Cavity method for force transmission in jammed disordered packings of hard particles

Lin Bo,1 Romain Mari,1 Chaoming Song,2 and Hernan A. Makse1

1Levich Institute and Physics Department, City College of New York, New York, NY 10031, USA2Physics Department, Northeastern University, Boston, MA, 02115, USA

(Dated: September 30, 2013)

The force distribution of jammed disordered packings has always been considered a central objectin the physics of granular materials. However, many of its features are poorly understood. In partic-ular, analytic relations to other key macroscopic properties of jammed matter, such as the contactnetwork and its coordination number, are still lacking. Here we develop a mean-field approach tothis problem, based on the consideration of the contact network as a random graph where the forcetransmission becomes a constraint optimization problem. We can thus use the cavity method de-veloped in the last decades within the statistical physics of spin glasses and hard computer scienceproblems. This method allows us to compute the force distribution P(f) for random packings of hardparticles of any shape, with or without friction. We find a new signature of jamming in the smallforce behavior P(f) ∼ fθ, whose exponent has attracted recent active interest. We find a finite valuefor P(f = 0), along with θ = 0 over an unprecedented six decades of force data, which agrees withexperimental measurements on emulsion droplets. Furthermore, we relate the force distribution toa lower bound of the average coordination number zmin

c (µ) of jammed packings of frictional sphereswith coefficient µ. This bridges the gap between the two known isostatic limits zc(µ = 0) = 2D(in dimension D) and zc(µ→∞) = D + 1 by extending the naive Maxwell’s counting argument tofrictional spheres. The framework describes different types of systems, such as non-spherical objectsand dimensions, providing a common mean-field scenario to investigate force transmission, contactnetworks and coordination numbers of jammed disordered packings.

PACS numbers:

I. INTRODUCTION

Mechanically stable packings of granular media are im-portant to a wide variety of technical processes [1]. Oneapproach to characterize jammed granular packings is viathe interparticle contact force network. In turn, this net-work determines the probability density of inter-particlecontact forces P(f) and the average coordination numberz. While the force network has been studied for years,there is yet no unified theoretical framework to explainthe common observations in granular packings, rangingfrom frictional to frictionless systems, from spherical tonon-spherical particles and in any dimensions.

Experimental force measurements [2–8] and simula-tions [9–12] have shown that the interparticle forcesare inhomogeneously distributed with common features:P(f) near the jamming transition has a peak at smallforces and an exponential tail in the limit of large f .It is argued that the development of a peak is a signa-ture of the jamming transition [10]. Some parts of thequalitative behavior of P(f) are correctly captured bysimplified mean-field models, such as the q-model [3, 13]and Edwards’ model [7, 8]. Both of them describe theexponential decay at large force and a power law be-havior P(f) ∼ fθ for small forces [7, 8, 14]. How-ever, the exponent obtained by the q-model (an inte-ger θ ≥ 1 [3, 14]) is larger than the experimental valueθ ' 0 [2] or the one obtained by recent numerical simula-tions accessing the low force limit with increasing accu-racy 0 . θ . 0.5 [15, 16], while the exponent obtained byEdwards’ model θ = 1/(z − 2) depends strongly on thedimension (through z), when simulations show it does

not [15]. Other approaches based on entropy maximiza-tion similar to Edwards’ statistical mechanics [17] alsorecover the large force exponential decay [18–22]. Someof those works however predict a Gaussian tail [23], andso does a mean-field theory based on replica theory ofspin glasses [15, 24, 25].

The average coordination number per particle z is an-other key signature of jamming. Close to jamming, manyobservables (pressure, volume fraction, shear modulus orviscosity [26] to name a few) scale with the distance z− zcto the average coordination number zc at the transition.Understanding the value of zc is thus of primary im-portance. The case of frictionless spheres, for which,zc = 2D where D is the dimension, has been rationalizedbased on counting arguments leading to the isostatic con-jecture [27–29]: a lower bound zc ≥ 2D is provided byMaxwell’s stability argument [30], and an upper boundzc ≤ 2D is given by the geometric constraint of havingthe particles exactly at contact, without overlap. Theproblem, however, turns out to be more complicatedwhen friction is considered, and no method is knownso far to predict zc in such a case. The (naive) gen-eralization of the counting arguments gives the boundsD+1 ≤ zc ≤ 2D in frictional packings, independently ofthe value of the interpaticle friction coefficient µ. Indeed,there is a range of zc obtained numerically and experi-mentally [22, 29, 31–38]. However, for small µ, this rangenever extends to the predicted lower bound, as packingswith low friction coefficient lie close to zc = 2D.

Here we present a theoretical framework at a mean-field level to consider force transmission as a constraintoptimization problem on random graphs, and study

2

this problem with standard tools, namely the cavitymethod [39]. We first obtain the force distribution forspheres, in frictionless and frictional cases, both in twoand three-dimensions. Besides showing the experimen-tally known exponential fall-off at large forces, these dis-tributions bring a new insight in the much less knownsmall force regime. In particular, we find for friction-less spheres in both 2 and 3 dimensions a finite value forP(f) in f = 0, leading to an exponent θ = 0 for smallforces that extends over four decades, beyond currentlyperformed simulated data [15, 16].

We also show how the frictional coefficient µ affectsthe average number of contacting neighbors zc at thejamming transition, and we find a lower bound zmin

c (µ)for this number (which is also a lower bound on z, sincez ≥ zc in a packing). We achieve this by generalizingin a careful way the naive Maxwell counting arguments,considering the satisfiability of force and torque balanceequations. Linking zc to the behavior of force and torquebalances is not a new idea, as it was already suggested bySilbert et al. [32]. Furthermore the generalized isostatic-ity picture [34] gives a bound on the number of fully mo-bilized forces (ie the number of tangential forces whichare taking the maximally value allowed by the Coulomblaw) based on the value of zc. However, none of theseworks derive a bound for zc itself, at a given µ. Thebound zmin

c (µ) that we obtain interpolates smoothly be-tween the two isostatic limits at µ = 0 and µ→∞.

II. METHODS

A. Force balance as a satisfiability problem

Similarly to the global rigidity condition, force andtorque balance are entirely constrained by the contactnetwork of the packing. If we define ~r ai as the vectorjoining the center of particle a and the contact i on it,the contact network is uniquely defined by the complete

set ~r ai . Calling ~f ai the force acting on particle a fromcontact i, force and torque balances read:∑

i∈∂a

~f ai = 0, ~r ai · ~f ai < 0,

∀ a∑i∈∂a

~r ai × ~f ai = 0, f ti ≤ µf ni ,(1)

where the notation ∂a denotes the set of contacts ofparticle a. We explicitly take into account friction bydecomposing the force into normal and tangential parts~f ai = −f ni n ai + f ti t

ai , where n ai and t ai are normal and

tangential unit vectors to the contact, respectively. The

inequality ~r ai · ~f ai < 0 ensures the repulsive nature of thenormal force. The last inequality ensures Coulomb’s lawwith friction coefficient µ.

The constraints induced by force and torque balanceson the forces fi are not always satisfiable. In the caseof frictionless spheres, we can recover the known z ≥ 2D

directly from force balance alone. The naive Maxwellcounting argument [29] applied to frictionless spheresreduces Eq. (1) to a set of linear equations by takinginto account only force balance and neglecting the re-pulsive nature of the forces. Maxwell argument con-siders the minimal number of forces needed to satisfyEq. (1) which gives, per sphere, D equations and z/2variables (forces), implying z ≥ 2D to have a solution.Below this threshold, there is generically no solution toEq. (1). To accurately extend this counting argumentto more general conditions (frictional, repulsive, and/ornon-spherical particles), one must take into account allthe constraints in Eq. (1), including the repulsive natureof the forces and the Coulomb condition for frictionalpackings. Indeed, the naive Maxwell argument, neglect-ing those constraints, concludes z > zmin

c = D+1 for anyfrictional packing of spheres, ignoring the dependance onthe friction coefficient [40]. On the other hand, belowwe show that including the above mentioned constraintswe obtain an accurate lower bound zmin

c (µ), explicitelydepending on µ.

We tackle the problem of satisfiability of force andtorque balances Eq. (1) by looking at the contact net-work in an amorphous packing as an instance of randomgraph. As depicted in Fig. 1A, starting from a packingof N particles, we explicitly construct a so-called factorgraph [41], considering the M = zN/2 contacts as ‘sites’,and the N particles as ‘interaction nodes’. Each site ibears two vectors ~r ai and ~r bi and two opposite forces~f ai = −~f bi (one per particle involved in the contact).Note that n ai is uniquely determined by the contact net-work ~r ai and represents the ‘quenched’ disorder in thesystem, whereas t ai is free to rotate in the plane tangentto contact. On each interaction node (particle) a, we en-force force balance, torque balance, repulsive interactionsand Coulomb friction conditions on its za neighboringsites by an interaction function,

χa(fn, f t, n a, t a∂a) =δ( ∑i∈∂a

~f ai

)δ( ∑i∈∂a

~r ai × ~f ai

)×∏i∈∂a

Θ(f ni )Θ(µf ni − f ti ).

(2)We define the partition function Z and entropy S for theproblem of satisfiability of force and torque balances fora fixed realization of the quenched disorder nai as (seeSI for more details):

eS = Z =

∫ M∏i=1

df ni df ti dt ai dt bi δ(tai + t bi )

δ(∑i f

ni −Mp)

∏Na=1 χa(fn, f t, n a, t a∂a).

(3)Without loss of generality, we work in the constant pres-sure p ensemble: if we find a solution to the force andtorque balances problem having a pressure p′, we canalways find one solution with pressure p by multiplyingall forces by p/p′. If the entropy is finite, there existsa solution to force and torque balances. The satisfiabil-

3

Qb!i(fni , f t

i )Qi(fni , f t

i )Qa!i(fn

i , f ti )

Qc!j(fnj , f t

j )

a

b

c

i

j

a

interaction node (particle) site (force)

b

a

cj

i

(A) (B)

FIG. 1: Factor graph and variables. (A) Building thefactor graph of contacts from a packing [41]. The contactsbecomes sites (closed circles) with associated variables beingforces. The particles are reduced to interaction nodes (opensquares) dealing with force/torque balance Eq. (1). (B) Partof factor graph used to compute Qi and Qa→i by using Eq. (5)and Eq. (6). On the edges around particle a, the arrows indi-cate that we consider the force probabilities Q→ as ‘messages’going through contacts in the contact network. In our algo-rithm, Qa→i is iteratively updated by the uncorrelated forceprobability Qc→j on neighboring edges with the force/torquebalance constraint χa(f n, f t, n a, t a∂a) on that particle a,verifying the cavity equation Eq. (6). The marginal probabil-ity Qi(fni , f

ti ) on site i is calculated as a product of Qa→i and

Qb→i as in Eq. (5).

ity/unsatisfiability threshold of force and torque balancesis the coordination number zmin

c (µ) that separates the re-gion of finite S from the region for which S → −∞, cor-responding to an underdetermined/overdetermined set ofEq. (1), respectively.

Within this framework, the satisfiability problemEq. (1) is one of the well studied class of constraint satisfi-ability problems (CSP) defined on random networks [41].These problems are ubiquitous in statistical physics andcomputer science and have attracted a lot of attentionin recent years. Thus powerful methods from statisticalphysics have been developed to study them [41].

We will work here with random graphs, retaining fromactual packings the distribution of coordination num-ber R(z) and a joint distribution of contact positionsaround one particle Ω(~r a1 , . . . , ~r

aza), which enforces the

non-overlap between neighbors of a given particle. Areal contact network of a two or three-dimensional pack-ing shows some finite dimensional structure, of course,and treating it as random graphs can only be an approx-imation. This amounts to a mean-field approximation,neglecting correlations between the different contactingforces acting on one particle. This approximation is rou-tinely used in the context of spin-glasses for example [39].From this point of view, we stand on the same ground asthe q-model approach [3, 13], and the Edwards’ approachof Brujic et al. [7, 8, 42].

B. Force distribution and satisfiability of force andtorque balances

As the simplest case, we restrict the description of thissection to packings of spheres, with obvious generaliza-tion to non-spherical objects. For a given disorderedpacking, each particle a has unique surroundings, dif-ferent from its neighbors or other particles in the pack-ing. These surroundings are defined by the contact num-ber za, contact vectors ~r ai .If the system is underdeter-mined, several sets of forces in the system satisfy forceand torque balance, and each contact force has a certainprobability distribution Qi(fni , f

ti ).

The local disorder makes each contact unique, and theprobability distributions of forces Qi(fni , f

ti ) are different

from contact to contact. We define the overall force dis-tribution in a packing P(fn, f t) as an average over theprobability distributions of forces over the contacts:

P(fn, f t) ≡ 〈Qi(fn, f t)〉 =1

M

∑i

Qi(fn, f t). (4)

Next, we show that on a random graph, we can accessthe distributions Qi(fni , f

ti ) with a self-consistent set of

local equations using the cavity method [41]. In this de-scription, we work at fixed pressure p, ie we considerany two solutions differing only by an overall rescalingof the pressure to be only one genuine solution (see SIfor the detailed implementation in the following cavityformalism). Each contact is linked to two particles, aand b. We denote Qa→i(fni , f

ti ) the probability distribu-

tion of the force ~fi of a site (contact) i, if i is connectedonly to the interaction node (particle) a, that is, if weremove (dig a cavity) particle b from the packing. Themain assumption of the cavity method is to consider thatQa→i(fni , f

ti ) and Qb→i(fni , f

ti ) ar uncorrelated. There-

fore, we can write the probability of forces at contact ias:

Qi(fni , fti ) =

1

ZiQa→i(fni , f

ti ) Qb→i(fni , f

ti ), a, b = ∂i

(5)with Zi the normalization.

Under the mean-field assumption a set of local equa-tions (called cavity equations) relates the Q→’s, as de-picted in Fig. 1B:

Qa→i(fni , fti ) =

1

Za→i

∫dt ai

∏j∈∂a−i

dfnj df tj dt aj

×χa(fn, f t, n a, t a∂a)∏

c=∂j−aQc→j(fnj , f

tj )

≡ Fa→i(

Qc→j),(6)

where the notation ∂x−y stands for the set of neighborsof x on the graph except y, and Zi→a is the normaliza-tion. Crucially, we do not average over the contact direc-tions n a∂a at this stage (whereas the q-model [3, 13]and Edwards’ model [7, 8] do). This implies that every

4

link a → i has a different distribution, due to the lo-cal ‘quenched’ disorder provided by the contact networkn a∂a and contact number za. Hence, finding a set ofQ→ that are solutions of Eq. (6) allows to get the dis-tribution of forces on every contact individually, throughthe use of Eq. (5).

Looking for a solution of Eq. (6) for a given instanceof the contact directions (meaning for one given packing)is possible. These equations are commonly encounteredas ‘cavity equations’ in the context of spin glasses or op-timization problems defined on random graphs [41], andthey can be solved by message passing algorithms likeBelief Propagation. Here we follow another route, sincewe are interested in P(fn, f t) for not only one packingbut over the ensemble of all random packings. Thus, westudy the solutions of the cavity equations in the thermo-dynamic limit to provide typical solutions for large pack-ings. As in statistical mechanics, the partition functionwill be dominated by the relevant typical configurationswhich we expect will be realized in experiments.

In the thermodynamic limit, the set of Q→’s that aresolutions of Eq. (6) are distributed according to the prob-ability Q(Q→):

Q(Q→) ≡ 1

M

∑a→i

δ[Q→ −Qa→i] (7)

In this case, we can replace the sum over a→ i by acontinuum description of the Q→’s based on their distri-bution Q(Q→). The probability that a given Q→ is setby a cavity equation Eq. (6) involving z − 1 contact isproportional to zR(z) Ω(ni, nj). Thus, averaging overthe ensemble of random graphs, Eq. (7) becomes a self-consistent equation [39, 41]:

Q(Q→) =1

Z∑z

zR(z)

∫Ω(n, nj)

z−1∏j=1

dnj

×DQ→j Q(Q→j)δ[Q→ −F→

(Q→j

)].

(8)where Z is the normalization. Note that the value of theintegral does not depend on the choice of n. Once a solu-tion to Eq. (8) is known, we deduce the force distributionP(fn, f t) in the overall packing as the average of all theseprobability distributions and contacts:

P(fn, f t) = 〈Qi(fn, f t)〉

=1

ZP

[∫DQ→Q(Q→)Q→(fn, f t)

]2(9)

where ZP is the normalization to ensure∫

P(fn, f t) = 1.Equation (8) stands out as the main and crucial dif-

ference with previous approaches, in particular the q-model [3, 13] and Edwards’ description [7, 8]. Althoughthese approaches also neglect correlations, our work doesnot reduce to those models due to a fundamentally dif-ferent way of treating the disorder in the packing. Here,we consider a site-dependent Qi(fni , f

ti ), where the Ed-

wards’ model and q-model create all sites equal. Thus in

our method the average over the packing configurationsis not done at the same level as the average over forces.That is, we perform a quenched average over the disorderof the graph. As random packings in two or three dimen-sions have a rather small connectivity, the fluctuationsin the environment of one particle are large: no parti-cle stands in a ‘typical’ surrounding. Hence, the averageover the ‘quenched’ disorder (the packing configurations)must be done with care. Averaging directly Eq. (6) (aso-called ‘annealed’ average in spin-glass terminology), asthe previously cited approaches do [3, 7, 8, 13], amountsto neglect the site to site fluctuations. Performing a‘quenched’ average as in Eq. (8), however, allows to takeinto account these fluctuations correctly [39], and leadsto a force distribution which is the average force distribu-tion over the ensemble of possible packings, as opposedto the force distribution of an averaged packing.

This issue becomes also crucial to the study of the sat-isfiability transition of force and torque balances. Forexample, the q-model describes a force distribution ina packing of frictionless spheres (ie it finds solution toforce balance), even in cases where we know there is nosolution, such as when z < 6 in 3-D frictionless systems.This can be understood by looking at the entropy definedin Eq. (3). The annealed average over disorder done inthe q-model amounts to compute the averaged partitionfunction Z, and get the entropy through San = lnZ,with Z defined in Eq. (3). But one can show that Z isalways finite for z ≥ 2. Indeed, for z = 2, an infinite rowof perfectly aligned spheres will satisfy force and torquebalances and will contribute to the partition function. Astraightforward generalization of this example shows thatfor z ≥ 2, the annealed entropy is finite. On the contrary,the quenched average amounts to compute the averagedentropy Squ = lnZ. Now, for our frictionless sphere ex-ample, typical configurations with z < 6 cannot satisfyforce balance, and their diverging negative entropy willdominate the average Squ. Hence, the quenched averagecorrectly captures the satisfiability/unsatisfiability tran-sition at z = 6 while the annealed average does not.

Equation (8) is typically hard to solve, since it is aself-consistent equation for a distribution of distributionsQ(Q→). For this purpose, we use a Population Dynamicsalgorithm (details in SI), familiar to optimization prob-lems [41]. This method consists to describe the distri-bution Q via a discrete sampling (a ‘population’) madeof a large number of distributions Q→. Applying itera-tively Eq. (8), we find, if it exists, a fixed-point of thedistribution Q(Q→).

It is interesting to discuss the different types of solu-tions expected from Eq. (8). For a given contact network,if the system is satisfiable and underdetermined, henceadmits an infinite set of solutions for force and torquebalances, the distributions Qi(fn, f t) should be broad,allowing each force to take values in a non-vanishingrange. This means that on each contact, Qa→i(fn, f t)and Qb→i(fn, f t) should be broad and overlapping. Ifthe system is neither under- nor overdetermined (i.e. iso-

5

static), there is only one solution to force and torquebalances for every site i, and each Qi(fn, f t) is a Diracδ-function centered on the solution. If the system isoverdetermined or unsatisfiable, there is typically no so-lution to Eq. (8), meaning that an algorithm designedto solve it would not converge. In practice, since weperform a population dynamics algorithm to averageover all possible packings, if one start with a set ofbroad Qa→i(fn, f t) as a guess for the solution, bothisostatic and overdetermined ensembles show that allQa→i(fn, f t) shrink to δ-distributions, while under-determined ensemble always gives broad (not vanishing)probabilities. Therefore the threshold zmin

c (µ) of the sat-isfiability/unsatisfiability transition for force transmis-sion can be located by measuring the width of the forcedistributions Qa→i(fn, f t).

The location of this transition, in turn, constitutesa lower bound for the possible coordination numberzminc (µ), which extends Maxwell’s counting argument forµ = 0 to any friction. An additional quantity availableis the force distribution itself, as a function of zmin

c (µ).Therefore, our approach explicitely relates two essentialproperties of the jamming transition: the average coor-dination number and the force distribution.

III. RESULTS

A. Force distribution for frictionless spherepackings

We start by computing the force distribution P(f =fn) for two and three-dimensional frictionless spherepackings, when we fix the average contact number z =zc = 4 and 6 respectively, by solving SI-Eq. (7). Re-sults in Fig. 2 reproduce the exponential tail of the forcedistribution at large forces, as seen in numerical simu-lations [9–12, 32, 33, 43, 44] and experiments [3–8, 12].The force distribution we obtain can be well fitted withP (x) = [7.84x2 + 0.86 − 0.75/(1 + 4.10x)]e−2.67x for 2-D (Fig. 2A) and P (x) = [7.45x2 + 1.20 − 1.06/(1 +2.33x)]e−2.65x for 3-D (Fig. 2B), with x = f/〈f〉. Bothfitting functions are close to the empirical fit P (x) =[3.43x2 + 1.45 − 1.18/(1 + 4.71x)]e−2.25x to the forcedistribution of dense amorphous packings generated byLubachevsky-Stillinger algorithm in 3-D by Donev etal. [43]. More interestingly, our method allows to accessthe small force region with unprecedented definition. Wegather data down to 10−6 times the peak force (whichis of the order of the pressure). This range is way be-low what is accessible with state of the art simulationsof packings [9–11, 15, 16, 32, 33, 44]. The reason for thisis that we avoid two problems: (i) as we have no realpacking to generate, we can easily generate a huge num-ber of forces (we used 106 to compute our P(f), comparewith few tens of thousands that can be generated in sim-ulated packings [9–11, 32]), and (ii) we can work exactlyat the jamming transition point, as we set z = 2D, which

10-4

10-3

10-2

10-1

100

10-6 10-5 10-4 10-3 10-2 10-1 100

P(f/<

f>)

f/<f>

(A) 2D frictionless (<zc=4)

e=0 slope=0.52

10-2

10-1

100

0 1 2 3 4

10-4

10-3

10-2

10-1

100

10-6 10-5 10-4 10-3 10-2 10-1 100

P(f/<

f>)

f/<f>

(B) 3D frictionless (<zc=6)

e=0 slope=0.52

10-2

10-1

100

0 1 2 3 4

FIG. 2: Force distribution, P(f) in frictionless spheres pack-ing in (A) 2-dimensional and (B) 3-dimensional systems. Re-sults obtained from the cavity method (open triangles) showa flat regime with exponent θ = 0 at small forces and 0.52in the intermediate region in both cases. In inset, log-linearplot of the same distribution exhibits exponential tail at largeforces. The red solid lines correspond to the fitting functions,as defined in the text.

contrasts with actual numerical or experimental studieswhere the limit of vanishing pressure with z = 2D is verychallenging.

The behavior of P(f) at small forces has recently at-tracted attention [15, 16, 33, 44]. Wyart [16] pointedout a relation between the small force scaling P(f) ∼ fθand the distribution function of the gaps h between par-ticles close to contact g(h) ∼ h−γ via the inequalityγ ≥ 1/(2 + θ). Few empirical data exist so far for theθ exponent, even if some recent efforts greatly improvedavailable values [15, 44].

Here we find for frictionless spheres a distribution ofcontact forces having a finite value for f = 0. This trans-lates as an exponent θ = 0 over four decades of data inboth 2-D and 3-D packings (Fig. 2). Such a finite valuefor the force distribution was observed in experiments onconcentrated emulsion droplets designed to capture theirsmall deformation using confocal microscopy of fluores-

6

cent dyes to highlight the contact network [2], with whichour prediction agrees. On the theoretical side, severalvalues of θ have been predicted: the replica theory alsopredicts a finite value for P(f = 0) [15, 24, 25] (it predictsa Gaussian for P(f)); the q-model can give several val-ues for θ, depending on the underlying assumptions, butin any case predicts θ ≥ 1 [3, 13]; and Edwards’ modelpredicts θ = 1/(z−2) [7, 8]. The differences between Ed-wards’ and q-model approaches, which mostly stem fromthe treatment of local disorder in the contact normal di-rections, indicate that the way to deal with this disorderis crucial for a correct description of the small force be-havior. According to the inequality γ ≥ 1/(2 + θ) [16],our prediction of θ = 0 gives γ ≥ 1/2, which would meanthat the commonly reported value of γ = 1/2 [31] actu-ally saturates this bound.

Interestingly, we find an intermediate regime for fslightly smaller than its average value 〈f〉, for whichP(f) ∼ f0.52 (Fig. 2). This regime, extending fromf ' 10−2〈f〉 to f ' 〈f〉 is the one probed by most exper-iments and simulations. We suggest that the discrepancybetween the several values of θ reported so far might stemfrom the existence of this regime and the crossover to theθ = 0 regime at smaller f .

B. Calculation of zminc (µ) for frictional sphere

packings

We turn to the determination of the force distributionfor arbitrary friction coefficient µ and a lower bound onzc for the existence of random packings of sheres at agiven µ. This threshold corresponds to the point wheresolutions of the cavity equations Eq. (6) and Eq. (8) nolonger exist.

We search for the existence of a solution by applyingiteratively Eq. (6) to a population of force distributionsQ→(fn, f t), according to the Population Dynamics al-gorithm (SI). A solution exists if this process leads to aconverged solution Q(Q→). The convergence of Q(Q→)is hard to get numerically, as it requires a very largepopulation of Q→ to describe the Q→ space preciselyenough. However, for our purpose, we do not need todescribe Q(Q→) in detail, since we just need to know ifit exists. We thus adopt a simpler criterion to test thisexistence. It is based on the convergence of the averagewidth of the distribution Q→. If this width convergesto a finite value, a solution to the cavity equations exists(satisfiability), whereas if it vanishes, no solution exists(unsatisfiability), as described in the previous section.This convergence can be studied with a smaller popula-tion, and it occurs more rapidly than the convergence ofQ(Q→). Still, the computational cost is high, and wehere apply the method to 2-D packings only.

We define the ‘width’ of Q→(fn, f t) on fn and f t byWn and Wt respectively as the difference of two extremevalues of fn and f t at which Q→(fn, f t) is equal to 10−3

(see SI-Fig. 1B). For frictionless packings, we calculate

10-4

10-3

10-2

10-1

100

100 101 102

< Wn >

Time Steps

(A)2D frictionless

<z = 3.9<z = 4.0<z = 4.1data discretization

3.0

3.2

3.4

3.6

3.8

4.0

4.2

0 10-2 10-1 100 101

< z cmin(µ)

µ

(B)

10-1

100

10-5 10-1 103

< z cmin(0)-< zcmin(µ)

< z cmin(µ)-< zcmin(')

µ

_=0.67 `=0.35

FIG. 3: (A) Evolution of average width 〈Wn〉 of a large popu-lation Q→ versus time steps in population dynamics in 2-Dfrictionless packing. zmin

c (µ = 0) is found at 4.0, as expectedfrom counting argument. (B) Linear-Log plot of zmin

c (µ)vs. µ for various friction coefficients in 2D disks packings.zminc (µ) shows a monotonic decrease with increasing µ fromzminc (µ = 0) = 2D = 4 to zmin

c (µ =∞) & D + 1 = 3. Er-ror bar indicates the range from the largest zmin

c (µ) havingno solution to the smallest zmin

c (µ) having solution. In theinset, two power law scaling relations zmin

c (0) − zminc (µ) ∼

µα, zminc (µ) − zmin

c (∞) ∼ µ−β are found with α = 0.67,β = 0.35, respectively.

the average width over the sites as 〈Wn〉, whereas infrictional case, we calculate 〈Wn〉 and 〈Wt〉, the aver-age width on variable fn and f t, respectively. Fig. 3Ashows the evolution of the average width of distributionsat different z versus time steps in population dynamicsfor the particular case of 2D frictionless packings (othersystems are shown in SI-Fig. 1). Results indicate thatthe final population of distributions Q→ after tmax it-erations have dramatically different shapes at various z.We find that the population Q→ rapidly tends to a setof non-overlapping Dirac peaks when z is small (see SI-Fig. 1A). In this case, the average width of fields (shownin Fig. 3A for z = 3.9) decreases as a function of timestep in the Population Dynamics iteration and finally

7

goes below data discretization (the interval of force tointegrate the force distribution), leading to no solutionof Qi(fn, f t) according to Eq. (5). In this case the sys-tem is overdetermined in the sense that z contacts perparticle are not enough to stabilize the whole packing.In contrast, when z is increased to z = 4.5 > zmin

c (µ),the distributions in Q→ become broad (see SI-Fig. 1B).The average of width converges to a finite value as seenin Fig. 3A, allowing a non-vanishing force distributionQi(fn, f t) at each contact. In this case a set of solutionsto force and torque balances exists. The point where theaverage widths of the distributions 〈Wn〉 and 〈Wt〉 vanishis zmin

c (µ).Frictionless packings show a transition point zmin

c (µ =0) = 2D, in D = 2 and D = 3, as expected. The fullcurve zmin

c (µ) is shown in Fig. 3B for D = 2. We observea monotonic decrease with increasing µ from 2D = 4at µ = 0, a well-known behavior of frictional packings,previously found in numerous studies, both experimen-tally and numerically [22, 29, 31–38]. Notice that thecritical contact number we obtain at infinite friction isslightly above the Maxwell argument D + 1; also a typ-ical feature [31, 34, 36]. This is interesting, as it meansthat the naive counting argument, ignoring the repul-sive nature of the forces, fails to reproduce the correctbound for such a simple case as sphere packings withµ→∞, where neither Coulomb condition nor non-trivialgeometrical features complexify the picture. The factthat the naive Maxwell counting argument still gives thecorrect bound for frictionless sphere packing can there-fore be seen as a quite fortunate isolated prediction. Inthe inset of Fig. 3B, two power law scaling relationszminc (0) − zmin

c (µ) ∼ µα, zminc (µ) − zmin

c (∞) ∼ µ−β arefound with α = 0.67, β = 0.35 respectively. Our result αagrees well with previous simulation of 2-D polydispersepackings α = 0.70 [34], and close to the prediction of 2-Dmonodisperse packing α = 1 by Wang et al [36], while βis much smaller than their predicted value β = 2 and theresult of β = 1.86 obtained from simulation [36].

C. Frictional sphere packings: joint forcedistribution

Similar to the frictionless case, the cavity method cangenerate the joint force distribution Pµ(fn, f t) for fric-tional sphere packings with friction coefficient µ.

The simplest case of infinite friction, Pµ=∞(fn, f t) for2-D and 3-D sphere packings is shown in Fig. 4A andFig. 4B respectively, and follow similar behavior as mea-sured in previous numerical studies [36]. In particular, werecover the non-trivial qualitative change shown in [36]between 2-D and 3-D: while in 3-D, Pµ=∞(fn, f t) 'Pµ=∞(f t, fn), in 2-D, this symmetry is clearly broken.This is a consequence of the more symmetrical role tan-gential and normal forces play in 3-D with as many torquebalance as force balance equations, whereas in 2-D, thereare twice less torque balance than force balance equa-

tions.Furthermore, we obtain the distributions of the nor-

mal and tangential components as plotted in Fig. 4C andFig. 4D. The normal force distribution have slight peaksaround the mean and approximate exponential tails atlarge forces. Below the mean, the normal force distribu-tion for infinite friction has a nonzero probability at zeroforce whereas it shows a dip towards zero for µ = 0.2 (seeSI-Fig. 2 for results of µ = 0.2 in 2-D). The tangentialforce distribution also has an exponential tail, however, itdecreases monotonically without an obvious rise at smallforces. Our results of the probability distribution of nor-mal forces and tangential forces agree with previous ex-perimental measurements in 2-D frictional discs packing[45].

When the friction coefficient is finite (see SI-Fig. 2A),the pattern inside the Coulomb cone looks similar tothe one obtained at infinite friction. Quite interestingly,we do not observe any excess of forces at the Coulombthreshold f t = µfn, implying that there are no slidingcontacts. This offers a theoretical explanation to a sin-gular fact already observed in simulations: control pa-rameters (essentially volume fraction and friction coeffi-cient) being equal, the percentage of plastic contacts ina packing depends on the preparation protocol [33, 46].Our formalism takes into account those different packings(hence protocols) by performing a statistical average overpossible packings, and the outcome shows that packingswithout plastic contacts are dominant. In this regards,the fragility associated with the large number of plasticcontacts in many experimentally or numerically gener-ated packings could be mostly attributed to the prepa-ration protocol, rather than to an inherent property ofrandom packings of frictional spheres.

IV. DISCUSSION

In conclusion, we develop a theoretical framework byusing the cavity method, introduced initially for thestudy of spin-glasses and optimization problems, to ob-tain a statistical physics mean-field description of theforce and torque balances constraints in a random pack-ing. This allows us to get the force distribution and thelower bound on the average coordination number in fric-tional and frictionless spheres packings.

We find a signature of jamming in the finite valueP(f = 0) of the force distribution at small force, whichagrees with experimental measurements on emulsiondroplets [2]. We also notice that there is a power law riseP(f) ∼ f0.52 in the intermediate region of P(f). Thus itis likely that one obtains an exponent 0 < θ < 0.52 if thesimulations or experiments can not achieve data down tolow enough forces. For frictional packings, we can accessthe complete joint distribution Pµ(fn, f t).

Concerning the average coordination number, wedescribe its lower bound zmin

c (µ), which interpo-lates smoothly between the isostatic frictionless case

8

2D (μ=∞)

(A) P∞( f n, f t )

f n/<f n>

f t /<f t >

3D (μ=∞)

(B) P∞( f n, f t )

f n/<f n>

f t /<f t >

10-2

10-1

100

0 0.5 1 1.5 2 2.5 3 3.5 4

P '(f/<fn>

)

f/<f n>

(C)

slope=-1.38slope=-3.39

2D (µ=')f nf t

10-2

10-1

100

0 0.5 1 1.5 2 2.5 3 3.5 4

P '(f/<fn>

)

f/<f n>

(D)

slope=-1.68slope=-3.89

3D (µ=')f nf t

FIG. 4: The joint force distribution P∞(fn, f t) at µ = ∞ in (A) 2-D disks packing and (B) 3-D spheres packing. Plots ofthe probability distribution of normalized normal forces and frictional forces at µ = ∞ in (C) 2-D disks packing and (D) 3-Dspheres packing.

zminc (0) = 2D, and a large µ limit zmin

c (∞) slightlyabove D + 1. This confirms that there is no disconti-nuity at µ = 0. We predict two scalings for small andlarge friction coefficients as zmin

c (0) − zminc (µ) ∼ µ0.67

and zminc (µ)− zmin

c (∞) ∼ µ−0.35.The statistical mechanics point of view on force and

torque balances for random packings thus proves fruit-ful. Many features of these systems can be inferred fromthose simple considerations. The use of the cavity tech-nique enables us to tackle this problem with a correcttreatment of the disorder, leading to several new results.This formalism will be extended to packings of more gen-eral shapes and could be used to predict other propertiesof disordered packings, like the yield stress for instance.Granular materials are not the only systems subject to

force and torque balance, and this constraint is seen inall overdamped systems, among which suspensions at lowReynolds number constitute an important example, bothconceptually and practically. We hope that our resultswill motivate investigations in this direction.

Acknowledgments

We gratefully acknowledge funding by NSF-CMMTand DOE Office of Basic Energy Sciences, Chemical Sci-ences, Geosciences, and Biosciences Division. We thankF. Krzakala and Y. Jin for interesting discussions.

[1] R.P. Behringer and J.T. Jenkins, Powders & Grains 97,(Balkema, Rotterdam,1997).

[2] J. Zhou, S. Long, Q. Wang and A.D. Dinsmore, Measure-ment of forces inside a three-dimensional pile of friction-less droplets, Science 312, 1631–1633 (2006)

[3] C.-h. Liu, S.R. Nagel, D.A. Schecter, S.N. Coppersmith,S. Majumdar, O. Narayan and T.A. Witten, Force fluc-tuations in bead packs, Science 269, 513–515 (1995)

[4] D.M. Mueth, H.M. Jaeger, S.R. Nagel, Force distributionin a granular medium, Phys. Rev. E 57, 3164–3169 (1998)

[5] G. Løvoll, K.J. Maløy and E.G. Flekkøy, Force measure-ments on static granular materials, Phys. Rev. E 60,

5872–5878 (1999)[6] J.M. Erikson, N.W. Mueggenburg, H.M. Jaeger and S.R.

Nagel, Force distributions in three-dimensional compress-ible granular packs, Phys. Rev. E 66, 040301 (2002)

[7] J. Brujic, S.F. Edwards, I. Hopkinson and H.A. Makse,Measuring the distribution of interdroplet forces in a com-pressed emulsion system, Physica A 327, 201–212 (2003)

[8] J. Brujic, S.F. Edwards, D.V. Grinev, I. Hopkinson, D.Brujic and H.A. Makse, 3d bulk measurements of the forcedistribution in a compressed emulsion system, FaradayDisc. 123, 207–220 (2003)

[9] F. Radjai, M. Jean, J.J. Moreau and S. Roux, Force

9

distributions in dense two-dimensional granular systems,Phys. Rev. Lett. 77, 274–277 (1996)

[10] C.S. O’Hern, S.A. Langer, A.J. Liu and S.R. Nagel, Forcedistributions near jamming and glass transitions, Phys.Rev. Lett. 86, 111–114 (2001)

[11] A.V. Tkachenko and T.A. Witten, Stress in frictionlessgranular material: Adaptive network simulations, Phys.Rev. E 62, 2510–2516 (2000)

[12] H.A. Makse, D.L. Johnson and L.M. Schwartz, Packingof compressible granular materials, Phys. Rev. Lett. 84,4160–4163 (2000)

[13] S.N. Coppersmith, C. -h. Liu, S. Majumdar, O. Narayanand T.A. Witten, Model for force fluctuations in beadpacks, Phys. Rev. E 53, 4673–4685 (1996)

[14] J.E. Socolar, Average stresses and force fluctuations innoncohesive granular materials, Phys. Rev. E 57, 3204(1998)

[15] P. Charbonneau, E.I. Corwin, G. Parisi, F. Zamponi,Universal microstructure and mechanical stability ofjammed packings, Phys. Rev. Lett. 109, 205501 (2012)

[16] M. Wyart, Marginal stability constrains force and pairdistributions at random close packing, Phys. Rev. Lett.109, 125502 (2012)

[17] S.F. Edwards and R. Oakeshott, Theory of powders,Physica A 157, 1080–1090 (1989)

[18] N. Kruyt and L. Rothenburg, Probability density func-tions of contact forces for cohesionless frictional granularmaterials, Int. J. Solids Struct. 39, 571–583 (2002)

[19] K. Bagi, Statistical analysis of contact force componentsin random granular assemblies Granul. Matter 5, 45–54(2003)

[20] J. Goddard, On entropy estimates of contact forces instatic granular assemblies, Int. J. Solids Struct. 41, 5851–5861 (2004)

[21] S. Henkes and B. Chakraborty, Statistical mechanicsframework for static granular matter, Phys. Rev. E79,061301 (2009)

[22] J.H. Snoeijer, T.J.H. Vlugt, M. van Hecke and W. vanSaarloos, Force network ensemble: A new approach tostatic granular matter, Phys. Rev. Lett. 92, 054302(2004)

[23] B.P. Tighe, A.R.T. van Eerd and T.J.H. Vlugt, Entropymaximization in the force network ensemble for granularsolids, Phys. Rev. Lett. 100, 238001 (2008)

[24] G. Parisi and F. Zamponi, Mean-field theory of hardsphere glasses and jamming Rev. Mod. Phys. 82, 789–845 (2010)

[25] L. Berthier, H. Jacquin and F. Zamponi, Microscopic the-ory of the jamming transition of harmonic spheres, Phys.Rev. E 84, 051103 (2011)

[26] E. Lerner, G. During and M. Wyart, Toward a micro-scopic description of flow near the jamming threshold,Europhys. Lett. 99, 58003 (2012)

[27] S. Alexander, Amorphous solids: their structure, latticedynamics and elasticity, Phys. Rep. 296, 65–236 (1998)

[28] A.V. Tkachenko and T.A. Witten, Stress propagation

through frictionless granular material, Phys. Rev. E 60,687–696 (1999)

[29] C.F. Moukarzel, Isostatic phase transition and instabilityin stiff granular materials, Phys. Rev. Lett. 81, 1634–1637 (1998)

[30] J.C. Maxwell, On the calculation of the equilibrium andstiffness of frames, Philos. Mag. 27, 294–299 (1864)

[31] L.E. Silbert, D. Ertas, G.S. Grest, T.C. Halsey and D.Levine, Geometry of frictionless and frictional spherepackings, Phys. Rev. E 65, 031304 (2002)

[32] L.E. Silbert, G.S. Grest and J.W. Landry, Statistics ofthe contact network in frictional and frictionless granularpackings, Phys. Rev. E 66, 061303 (2002)

[33] H.P. Zhang and H.A. Makse, Jamming transition inemulsions and granular materials, Phys. Rev. E 72,011301 (2005)

[34] K. Shundyak, M. van Hecke and W. van Saarloos, Forcemobilization and generalized isostaticity in jammed pack-ings of frictional grains, Phys. Rev. E 75, 010301 (2007)

[35] C. Song, P. Wang and H.A. Makse, A phase diagram forjammed matter, Nature 453, 629–632 (2008)

[36] P. Wang, C. Song, C. Briscoe, K. Wang and H.A. Makse,From force distribution to average coordination number infrictional granular matter, Physica A 389, 3972 – 3977(2001)

[37] L.E. Silbert, Jamming of frictional spheres and randomloose packing, Soft Matter 6, 2918–2924 (2010)

[38] P. Stefanos, C.S. O’Hern and M.D. Shattuck, Isostatic-ity at frictional jamming, Phys. Rev. Lett. 110, 198002(2013)

[39] M. Mezard, G. Parisi and M.A. Virasoro, Spin Glass The-ory And Beyond, (World Scientific, Singapore, 1986).

[40] S.F. Edwards and D.V. Grinev, Statistical mechanics ofstress transmission in disordered granular arrays, Phys.Rev. Lett. 82, 5397–5400 (1999)

[41] M. Mezard and A. Montanari, Information, Physics, andComputation, (Oxford University Press, USA, 2009)

[42] H.A. Makse, J. Brujic and S.F. Edwards, Statistical me-chanics of jammed matter, The Physics of Granular Me-dia, edited by H. Hinrichsen and DE Wolf (Wiley-VCH,2004).

[43] A. Donev, S. Torquato, F.H. Stillinger, Pair correlationfunction characteristics of nearly jammed disordered andordered hard-sphere packings, Phys. Rev. E 71, 011105(2005)

[44] E. Lerner, G. During and M. Wyart, Low-energy non-linear excitations in sphere packings, Soft Matter 9,8252–8263 (2013)

[45] T.S. Majmudar and R.P. Behringer, Contact force mea-surements and stress-induced anisotropy in granular ma-terials, Nature 435, 1079–1082 (2005)

[46] A.P.F. Atman, P. Claudin, G. Combe and R. Mari, Me-chanical response of an inclined frictional granular layerapproaching unjamming, Europhys. Lett. 101, 44006(2013)