalized Edwards ensemble^{5,21,24,25} characterized by φ and P have basin volumes that are similar, if not identical, over the full range of φ that we have explored (see scatter plot in Fig. 2b), indicating that equiprobability in the stress–volume ensemble^{5,24} is a more robust formulation of the Edwards hypothesis. This observation is consistent with recent experiments^{26}.
Although, the equiprobability of jammed states at a given packing fraction was posited by Edwards for jammed packings of hard particles, our analysis shows that for soft particles, the Edwards hypothesis is valid only for the marginally jammed states at φ_{N→∞}^{∗} = φ_{N→∞}^{J}, where the jamming probability vanishes, the entropy is maximized, and relative pressure fluctuations diverge. We have shown not only that there exists a practical ‘Edwardsian’ packing generation protocol, capable of sampling jammed states equiprobably, but we have uncovered an unexpected property of the energy landscape for this class of systems. At this stage, we cannot establish whether the same considerations are valid in three dimensions, although the already proven validity of equation (16) in three dimensions would suggest so^{13}. The exact value of the entropy at unjamming, whether finite or not, also needs to be elucidated. The implications for ‘soft’ structural glasses is apparent: at φ^{J} the uniform size of the basins implies that the system, when thermalized, has the same probability of visiting all of its basins of attraction; hence, there are no preferred inherent structures. This could be a signature of the hardsphere transition occurring at the same point^{27}. Our approach can, therefore, be extended to spinglasses and related problems, and it would be clearly very exciting to explore the analogies and differences between ‘jamming’ in various systems for which the configuration space can break up into many distinct basins.
Methods
Packing preparation protocol.
In this section we describe the algorithm that we have used to sample the phase space of jammed packings. This procedure samples each configuration proportionally to the volume of its basin of attraction.
Hardsphere fluid sampling.
We start by equilibrating a fluid of N hard discs (that serve as the cores of the particles with soft outer shells) at a volume fraction φ_{HS} in a square box with periodic boundary conditions. The particle radii are sampled from a truncated Gaussian distribution with mean μ = 1 and standard deviation σ = 0.1. We achieve equilibration by performing standard Markov chain Monte Carlo simulations consisting of singleparticle displacements and particle–particle swaps, as in ref. 13. To ensure statistical independence, we draw fluid configurations every n_{MC} steps, where n_{MC} is (precomputed by averaging over multiple simulations) the total number of Markov chain Monte Carlo steps necessary for each individual particle to diffuse at least a distance equal to the largest diameter in the system.
Soft shells and minimization.
We next take each equilibrated harddisc fluid configuration and inflate the particles (instantaneously) with a WCAlike soft outer shell^{30}, to reach the target soft packing fraction φ_{SS} > φ_{HS}. Each hard sphere is inflated proportionally to its radius, so that the softsphere radius is given by
where d is the dimensionality of the box (2 in our case), and r_{SS} and r_{HS} are the soft and hardsphere radii respectively. Clearly, this procedure does not change the polydispersity of the sample. The radii are identical across volume fractions and system sizes, and the harddisc fluid density is chosen so that the radius ratio of hard to soft discs is (0.88/0.7)^{1/2} ≈ 1.121.
Next, particle inflation is followed by energy minimization using FIRE^{31,32}, to produce mechanically stable packings at the desired soft volume fractions φ. This protocol has the advantage of generating packings sampled proportionally to the volume of their basin of attraction. In our simulations, we considered all mechanically stable packings, irrespective of the number of ‘rattlers’. To guarantee mechanical stability we required that the total number of contacts is sufficient for the bulk modulus to be strictly positive, N_{min} = d(N_{nr} − 1) + 1 (ref. 33), where N_{nr} is the number of nonrattlers and d is the dimensionality of the system.
Our implementation of FIRE enforces a maximum step size (set to be equal to the softshell thickness) and forbids uphill steps by taking one step back every time the energy increases (and restarts the minimizer in the same fashion as the original FIRE implementation). We use a maximum time step Δt_{max} = 1, although the maximum step size is directly controlled in our implementation. All other parameters are set as in the original implementation^{31}.
HSWCA potential.
We define the WCAlike potential around a hard core as follows: consider two spherical particles with a distance between the hard cores r_{HS}, implying a softcore contact distance r_{SS} = r_{HS}(1 + θ), with θ = (φ_{SS}/φ_{HS})^{1/d} − 1. We can then write a horizontally shifted hardsphere plus WCA (HSWCA) potential as
where σ(r_{HS}) = (2θ + θ^{2})r_{HS}^{2}/2^{1/6} guarantees that the potential function and its first derivative go to zero at r_{SS}. For computational convenience (avoidance of squareroot evaluations), the potential in equation (4) differs from the WCA form in that the interparticle distance in the denominator of the WCA potential has been replaced with a difference of squares.
A power series expansion of equation (4) yields
hence, in the limit of no overlap the pair potential is harmonic.
We numerically evaluate this potential, matching the gradient and linearly continuing the function u_{HSWCA}(r) for r ≤ r_{HS} + ɛ, where ɛ > 0 is an arbitrary small constant, such that minimization is still meaningful if hardcore overlaps do occur.
Our choice of potential is based on the fact that: the hard cores greatly reduce the amount of configurational space to explore, replacing expensive energy minimizations (to test whether the random walker has stepped outside the basin) with fast hardcore overlap rejections; and the hard cores exclude highenergy minima (jammed packings) that are not ‘hardspherelike’.
Total accessible volume.
The basins of attraction of energy minima tile the ‘accessible’ phase space (schematically shown in Fig. 1b, c). This inaccessible part of the phase space arises due to hardcore constraints and the existence of fluid states (see, for example, ref. 13). The total phasespace volume is equal to V_{box}^{N}. The inaccessible part of this volume arising from the hardcore constraints (shown as hatched areas in Fig. 1) is denoted by V_{HS}, and the part corresponding to the coexisting unjammed fluid states is denoted by V_{unj} (shown as blue regions with solid dots in Fig. 1b, c). V_{unj} is significant only for finitesize systems at or near unjamming. We denote the space tiled by the basins of mechanically stable jammed packings by V_{J}. We then have V_{J} = V_{box}^{N} − V_{HS} − V_{unj}. In practice we compute V_{J} using the following equation
where f_{ex}(φ_{HS}) is the excess free energy, that is, the difference in free energy between the hardsphere fluid and the ideal gas, computed from the Santos–Yuste–Haro equation of state^{34} as in ref. 13, and p_{J}(φ_{SS}) is the probability of obtaining a jammed packing at soft volume fraction φ_{SS} with our protocol, shown in the inset of Fig. 3b.
Counting by sampling.
We briefly review our approach to computing the number Ω of distinct jammed packings for a system of N soft discs at volume fraction φ. We prepare packings by the protocol described above, which generates jammed structures (energy minima) with probability p_{i} proportional to the volume of their basin of attraction v_{i}. We define the probability of sampling the ith packing as
where V_{J} is the total accessible phase space, such that
Details of the computation of v_{i} are discussed in refs 13,16. To find Ω, we make the simple observation
from which it follows immediately that
The ‘Boltzmannlike’ entropy, suggested in a similar form by Edwards^{1}, is then
where the lnN! correction ensures that two systems in identical macrostates are in equilibrium under exchange of particles^{17,18,19}.
Note that 〈v〉 is the unbiased average basin volume (the mean of the unbiased distribution of volumes). We distinguish between the biased, (as sampled by the packing protocol), and the unbiased, , basin logvolumes distributions (F = −lnv_{basin}). Since the configurations were sampled proportionally to the volume of their basin of attraction, we can compute the unbiased distribution as
where is the normalization constant, such that
Since small basins are much more numerous than large ones, and grossly undersampled, it is not sufficient to perform a weighted average of the sampled basin volumes. Instead, to overcome this problem, one can fit the biased measured basin logvolumes distribution with an analytical (or at least numerically integrable) distribution, and perform the unbiasing via equation (12) on the best fitting distribution. Different approaches to modelling this distribution give rise to different analysis methods, which all yield consistent results as shown in ref. 13. In this work we follow ref. 13 and fit using both a (parametric) generalized Gaussian model^{35} (see Supplementary Equation 21) and a (nonparametric) kernel density estimate with Gaussian kernels^{36,37} and bandwidth selection performed by crossvalidation^{13,38}, yielding consistent results in agreement with ref. 13. Before performing the fit we remove outliers from the freeenergy distribution in an unsupervised manner, as discussed in the ‘Data analysis’ section of the Methods.
No such additional steps are needed to compute the ‘Gibbslike’ version of the configurational entropy, in fact
is simply the arithmetic average of the observed volumes: the sample mean of F = −lnv_{basin} is already correctly weighted because our packing generation protocol generates packings with probability p_{i}.
Powerlaw between pressure and basin volume.
A powerlaw relationship between the volume of the basin of attraction of a jammed packing and its pressure was first reported in ref. 13. In what follows we provide insight into this expression on the basis of this work’s findings. We observe that distributions of basin negative logvolumes, F = −lnv_{basin}, and logpressures, Λ = lnP, are approximately normally distributed (see Supplementary Figs 1 and 9). We therefore expect their joint probability to be well approximated by a bivariate Gaussian distribution (when listing a function’s arguments we place parameters that are held constant before the semicolon), with mean μ = (μ_{F}, μ_{Λ}) and covariance matrix (ref. 39). This is consistent with the elliptical distribution of points in Fig. 2b. For a given random variable X, with an (observed/biased) marginal distribution , the mean is given by . Similarly, the (biased) conditional expectation of F for a given Λ is then^{39}
This is simply the linear minimum mean square error regression estimator for F, that is, the linear estimator that minimizes . The expectation of the dimensionless free energy (ref. 40) is the average basin negative logvolume at volume fraction φ and logpressure Λ. Here the average is also taken over all other relevant, but unknown, order parameters Γ, such that . In other words, we write the expectation of F at a given pressure as the (biased) average over the unspecified order parameters Γ. An example of such a parameter would be some topological variable that makes certain topologies more probable than others. Note that F(φ, Λ; Γ) is narrowly distributed around . To simplify the notation we write . We can thus rewrite the power law reported in ref. 13 as
where f = F/N is the basin negative logvolume per particle and λ ≡ 1/κ is the slope of the powerlaw relation, which depends crucially on the packing fraction φ. The last equality in equation (16) highlights how λ(φ) controls the contributions of the fluctuations of the logpressures ΔΛ ≡ Λ − μ_{Λ}(φ) to changes in the basin negative logvolume. Note that one can rewrite the ratio of fluctuations as σ_{fΛ}^{2}/σ_{Λ}^{2} = ρ_{fΛ}σ_{f}/σ_{Λ} where ρ_{fΛ} = σ_{fΛ}^{2}/(σ_{f}σ_{Λ}) is the linear correlation coefficient of f and Λ. Finally, we can gain further insight into the powerlaw dependence by noting that
Data analysis.
Reduced units. While presenting data from our computations, we express pressure and volume in reduced units as P/P^{∗} and v/v^{∗} respectively. The unit of volume is given by v^{∗} ≡ π〈r_{HS}^{2}〉, where 〈r_{HS}^{2}〉 is the mean squared hardsphere radius. The unit of pressure is then P^{∗} ≡ ε/v^{∗}, where ε is the stiffness of the softsphere potential, defined in equation (4). The pressure is computed as where is the virial stress tensor and V_{box} is the volume of the enclosing box.
Summary of calculations.
For the basinvolume calculations we consider systems of N = 64 discs sampled at a range of 8 volume fractions 0.828 ≤ φ ≤ 0.86 and for each φ we measure the basin volume for about 365 < M < 770 samples.
For the finitesize scaling analysis of the relative pressure fluctuations we study system sizes N = 32, 48, 64, 80, 96 and 128 for 48 volume fractions in the range 0.81 ≤ φ ≤ 0.87. For each system size we generate up to 10^{5} harddisc fluid configurations and compute the pressure for between approximately 10^{3} and 10^{4} jammed packings (depending on the probability of obtaining a jammed packing at each volume fraction).
Simulations were performed using the open source libraries PELE^{41} and MCPELE^{42}.
Outlier detection and robust covariance estimation.
Before manipulating the raw data we remove outliers from the joint distribution following the distancebased outlier removal method introduced in ref. 43. This is applied in turn to each dimension, such that we choose to keep only those points for which at least R = 0.5 of the remaining data set is within D = 4σ (compared with the much stricter R = 0.9988, D = 0.13σ required to exclude any points further than μ − 3σ for normally distributed data^{43}). On our data sets we find that this procedure removes typically none and at most 0.8% of all data points.
Mean and covariance estimates of are computed using a robust covariance estimator, namely the minimum covariance determinant (MCD) estimator^{37,44} with support fraction h/n_{samples} = 0.99. The MCD estimator defines μ_{MCD}, the mean of the h observations for which the determinant of the covariance matrix is minimal, and , the corresponding covariance matrix^{45}. We use these robust estimates of the location and of the covariance matrix (computed over 1,000 bootstrap samples^{28}) to fit our observations by linear minimum mean square error^{39} (see Fig. 2).
Before fitting (required to compute Ω), we perform an additional step of outlier detection based on an elliptic (Gaussian) envelope criterion constructed using the MCD estimator. We assume a support fraction h/n_{samples} = 0.99 and a contamination equal to 10% (ref. 37). We compute S_{G} and S_{B} from the resulting data sets. The procedure is strictly unsupervised and allows us to achieve robust fits despite the small sample sizes. We fit using both a (parametric) generalized Gaussian model^{35} and a (nonparametric) kernel density estimate with Gaussian kernels^{36,37} and bandwidth selection performed by crossvalidation^{13,38}.
Error analysis.
Errors were computed analytically where possible and propagated using the ‘uncertainties’ Python package^{46}. Alternatively, intervals of confidence were computed by bootstrap for the covariance estimation^{28} and by BCa bootstrap otherwise using the ‘scikitbootstrap’ Python package^{29,47}.
Data availability.
The data that support the plots within this paper and other findings of this study are available from the University of Cambridge data repository doi: http://dx.doi.org/10.17863/CAM.9985.
Additional Information
Publisher’s note: Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
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Acknowledgements
S.M. acknowledges financial support by the Gates Cambridge Scholarship. K.J.S. acknowledges support by the Swiss National Science Foundation under grant no. P2EZP2152188 and no. P300P2161078. D.F. acknowledges support by EPSRC Programme Grant EP/I001352/1 and EPSRC grant EP/I000844/1. K.R. and B.C. acknowledge the support of NSFDMR 1409093 and the W. M. Keck Foundation.
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S.M., D.F. and B.C. designed the study. S.M. and K.J.S. developed the software and performed the numerical simulations. S.M., K.R., B.C. and D.F. analysed the data and wrote the manuscript.
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Martiniani, S., Schrenk, K., Ramola, K. et al. Numerical test of the Edwards conjecture shows that all packings are equally probable at jamming. Nature Phys 13, 848–851 (2017). https://doi.org/10.1038/nphys4168
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DOI: https://doi.org/10.1038/nphys4168
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