EMERGING AREA www.rsc.org/softmatter | Soft Matter

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Statistical ensemble approach to stress transmission in granular packings†

Bulbul Chakraborty*

Received 4th January 2010, Accepted 28th April 2010

First published as an Advance Article on the web 25th May 2010

DOI: 10.1039/b927435a

In this emerging-area review article, I discuss the application of entropic principles to understanding

stress fluctuations in dry granular matter close to jamming. The paper reviews recent work that clarifies

the conditions necessary for the definition of intensive variables such as compactivity and angoricity of

granular packings, and tests of the canonical ensembles resulting from these definitions. The stress

ensemble, in which the compactivity is infinite and fluctuations are controlled only by the angoricity,

has been used to calculate spatial and temporal correlation functions of stress, and these are discussed

in detail. The review closes with an outlook for the future.

1 Introduction

Granular materials respond to external stresses in a manner

fundamentally different from elastic materials.1,2 One striking

demonstration of this difference is the phenomenon of Reynolds

dilatancy,3 and another is that they form force networks in

response to applied stress.4,5 Studies of force networks6,7 have

demonstrated that their geometrical and mechanical properties

are acutely sensitive to preparation procedures, especially near

the jamming transition.8 Since granular systems do not equili-

brate spontaneously, it is not surprising that the response

depends on the method of preparation: there are many meta-

stable states corresponding to a given set of macroscopic

parameters such as volume or imposed stress. Given the meta-

stability, and history dependence, is it possible to construct

a statistical framework that relates the collective, macroscopic

behavior of granular materials to their microscopic properties?

Granular materials are known to reach reproducible states

under certain dynamical protocols.9,10 These states are stable to

perturbation, and have well-defined values of macroscopic

quantities such as volume. These are the only type of states that

may be analyzed using a statistical approach. The question still

remains whether the statistical approach is useful, and what is the

appropriate statistical framework.

More than a decade ago, Edwards and collaborators

proposed11 that the dynamics of granular materials is controlled

by the mechanically stable configurations, referred to as blocked

states. Moreover, it was argued that the blocked states of infi-

nitely rigid grains is described by a statistical framework where

the volume of the state is the analog of the energy function in

thermal systems. In the Edwards ensemble, all blocked states of

the same volume are assumed to be equiprobable (the micro-

canonical hypothesis). The analog of temperature is the com-

pactivity, which is a measure of changes in entropy with volume,

and controls volume fluctuations.

The Edwards idea was generalized to include forces on grains

through the introduction of the force-moment tensor.12–15

Martin Fisher School of Physics, Brandeis University, Waltham, MA,02454-9110, USA

† This paper is part of a Soft Matter themed issue on Granular andjammed materials. Guest editors: Andrea Liu and Sidney Nagel.

2884 | Soft Matter, 2010, 6, 2884–2893

This article reviews the emerging area of application of statistical

ensembles to stress fluctuations in granular packings. The article

is not meant to be a comprehensive review of entropic ideas in

granular materials but is a personal recounting of results that

have been obtained within the last few years, which have opened

up new avenues for analyzing and predicting the response of

granular materials to external stresses.

The review is organized as follows. Section 2 provides a brief

summary of the work associated with the Edwards ensemble.

Section 3 introduces the generalizations of Edwards ensemble,

including the stress ensemble. Section 4 discusses the testing of

the stress ensemble. Section 5 presents results obtained through

the application of the stress ensemble, and Section 6 provides

a summary of achievements, outstanding questions, and outlook

for the future.

2 Edwards ensemble

The Edwards ensemble11 was constructed to describe the statis-

tics of the blocked (mechanically stable) states of infinitely rigid

grains. The statistical framework is based on three postulates: (1)

dynamics of granular materials are controlled by the statistics of

the blocked states, (2) the volume function, which is a function of

the positions of the infinitely rigid grains plays the role of the

Hamiltonian, and (3) all blocked states with free volume V are

accessed with the same probability, which is the analog of the

microcanonical postulate of equilibrium statistical mechanics.

Maximizing the entropy leads to the canonical distribution for

the probability to access the state n of a subsystem that is within

a much larger packing with volume V:

Pn f exp(–Wn/X) (1)

Here, Wn is the volume of the subsystem in state n, which is

a function of the positions of the grains. The intensive quantity

X, referred to as the compactivity, is the analog of temperature.

Just as temperature controls energy fluctuations in thermal

systems, the compactivity controls volume fluctuations in the

Edwards ensemble. The definition of compactivity is analogous

to that of temperature:1

X¼ vSðVÞ

vV, where S(V) is the entropy of

blocked states with volume V.16 If eqn (1) applies, then the

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probability of finding a packing with volume V at a compactivity

X is given by the canonical distribution,

PðVÞ ¼ 1

ZðX ÞUðVÞ expð� V=XÞ (2)

the analog of the Boltzmann distribution. In the above equation,

U(V) ¼ eS(V) ¼P

nd(V �Wn) is the density of states and Z(X) is

the partition function or generating function that generates all

correlations functions at a given X. It should be emphasized that

the partition function involves a sum over mechanically stable

states only. Thermal systems are characterized by an energy

function that leads to a well-defined density of states. Under the

equiprobability assumption, the density of states, U(V) is simi-

larly determined from the volume function, Wn ^ W({~ri}) of the

positions,~ri, of the grains.

Since the original proposal, much effort has been devoted to

constructing accurate volume functions for infinitely rigid grains

that are amenable to analytical calculations, and testing the

equiprobability assumption. Approximate, analytical forms of

W({~ri}) have been used to calculate the entropy as a function

of packing fraction,15,17 universal forms of the distribution of

volumes,18 and a phase diagram of jammed states in a (packing

fraction)-(contact-number) space.17

The equiprobability hypothesis has been tested through

simulations,19–22 and experiments.23,24 Some studies have been

direct investigations of the free volume distribution in pack-

ings.18,23,25–27 A lot of the tests have been indirect, generally

through the fluctuation–dissipation theorem (see ref. 20 and 19).

Experiments on fluidized beds have been used to carefully

characterize volume fluctuations.27 Many of these studies

support the Edwards hypothesis but others do not. Direct tests of

the equiprobability assumption22 through simulations of small

packings suggest that the equiprobability assumption is not

robust. Experiments on granular segregation in flowing mixtures

have shown that segregation is driven by an effective temperature

related to configurational entropy rather than the kinetic

temperature obtained from velocity fluctuations.28 These exper-

iments indicate that entropic principles are operative in

controlling the dynamics of granular media.

In recent work,29 experiments and simulations have been used

to directly test the canonical form of the volume distribution, eqn

(2). The results show that the distribution can indeed be written

in this very special form, which is a non-trivial result that

supports the entropic principles behind the Edwards hypothesis.

As the authors of ref. 29 point out, testing eqn (2) does not test the

equiprobability hypothesis.

The first of the three postulates of the Edwards ensemble, is

a statement about the importance of blocked states in controlling

the dynamics of slowly driven granular media. The other two are

specific assumptions about conservation laws and the sampling of

the phase space of blocked states. Recent interest has focused on

these two aspects, especially in the context of extending the

ensemble framework to stiff but not infinitely rigid grains. The

basic aim of any ensemble approach is to predict the probability of

occurrence of a microscopic state, given a set of macroscopic,

measurable quantities such as the volume and the external stress.

A static granular packing is characterized by the positions of the

grains, {~ri}, and the set of contact forces, {~f ij}. It is, therefore

natural to ask whether (a) the complete description of

This journal is ª The Royal Society of Chemistry 2010

configurational fluctuations within the ensemble of blocked states

requires intensive quantities in addition to or in place of the

compactivity, and (b) if the equiprobability assumption is neces-

sary for the definition of such intensive quantities. The purpose of

this review article is to discuss the statistical ensembles for blocked

states, which have emerged since the original Edwards framework

with emphasis on the underlying theoretical framework, and

applications to stress transmission in granular materials.

3 Generalized ensembles for blocked states

3.1 Angoricity–compactivity ensemble

For infinitely rigid grains, Blumenfeld and Edwards proposed

that contact forces, ~f ij, should be included as additional statis-

tical variables through the introduction of the force moment

tensor.12 The basic principle underlying this generalized ensemble

is that, in addition to volume, granular steady states could also be

characterized by the force-moment tensor, S, and that under

repeated shakings the packings move between microscopic states

that have the same values of V, and S.12 In addition to com-

pactivity, there is an intensive variable that is conjugate to S,

which is the angoricity tensor, (a)�1 identified by Edwards.12 The

canonical distribution for this generalized ensemble is:

Pn ¼1

ZðX ; aÞ eð�a:SnÞeð�Wn=X Þ (3)

The force moment tensor, Sn ¼P

<ij >n(~rj � ~ri)n~f n

ij, is deter-

mined by the set of positions, {~ri}, and the contact forces, {~f ij}

characterizing the microscopic, mechanically stable, state, and

the sum is over all the geometrical contacts.

As in the original Edwards ensemble, equiprobability was

assumed in defining the angoricity, which is tensorial analog of

compactivity: ða�1Þpq ¼vSðSÞvSpq

. The implications of the tensorial

nature of a will be discussed further in the context of the stress

ensemble.

Makse and collaborators have applied the angoricity–com-

pactivity ensemble to construct equations of state for frictional

hard spheres.15 In their work, the effect of mechanical stability

and contact forces was incorporated into the formulation of

a mesoscopic volume function, and it was shown that the

mesoscopic entropy density was maximum at the random-loose-

packing fraction and decreased to zero at the random-close-

packing value.15

Kruyt and Rothenburg30 and Metzger et al.31,32 use

a maximum-entropy approach with a multi-component

Lagrange multiplier very similar to a to enforce that the total

stress is conserved, and so work in the canonical a ensemble as

well. The authors assume a product distribution for the forces

and calculate the force distribution given the distribution of

contact angles and distances between grains.

3.2 Force-network ensemble

The hardness of deformable grains can be represented by

a parameter 3 ¼\rij.

\fij.

dfij

drij

, which is large if small changes in

deformation leads to large changes in the contact force (normal

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component).33 For large 3, therefore, there is a decoupling of the

equations of mechanical stability and the force-law relations.33

This observation led to the formulation of the force network

ensemble (FNE).33 In this framework, force networks are con-

structed on a given geometry by assuming equiprobability in force

space, and enforcing local mechanical equilibrium constraints

with a fixed global stress tensor. The geometry, necessarily, needs

to be hyperstatic since otherwise there is a unique set of forces

corresponding to a given geometry.34 The FNE is a minimal

model for fluctuating stresses in granular media, and has been

widely used as a testing ground for statistical frameworks.

Tighe et al.35 simulate the FNE on the strongly hyperstatic

hzi ¼ 6 > ziso triangular lattice and find a force distribution that

decays faster than exponential for a system under isotropic

compression, but an exponential decay for a sheared system. This

result is in qualitative agreement with experimental observa-

tions.8 Recent work by Tighe et al.36,37 introduce an area

conservation law in force space to calculate the distribution of

individual contact forces, P(f), in a mean-field approach that

does not require the implementation of force-balance on every

grain. These calculations show that P(f), generically has

Gaussian rather than exponential tails. Snoeijer et al.38 derive an

analytical force distribution in the FNE for an isotropically

stressed triangular lattice, as well as for a general geometry. They

obtain a density of states which scales as hFiD, where D�N(hzi �ziso) is the number of excess force degrees of freedom in the

system.

Fig. 1 Schematic illustration of the conservation principle for a planar

packing under periodic boundary conditions. The left figure shows the

packing in real space and the components of the forces, ~Fx and ~Fy. The

figure on the right is the mapping to force space, which is spanned by

these vectors, and are shown to enclose a possible force tiling comprised

of the individual contact forces from a packing. Each tile in force space

represents the contact forces on a grain.36

3.3 Stress ensemble

The stress ensemble is based on the result,13,14 discussed in detail

below, that the phase space of mechanically stable states of

grains can be partitioned into disconnected sectors labelled by

the force-moment tensor S. This is a type of topological

conservation principle, which implies that states in different

sectors cannot be connected by any dynamics that respects force

and torque balance at every grain, and involve only local changes

of positions and forces. Under periodic boundary conditions, the

value of S is strictly preserved. For other boundary conditions,

S can be changed only through intervention at the boundaries. In

addition to the conservation principle, the stress ensemble relies

on the assumption that states in the same sector are sampled with

some broad, but not necessarily, flat probability measure.

Equiprobability is, however, not assumed.

This section reviews the analysis that leads to a generalized

definition of angoricity, and a canonical distribution function:

Pn ¼un

ZðX ; aÞ eð�a:SnÞeð�Wn=XÞ (4)

where we have retained the possibility of a non-flat measure

through un. The force moment tensor, Sn has been defined

following eqn (3), and is a function of positions and contact forces.

In the limit of X/N, the angoricity is the only relevant

intensive parameter, and in what follows, it is this limit that will be

referred to as the stress ensemble. In this limit, volume fluctuations

are completely unrestricted, and the distribution of local volumes

is flat. We have no a priori justification for working in this limit,

however, applications show that this limit describes stress fluc-

tuations under quasistatic conditions, as discussed in Section 5.

2886 | Soft Matter, 2010, 6, 2884–2893

3.3.1 Conservation principle. The conservation principle

underlying the stress ensemble follows from the conditions of

mechanical equilibrium, and its origins have been discussed in

detail in our previous work.13,14 In planar packings, the conser-

vation of the force-moment tensor S can be derived by using the

formulation of loop forces,39 or height map.40 A more macro-

scopic version follows simply from the observation that for

systems in mechanical equilibrium, the stress tensor is divergence

free: V$s ¼ 0. This is a completely general statement, and the

phase space of mechanically stable states of any system can be

partitioned into sectors labelled by S. The need for an ensemble,

however, only arises in systems where there are an exponentially

large number of microscopic states that belong to each sector

such that the sectors have finite entropy. This is true for granular

systems, and the situation is similar to glassy systems, which have

many metastable states.2 For non-disordered, thermal systems,

which do not have finite entropy at zero temperature, the clas-

sification of states according to S is redundant since energy-

based statistical mechanics provides a complete description.

In this review, I outline a derivation of the conservation

principle that complements our earlier derivations, and high-

lights a more basic conservation principle that is most easily

illustrated for two-dimensional packings, and clarifies the rela-

tionship between force-moment conservation, and the ‘‘area

conservation’’ principle that has been invoked to calculate P(f) in

the FNE for planar packings.36,37

Taking a planar packing, we can lay down a regular x�y grid

with the grid spacing being roughly the grain size, as shown

in Fig. 1. The force moment tensor can then be written as

S ¼P

m

Pnsm,n where the local force moment tensor sm,n is

obtained by summing over all the contacts that cross the top and

right boundaries of the grid (m, n), and the components of sm,n

are the forces per unit grid-length acting along and normal to the

boundaries.41 Performing the sum over n yields a one-dimen-

sional grid with effective forces Fx(m), and Fy(m). Since the

system is in mechanical equilibrium, these forces have to be

independent of m, Fx(m) ^ Fxx, and Fy(m) ^ Fx

y (cf. Fig. 1).42

Similarly, one can define forces Fyx, and Fy

y, as in Fig. 1.

Mechanical equilibrium also requires that Fxy ¼ Fy

x. For

a packing consisting of M �M grains, each of these forces scale

as M. Each packing can be represented in force space by

a parallelogram defined by two vectors, ~Fx ¼ (Fxx,Fy

x), and

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~Fy ¼ (Fxy,F

yy), but there can be many packings that correspond to

a given ~Fx, ~Fy. The partitioning of the phase space of packings by

the value of the extensive quantity, S ¼ M(~Fx,~Fy), therefore, is

founded on the conservation of quantities that are subextensive,

and scale only linearly with system size. Other combinations of~Fx and ~Fy that scale as M2 can therefore be used to label sectors.

An alternative is the set {M(Fxx + Fy

y), FxxFy

y � (Fxy)2}, which is

extensive, and correspond to the choice of TrS anddet S

M2. This

choice is precisely the one identified through the force-tiling

picture of FNE.36

It is interesting that the topological invariants in mechanically

stable packings scale only linearly with system size, a scenario

reminiscent of other constraint satisfaction problems such as

dimer covering of the honeycomb lattice,43 where the conserved

quantity is the number of dimers per row. These lattice models

are known to exhibit unusual phase transitions, and we are now

exploring connections between the unjamming transition and

freezing transitions in such graph coloring models.44 The defi-

nition of angoricity in the stress ensemble is based on the

conservation of S.

3.3.2 Angoricity without equiprobability. It has been

shown23,45 that if a dynamics conserves a physical quantity U, but

violates the microcanonical version of detailed balance, then the

probability Pn of finding a microscopic state is no longer uniform

and equal to the inverse of the density of states,1

UðUÞ. Instead, it

is given by Pn ¼ un/Zm(U), where Zm(U) ¼P

nund(Un � U) is the

microcanonical partition function. If the microscopic weights, un

are all equal, then Zm reduces to the density of states, and one

obtains the usual microcanonical ensemble.

The canonical ensemble framework can also be generalized to

include systems with varying weights un. The necessary condition

for the definition of intensive parameters, and concomitant

canonical distributions is that the stationary distribution,

Pn ¼ un/Zm(U), is factorizable.23,45 Under the equiprobability

assumption, factorizability translates to a factorization of the

density of states: U(U1 + U2) ¼ U(U1)U(U2). With varying

weights, the factorizability assumption becomes: un(U1 + U2) ¼un1

(U1)un2(U2). This is a nontrivial assumption about the statis-

tical weights of microstates, but is weaker than the equiprob-

ability assumption.

These general ideas can be applied to blocked states of gran-

ular systems with U replaced by the tensor S.14

Microcanonical definition of angoricity. If the total force

moment tensor S is fixed in a system that is then divided into two

subsystems, 1 and 2, the conditional probability of finding a force-

moment tensor S1 within subsystem 1 is defined by

P(S1) ¼ P(S1|S)P(S). We can use the definition of the

microcanonical partition function, Zm to write the conditional

probability as

P(S1|S) ¼ (Zm(S))�1P

nund(Sn1� S1)d(Sn2

� (S � S1)) (5)

If the frequency with which the subsystems are accessed

factorizes, i.e. if

un(S) ¼ un1(S1)un2

(S � S2) (6)

This journal is ª The Royal Society of Chemistry 2010

the conditional probability becomes

P�S1

��S� ¼ Zm

�S1

�Zm

�S� S1

�Zm

�S� (7)

Eqn (6) is the central assumption in the derivation of the stress-

ensemble, and it is conceptually equivalent to requiring that the

dynamics chooses state n1 in subsystem 1 independently of state

n2 of subsystem 2. Since the subsystems interact through their

shared boundary only, we expect the correction to eqn (6) to scale

as Oð1=ffiffiffiffiffiffiN1

pÞ, where N1 is the number of grains in subsystem 1.

However, If the system is correlated over length scales x $ffiffiffiffiffiffiN1

p,

we expect eqn (6) to break down as well. The validity of the

factorization assumption has been tested through tests of the

stress ensemble discussed in Section 4.

The most probable value S*1 at a given S is found by setting the

derivative of the conditional probability with respect to S1 to

zero. Since S1 is a tensor, this needs to be done for each

component separately. Using the logarithmic derivative to

simplify the calculation,

0 ¼v ln Zm

�S1

�vS

pq

1

jS

pq*

1

þv ln Zm

�S� S1

�vS

pq

1

jS

pq*

1

The first and the second term have to cancel each other, and

the microcanonical equivalent of the inverse temperature apq is

defined by

�a�

pq¼

v ln Zm

�S1

�vS

pq

1

jS

pq*

1

¼v ln Zm

�S2

�vS

pq

2

jS�S

pq*

1

(8)

For the components of a to act like an inverse temperature

they should be independent of the size of the subsystems. The

tests that have been carried out for simulations of granular

packings13 verify the size independence for systems larger than

a few grains, and these tests are summarized in the next section.

The canonical ensemble. To define the canonical ensemble,

a system with N grains, and fixed S, is divided into one very small

partition Sm, with m� N, and the remaining SN–m (note that we

still need m [ 1 for the factorization ansatz to hold). Focusing

on the small partition where the total stress can fluctuate while

the full system with fixed S acts as a reservoir, similar to the

thermal case, the analysis of the previous paragraph can be

repeated, and in addition, Zm,(N � m)(S� Sm) can be expanded to

first order in Sm. Using this expansion, it is easy to show that:

ln P�Sm

��S� ¼ ln Zm;m

�Sm

��Xd

p;q¼1

v lnZm;N

�S�

vSpq S

pq

m (9)

Defining the inverse angoricity tensor of the N-grain system

(the ‘‘thermal bath’’) by

apq ¼v ln Zm;N

�S�

vSpq (10)

the sum in eqn (9) becomes the total contraction apqSpqm ¼

Tr(aTSm). The order of the indices is irrelevant since S is

a symmetric tensor, and so is a through its definition. The

canonical probability distribution for the total force moment

tensor Sm is then

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Pcan�Sm

�¼ P

�Sm

��S� ¼ Zm;m

�Sm

�e�TrðaSmÞ

ZðaÞ (11)

The canonical partition function, given by

Zm

�a�¼Y

p; q.p

ðdS

pq

m Pcan�Sm

�(12)

acts as a normalization, and is the generating function of all

correlation functions.

The extended definition of angoricity introduces the possibility

of a protocol dependence of P(S) since the weights associated

with the microscopic states, un could depend on how the packing

is prepared. The utility of a statistical framework depends on

how sensitively the partition function depends on history. If the

sensitivity is large than the utility is limited since the full history

will be needed for any calculation, and the framework will not

have much predictive power.

To summarize, the stress ensemble formalizes the Edwards

ensemble approach for blocked states of granular packings by (a)

identifying a physical quantity that is conserved by a certain class

of dynamics, and (b) extends the ensemble framework to situa-

tions where the equiprobability assumption is violated. The

formulation in terms of the partitioning of phase space by S allows

a natural connection between the statistics of blocked states and

the dynamics of slowly driven granular media. This connection

has been exploited to provide a description of stress fluctuations in

sheared systems, and is briefly discussed in the section on appli-

cations of the stress ensemble. A concise statement of the

connection is that there are barriers that separate sectors with

different S, and these barriers diverge with system size if the

dynamics is restricted to the ensemble of blocked states. Under

slow driving, there will be rare events that violate the mechanical

equilibrium condition and allow transitions from one sector to

another. The statistics of blocked states, and the principles of the

stress ensemble can be used to calculate these rates, if there is

a separation of time scales between the dynamics that explores the

states in a sector and the transitions from one sector to another.

4 Tests of the stress ensemble

The stress ensemble framework implies (eqn (8)) that the angoricity

of a mechanically stable packing has to be the same everywhere

Fig. 2 The angoricity,1

avs. mean scalar stress G/N in N-grain systems with ha

N ¼ 4096 (harmonic), and N ¼ 1024 (hertzian). The angoricity values for gra

there is a linear relation between the angoricity and the scalar stress with a s

2888 | Soft Matter, 2010, 6, 2884–2893

inside a packing. In addition, one should demand (a) that the

angoricity is independent of the size of the regions into which the

packing is subdivided, as long as the size is large compared to a grain

size, and (b) that there should be a unique relationship between

a and the S of the packing, providing an equation of state analogous

to temperature-energy equations of state.

The equality of a inside a packing has been tested in iso-

tropically compressed packings of frictionless grains, generated

by simulations of systems with hertzian and harmonic contact

forces.13,14,46 The test was designed to (a) check that the distri-

bution of stresses followed the Boltzmann-like distribution:

PaðGÞ ¼ZmðGÞZðaÞ expð�aGÞ (13)

where the scalar a is the component of the inverse angoricity that

is conjugate to G ¼ TrS, (b) measure the a of packings, and (c)

measure the effective density of states, Zm(G).

Fig. 2, is a plot of a as a function of G of the packing, and the

size of the subsystem within which a was measured.46 The figure

demonstrates that a is approximately independent of the number

m of grains included in the subsystem, and that there is a linear

relation: a x2N

G, where N is the total number of grains in the

packing. These results were obtained by measuring the distri-

bution of the local values of Gm in an m-grain cluster, and

rescaling the distributions to test the form of the canonical

distribution, eqn (13).13,14 Since this method required over-

lapping distributions of Gm, the rescaling method could be

applied only over a limited range of G from 10�5 to z 10�3, as

shown in the figure. To test the ensemble over a larger range of

G and, therefore, a, the microcanonical partition function Zm,

which is the analog of the density of states was fitted directly to

a functional form by assuming the canonical distribution, and

the relationship between a and Zm. This method was used to test

the ensemble for the system with harmonic interactions over

G ranging from 10�5 to 100, and led to the equation of state

a ¼ N

Gð2þ aðzÞÞ, where a(z) is a function of z, the average

number of contacts in the packing, and goes to zero at the

isostatic point.13,14 The a obtained by this method was found to

be independent of the subsystems size for m $ 16.

Work in our group is now focusing on directly testing the

equality of a in (a) packings of photoelastic beads under isotropic

rmonic and hertzian interactions. From left to right: N¼ 1024 (harmonic),

in clusters ranging from m ¼ 3 – 256 are shown, and over this full range,

lope consistent with 1/2.

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compression and shear and (b) simulations of frictionless, and

frictional grains subjected to quasistatic shear. The tests for

sheared packings are based on measuring histograms of all the

independent components of S inside m-grain subregions, and

rescaling the distributions, as was done for the isotropic systems

shown in Fig. 2. Preliminary results show that mechanically

stable states created by shear also satisfy the assumptions of the

stress ensemble,47 as expressed in eqn (11). In addition, we have

indirect evidence that the equal angoricity assumption does hold

for sheared states through applications of the stress ensemble to

granular rheology, as discussed in the next section.

5 Applications of the stress ensemble

In parallel with testing the stress ensemble, we have proceeded

with using the statistical framework to calculate spatial and

temporal fluctuations of stress in static, and slowly driven

granular materials, and comparing these predictions to experi-

ments and simulations.

For frictionless, non-rigid grains, the contact forces are related

to the grain positions through a force law, for example a Hertzian

law relating deformation to force. In the presence of friction,

there is indeterminacy of the tangential forces through the

Coulomb condition, and forces have to be treated as additional

statistical variables. All statistical variables have to satisfy the

constraint of mechanical equilibrium since only such states

belong to the statistical ensemble of eqn (3). The calculation of

the partition function, Z(a), therefore, involves imposing non-

trivial constraints. For planar packings, there is an elegant

formulation in terms of loop forces39 that has been used to

construct field theories of stress transmission in granular

packings.14,40,48

For most of our calculations, we have adopted the FNE idea

that the forces and grain-positions can be treated as independent

statistical variables, and the partition function includes all states

that satisfy force and torque balance. It is certainly possible,

within our framework, to introduce the force-law as an addi-

tional constraint,40 and this has been done to a limited extent by

incorporating empirically measured equations of state into our

theoretical framework.48

If we assume a flat measure, un ¼ 1, then the canonical

partition function of the stress ensemble and the distribution of

contact forces, P(f) at the isostatic point can be calculated

exactly, and P(f) has an exponential form.14 It should be

emphasized that the canonical stress ensemble does not imply an

exponential form for P(f) except at the isostatic point, if we

assume a flat measure. An exponential distribution emerges at

the isostatic point because, with a flat measure, the forces are

independent random variables at this point. Similarly, in an ideal

gas, the Boltzmann distribution, e�bE, becomes an exponential

distribution of energies of individual particles because the energy

E is a sum of single particle energies. For interacting systems, e�bE

does not imply an exponential distribution of single particle

energies. With interactions, a calculation of P(f) is challenging, as

any one-body distribution is difficult to calculate for interacting

systems.49

The stress ensemble framework is ideal for calculating corre-

lation functions and response functions, and this has been the

focus of the work in our group. In this section, I briefly describe

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two such examples, (i) calculating spatial correlations of stress,48

and (ii) an application of the stress ensemble to granular

rheology.50

5.1 Spatial correlations of stress

Measurement of the spatial correlation of local, grain level

pressure in two-dimensional packings of photoelastic disks8

show that the correlations are short-ranged for isotropic pack-

ings and become long-ranged under shear. The appearance of

force chains in images correlates well with the appearance of

long-ranged correlations in the pressure.

In equilibrium systems, correlation functions are calculated

from exact or approximate calculations of the partition function.

The stress ensemble opens up the same avenue for calculating

correlations in granular packings. In equilibrium systems,

approximations calculations of partition function are often

based on defining ‘‘quasiparticles’’ in terms of which the system

becomes non-interacting. Such an approach has been applied to

granular materials in the context of the original volume ensemble

with ‘‘quadrons’’ defined as the quasiparticles in terms of which

the volume function becomes non-interacting (analogous to

a non-interacting Hamiltonian).26 Similar approaches that

convert the partition function to that of a non-interacting

systems have been pursued by other groups.17

To construct a theory for stress correlations in two-dimen-

sional packings, we have adopted an alternative, coarse-grained

approach, in which jammed configurations are represented by

a continuous scalar field, j(r), and the the partition function

Zm(S) is written in terms of an appropriate Ginzburg–Landau

functional based on general invariance arguments, and the strict

positivity of forces in dry granular materials. As expected, the

form of the field-theoretic action is the same as that of elasticity

theory, however, with the elastic stiffness constants being

determined by S:

Zm

�S�heSðSÞ ¼

ðDce�LS ½j�

LS½c� ¼ð

d2r

�K1

�S��

v2xc�2þK2

�S� �

v2yc�2

þK3

�S��

v2xc��

v2yc�þ.

o(14)

The field j(r) is the familiar Airy stress function from two-

dimensional elasticity theory,51 and all stress correlations func-

tions can be expressed in terms of the correlations of j.48

The idea of load-dependent stiffness constants in granular

materials is not new. This feature, however, appears naturally in

the stress ensemble formulation,48 and the stiffness constants are

determined by the entropy S(S). This connection allows us to

relate stress fluctuations to measurements of entropy through

distributions of local stresses. Since the functional integral in

eqn (14) is Gaussian, the entropy S(S) can be easily expressed as

a function of the stiffness constants, Ki(S).14 Under isotropic

compression, for example, there is a single stiffness constant,

K(G), and S(G) f �ln K(G).14 We used exact calculations at the

isostatic point,13 and simulation data for probability distribu-

tions, P(G) away from the isostatic point to obtain S(G).13,14 The

stiffness constant obtained from S(G) was then used to calculate

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the spatial stress correlations for isotropically compressed

states.14,48 The results predicted that very close to jamming,

KðGÞ ¼ ziso

G2, with corrections depending on the deviation of the

contact number from the isostatic value, ziso, away from

jamming. This scaling of the stiffness constant implies that the

strength of the spatial correlations at different G also scale in this

manner. Comparisons to simulations and experiments showed

that this scaling was obeyed over a large range of compression,48

and even far away from jamming if we retained the correction

terms.

Our exact calculation of K(G) at the isostatic point shows that

the source of theziso

G2scaling is the quenching of fluctuations of

local pressure, and the reduction of entropy, because of the

constraint of positivity of the contact forces.14 This observation

led us to propose that under pure shear characterized by

compression G, and normal stress s, the two independent stiff-

ness constants should scale as2

ðGþ sÞ2, and

2

ðG� sÞ2.48 The

elasticity theory then becomes anisotropic, and the predictions

for the spatial stress correlations are in semiquantitative agree-

ment with experiments and simulations.48 Although the structure

of the elasticity theory (eqn (14)) follows from invariance argu-

ments, the specific form of the stiffness coefficients was deduced

from the functional form of S(G). If we include corrections to the

equation of state13 that appear for significant grain deformations,

then the theory can describe a large range of compressions. Fig. 3

shows a comparison of the measured spatial correlations of stress

with the predictions of the theory.48

Theories of force chains in granular media have been based on

the nature of isostatic states.52,53 Force chains also naturally arise

in anistropic elasticity theories. The entropically formulated

elasticity theory indicates a relationship between these two

different viewpoints since it is the load dependence of the entropy

of isostatic packings that leads to the anisotropic theory.

The application of the stress ensemble to stress fluctuations in

two-dimensional packings strongly suggests that the response of

granular materials to external stress is controlled by the

entropy S(S).

Fig. 3 Decay of pressure correlations hdG(r)dG(0)i/h(dG(0))2i along compre

under pure shear for (a) simulations at s/G¼ 0.3 and m¼ 0 and (b) experiments

scaled by an overall constant to match correlations in the expanded direction a

insets show the infinite system results. (Reprinted from ref. 48).

2890 | Soft Matter, 2010, 6, 2884–2893

5.2 Granular rheology

Extending the generalized statistical ensembles to analyze gran-

ular dynamics has recently gained momentum.12,18,50,54 A model

of granular rheology has to take into account the disordered

nature of the packings and metastability. These are common

feature of all soft glassy materials, and a framework that has

been applied is that of soft glassy rheology (SGR).55 SGR builds

on the trap model of glassy dynamics in thermal systems,56 which

describes activated dynamics in a landscape with a broad

distribution of barrier heights, and mean-field connectivity. Both

the trap model and SGR incorporate spatial heterogeneity and

intermittency with mesoscopic subregions that move coherently

through the landscape, ie all particles in a subregion move

together from one metastable state to another.2 A subregion hops

from one metastable state to another activated either by

temperature (thermal glasses) or a ‘‘fluctuation temperature’’

(SGR). The fluctuation temperature is a measure of the fluctu-

ations resulting from coupling between subregions.

The stress ensemble suggests that angoricity is the natural

candidate for fluctuation temperature in granular packings.

Subregions move from one metastable state, characterized by S,

to another aided by fluctuations of stress. The strength of these

fluctuations is precisely what is measured by angoricity. We have

adapted the SGR framework to describe activated dynamics in

a stress landscape, with angoricity and strain rate controlling the

time evolution of the stress in a mesoscopic subregion.50,54 This

framework was used to analyze data from packings being

sheared in a Couette geometry.57 The angoricity-driven activated

mechanism provided an explanation for the observed loga-

rithmic strengthening with shear rate.54 Using the theoretical

framework to analyze the temporal fluctuations of the stress

resulted in a relationship between the angoricity, packing frac-

tion and shear rate.50 In particular, we analyzed the build-up and

release of stress in steady state using the SGR model in which the

distribution of barriers in the stress landscape is characterized by

an angoricity 1/a0. The escape over barriers is activated, with the

angoricity 1/a of the sample playing the role of temperature, and

the shear rate, _g, reducing the effective barrier that needs to be

overcome.55,50 Fig. 4 shows the results of fitting experimental

ssed (circles, dashed lines) and expanded (squares, solid lines) directions

at s/G¼ 0.51 and m¼ 0.7. Theoretical predictions (lines), which have been

t 0.5, are compared to (a) simulations and (b) experiments (symbols). The

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Fig. 4 Log–Log plot of experimental distribution of stress drops and its fitting to our theoretical form.50 The experimental and fitting parameters are:

(a) _g ¼ 0.3318 mHz, f � fc ¼ 0.0053, a ¼ 0.70, a0 ¼ 0.3247, (b) _g ¼ 0.9972 mHz, f � fc ¼ 0.0053, a ¼ 0.7, a0 ¼ 0.2660, (c) _g¼ 5.3155 mHz,

f � fc ¼ 0.0088, a ¼ 1.2, a0 ¼ 0.2131, and (d) _g ¼ 0.0657 mHz, f � fc ¼ 0.0137, a ¼ 0.4, a0 ¼ 0.2547.

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data to our theory over a wide range of _g and packing fractions f

using a and a0 as fitting parameters. The results of the fit lead to

an interesting relationship between x ¼ a0

a, and the scaling

combination _g(f � fc)0.4, where fc is the jamming threshold

below which the system cannot sustain shear. The scaling rela-

tion is intriguing because it suggests that x can be held constant

in systems with spatially varying shear rates and packing frac-

tions such as in shear zones, if _g and f follow the scaling form.

We are in the process of measuring a in experimental packings

with shear zones.

The idea of stress fluctuations playing the role of temperature

has been used previously to understand shear zones in granular

chute flows.58 The stress ensemble puts this connection on a more

rigorous footing. The coupling between stress and volume that is

encapsulated in the generalized stress ensemble offers a possible

route to understanding the mechanism behind Reynolds dilat-

ancy.

6 Emerging field

In the past, the main focus of research centered on statistical

ensemble ideas for granular matter had been (a) testing the

equiprobability hypothesis and (b) constructing simple models

for the volume function, W(~ri) for which partition functions

could be calculated exactly. Within the last couple of years, there

has been a shift in emphasis with groups using the generalized

statistical ensemble to analyze experimental and simulation data,

and using the results of the analysis to construct effective models

that have predictive power. Using the canonical, Boltzmann-like

distribution as a basis for analyzing fluctuations in granular

materials has emerged as a productive approach. We certainly

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need to perform more tests and understand the limitations of the

approach. A lot can be learned, however, by assuming the

applicability of the ensemble and making testable predictions.

The canonical distribution also provides a mechanism for

performing simulations based on stochastic dynamics rather

than Molecular Dynamics. The analogue of the Boltzmann

weight can be use to construct a Markov process that obeys

detailed balance and produces a the canonical distribution as the

stationary distribution. Algorithms that have proven useful for

understanding phase transitions in thermal systems can be

extended to the study of granular packings. As illustrated

through one of the examples in the previous section, effective

field theories can be constructed taking the Landau-type

approach based on symmetries and constraints.

The statistical approach lends a unifying framework to

concepts that existed as isolated suggestions. For example the

analysis of shear zones in terms of a stress-activated process58

finds a natural basis in the stress ensemble. The stress ensemble

with a distribution that involves exp(�a:S) leads to an expo-

nential distribution, P(f) of individual contact forces only if the

contact forces are independent variables with no correlation

between them, as in an ideal gas. As we have shown14 this

happens only under special circumstances such as at the isostatic

point of frictionless grains. Constraints of mechanical equilib-

rium normally introduce effective interactions between the

contact forces and change the exponential form of P(f).

In particular, the constraints introduce terms of the type

exp(– Kf2) in P(f). Observation of an exponential distribution,

therefore, says something about the nature of correlations.

The stress ensemble can be used to calculate fluctuation–

dissipation relations, and the associated effective tempera-

tures.21,59–61 These calculations will link the measurements of

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effective temperatures to the entropy of packings, which can be

independently measured, thus providing important consistency

checks, and new insight into the origin of effective temperatures

in slowly-driven granular materials.

The stress, or generalized Edwards ensemble has much in

common with statistical frameworks developed for glassy

systems.2 Collective properties in both are determined by the

complexity of the landscape of metastable states. Complexity in

spin glasses is measured by the logarithm of the number of local

free energy minima that have a given free energy density.2,62

An analogous definition applies to the potential energy land-

scape of supercooled liquids.63,64 For granular systems,

complexity is measured by the logarithm of the number of

mechanically stable states with a given force moment tensor,

which is the entropy S(S). As illustrated by the example of spatial

correlations of stress, discussed above, stress fluctuations are

controlled by the configurational entropy, S(S).

Unlike thermal ensembles, glassy and granular systems have

broken ergodicity with barriers separating metastable states.

In meanfield spin glass models, these barriers diverge with system

size. Within the ensemble of mechanically stable states, the

barriers between sectors with different S diverge with systems

size because of the topological conservation principle. It has long

been argued2 that these similarities can be exploited to analyze

the dynamics of granular systems. In our example of analyzing

stress statistics in a Couette geometry, the framework was bor-

rowed from soft glassy rheology and adapted to the stress

ensemble.

The application of statistical ensembles to calculate correlation

and response in granular materials is just beginning, and the field

is in its infancy. In the next few years, I expect there to be many

more applications of the statistical ensemble framework to

characterize fluctuations and non-equilibrium transitions in

granular assemblies.

7 Acknowledgements

BC acknowledges her collaborators, Silke Henkes, R. Behringer,

Dapeng Bi, Trush Majmudar, Gregg Lois, Jie Zhang, Mitch

Mailman, and Corey O’Hern. The discussion of the stress

ensemble is based on the PhD thesis work of Silke Henkes and

Dapeng Bi. She also acknowledges helpful discussions with Mike

Cates, Nick Read and Brian Tighe. This work was funded by the

grants NSF-DMR0549762, and NSF-DMR0905880.

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