We have looked into an experiment that has been termed the ‘‘canonical example’’ of jamming: granular material, clogging the outlet of a container as it is discharged by gravity. We present quantitative data of such
an experiment. The experimental control parameter is the ratio between the radius of the orifice and the radius of the beads. As this parameter is increased, the jamming probability decreases. However, in the range of parameters explored, no evidence of criticality—in the sense of a jamming probability that becomes infinitely
small for a finite radius—has been found. We draw instead a comparison with a simple model that captures the main features of the phenomenon, namely, percolation in one dimension. The model gives indeed a phase transition, albeit a special one

Garcimartín-Montero, Á. (Ángel)

Maza-Ozcoidi, D. (Diego)

Pugnaloni, L.A. (Luis A.)

Zuriguel-Ballaz, I. (Iker)

Montero, Á. (Ángel)

Dadun, University of Navarra

RAPID COMMUNICATIONSPHYSICAL REVIEW E 68, 030301~R! ~2003!Jamming during the discharge of grains from a silo described as a percolating transitionIker Zuriguel,1 Luis A. Pugnaloni,2 Angel Garcimartı´n,1,* and Diego Maza11Departamento de Fı´sica, Facultad de Ciencias, Universidad de Navarra, E-31080 Pamplona, Spain2Procter Department of Food Science, University of Leeds, Leeds LS2 9JT, United Kingdom~Received 23 December 2002; published 9 September 2003!We have looked into an experiment that has been termed the ‘‘canonical example’’ of jamming: granularmaterial, clogging the outlet of a container as it is discharged by gravity. We present quantitative data of suchan experiment. The experimental control parameter is the ratio between the radius of the orifice and the radiusof the beads. As this parameter is increased, the jamming probability decreases. However, in the range ofparameters explored, no evidence of criticality—in the sense of a jamming probability that becomes infinitelysmall for a finite radius—has been found. We draw instead a comparison with a simple model that captures themain features of the phenomenon, namely, percolation in one dimension. The model gives indeed a phasetransition, albeit a special one.DOI: 10.1103/PhysRevE.68.030301 PACS number~s!: 45.70.Ht, 45.70.MgThe suggestive idea that jamming is at the origin of a newstate of matter, amenable to be described in a thermodynamicformalism, has been recently proposed @1#. Our experimentis akin to the familiar experience of shaking a saltcellar; saltgrains plug the outlet of the container due to the formation ofarches. Shaking or tapping is needed to break up the arch andrestore the flow after each jam. A similar situation is found inmany industrial applications, such as dosage hoppers, and invarious transport phenomena. Ad hoc laws have been putforward based on experience and observation @2#, but thephysics of arching is still poorly understood. A new vision ofjamming has recently been put forward, relying on the con-cept of fragile matter, which encompasses not only granularmaterials but foams, unstructured glasses, and other systems@3#. The common feature they share is that they support cer-tain stresses, called compatible, developed as a response tothe external forces, but are unstable against incompatiblestresses.We have devised a simple arrangement to conduct a quan-titative study of the jamming that halts a granular flow. Ourassembly consists of a scaled cylindrical silo with a circularopening at the base. The bin is filled with monodisperse glassspheres ~radius standard deviation is about 1%!. Beads ofdifferent sizes were used in separate trials, but always with adiameter larger than 1 mm in order to reduce the relativeimportance of disturbances—such as humidity or electro-static interactions—against gravity. The diameter of the binis larger than 30 bead diameters; in this way, the finite size ofthe system can be neglected, as reported in the literature @4#and checked specifically for our setup @5#. The silo is alwaysfilled with the same procedure ~a distributed filling @6#, pour-ing the grains rapidly in the silo! in order to obtain a repeat-able compaction of the material ~a volume fraction of about0.59! from run to run. Besides, the pressure at the base of asilo is known to be almost constant as long as the height ofthe material above it exceeds a certain level @7#. Indeed, the*Author to whom correspondence should be addressed. Email ad-dress: angel@fisica.unav.es1063-651X/2003/68~3!/030301~4!/$20.00 68 0303results of our experiment are repeatable—irrespective of theamount of material in the silo—provided that the heightreached by the beads is larger than twice the diameter of thesilo. We refill the silo from time to time so that this conditionis met. We have used a transparent glass silo in order tocheck that a mass flow develops inside it @6#; a funnel wastherefore not formed during the experiments.When the experiment begins, beads pour freely from theoutlet due to gravity. Soon thereafter, however, if the diam-eter of the orifice does not exceed a few bead diameters, anarch is formed arresting the flow. An electronic balance be-neath the silo measures the number of grains s which fallbetween two successive jams. The size of the avalanche isstored in a computer and then the arch at the outlet is de-stroyed so as to trigger another avalanche. This is accom-plished by means of a jet of compressed air aimed at theorifice from beneath the silo. In this way, the compaction ofthe granular material remains approximately constant duringan experimental run—shaking would instead pack the grainstogether more densely @8#. Further information on the setupwill be provided elsewhere. We made sure that variablessuch as the pressure of compressed air or the material fromwhich the bin is made have a negligible effect on the datathat we are presenting @9#. The parameter that we control isthe ratio between the diameter of the outlet orifice and thediameter of the beads, called R in the following. Both thesize of the orifice and the size of the beads may be changed;only the ratio R is relevant.For a fixed value of R, the data collected after a largenumber of avalanches—several thousands typically—arebest viewed in the form of a histogram. In Fig. 1, nR(s),which is the number of avalanches consisting of s grains fora given R, is presented for R53 in a semilogarithmic plot.From the histograms of all the orifice sizes studied (1,R,4.5) it is possible to define the jamming probabilityJ(R ,N) as the probability of finding an avalanche of size lessthan or equal to N grains for a given R. That is to say,J(R ,N) is the probability that an orifice of relative size Rgets obstructed at least once before more than N grains fallthrough it. Then J can be evaluated as©2003 The American Physical Society01-1RAPID COMMUNICATIONSZURIGUEL et al. PHYSICAL REVIEW E 68, 030301~R! ~2003!J~R ,N !5(s50NnR~s !(s50‘nR~s !. ~1!We have obtained J for different values of R from theexperiment; the results are shown in Fig. 2. The curves in thegraph show that the bigger the size of the orifice, the smallerthe probability that it gets blocked before an avalanche largerthan N grains falls. Also, J grows with N for fixed R. Thesedata are consistent with those found in a two dimensionalhopper @10#, obtained with a fixed number of grains, muchsmaller than in our experiment.One could conceivably speculate from Fig. 2 that J wouldperhaps tend to a step function as N→‘ . The question iswhether for that limit the jamming probability is one below acertain size Rc , while it is zero above Rc . In other words,whether there is a finite value of R beyond which the orificewould not jam at all for an infinitely large silo. We can assessthis by evaluating the size of the orifice for which the jam-ming probability is one half—let us call it R0(N). This canbe easily obtained from the fit of J(R ,N) ~see Fig. 2!. Theparticular fitting function for J chosen to obtain R0(N) isalmost irrelevant because the slope of J is quite large at R5R0. In Fig. 3~a! we plot R0 versus N in semilogarithmicscale. It is unclear whether R0 saturates. Several functionscan be fit reasonably well to R0(N) @see Fig. 3~a!#, some ofwhich saturate for N→‘ , while others do not. Values of Nmuch bigger than 105 are experimentally inaccessible in ourlaboratory. Therefore we cannot conclude whether or notthere is a critical value for R0. The family of curves used tofit J also depends on a , which is related to the slope of J atFIG. 1. Histogram for the number of grains that flow betweentwo successive jammings. Data correspond to R53 ~the beads havea diameter of 2 mm, and the circular orifice is 6 mm wide!. Morethan 4000 events have been recorded. The line is a linear fit. Inset:first return map, i.e., the avalanche size nt vs the next avalanchesize nt11. Note that t is just a correlative index ordering the se-quence of avalanches.03030R0. The inverse of a is the width of the transition regionfrom J51 to J50. The values for 1/a are plotted in Fig.3~b!. It decreases with N but it is not possible to ascertain ifit cancels out for a finite value of N, as would be the case fora step function. Both parameters, R0 and a , are not related ina simple way.These results can be understood in the framework of aone-dimensional percolation model. Let us call pR the prob-ability that one grain gets past the outlet of size R withoutblocking it. Assuming that these events are independent, anFIG. 2. Jamming probability J as a function of R. Each seriescorresponds to a fixed N, i.e., the probability of the orifice gettingjammed before N grains pass through it. Data taken for several beaddiameters are displayed. Successive series, from left to right, corre-spond to N53, 10, 30, 100, 300, 1000, 3000, 10 000, and 30 000beads. The fit corresponds to a hyperbolic tangent with two adjust-able parameters: J(R ,N)5$12tanh@a(R2R0)#%/2. The two pa-rameters of this family of curves, a and R0, are shown in Fig. 3.Error bars are statistical.FIG. 3. ~a! R0 vs N. Note the logarithmic scale on the horizontalaxis. ~b! 1/a as a function of N in semilogarithmic scale. Both R0and a were obtained from the fits of Fig. 2.1-2RAPID COMMUNICATIONSJAMMING DURING THE DISCHARGE OF GRAINS FROM . . . PHYSICAL REVIEW E 68, 030301~R! ~2003!avalanche of size s would consist of one jamming event, snonjamming events, and another jamming event. The prob-ability of this sequence isnR~s !5~12pR!2pRs. ~2!This also holds if one considers the grains falling in clusters,or groups, rather than one at a time. In this case, pR is theprobability that a group of grains passes the outlet withoutjamming, i.e., without forming an arch. If outgoing groupshave k grains in average, the probability of finding an ava-lanche of s grains would be nR(s)5(12pR)2pRs/k . From Eq.2 one obtains log(nR)52log(12pR)1s log(pR), so log(nR) hasa linear dependence on s, exactly as observed experimentally.The value of pR can be determined from the histogram. Theexponential dependence of nR(s) on s precludes the interpre-tation of these data in the framework of self-organized criti-cality ~SOC! @11#, because there exists a characteristic ava-lanche size manifested in the exponential distribution ofevents. Conversely, experiments with rough grains in differ-ent experimental conditions @12# and numerical simulationsin a two-dimensional silo @13# yield a power lawdistribution—asignature of SOC.Incidentally, the exponential distribution is a compellingevidence that arch formation is an uncorrelated process.From the temporal point of view, this is also revealed by theautocorrelation function of the series of avalanches and bythe first return map ~see Fig. 1, inset!, which show no sign ofmemory between consecutive avalanches.From the avalanche size distribution given by Eq. ~2!, themean size avalanche S can be explicitly calculated, yieldingS5(s50‘s2nR~s !(s50‘snR~s !511pR12pR. ~3!The mean avalanche sizes obtained from our data areshown along with the above expression in Fig. 4. Indeed thehistogram given by Eq. ~2! and the mean avalanche size ~3!are those of a one-dimensional percolation model @14#,which leads to a second order phase transition, but only atpR51. This is marked by a divergence to infinity in themean avalanche size S, which is a measure of the suscepti-bility of the system. Let us remark that the limit pR51 isunattainable, so such a phase transition would never be ob-served. In practice, however, pR could be made high enough~if the radius of the outlet orifice is big enough!, so thatjamming would seldom be observed.We can put forward the idea that the jamming transitionbears an analogy with the glass transition. By increasing Rone selects larger bridges only, so 1/R could be thought of as03030the ‘‘temperature.’’ In the glass transition, the characteristicviscous time is certainly infinite for zero temperature; for afinite temperature, it can be extremely large but a power lawdivergence is not found @15#. In the same sense, the meanavalanche size ~or the susceptibility, for that matter! is infi-nite for 1/R→0, and could be very large for finite values, butthere would not exist a critical exponent for S.In this paper, we have shown that jamming events duringthe discharge of a silo of glass beads are uncorrelated andshow no evidence of self-organized criticality, in contrastwith other avalanche studies @12,13# which were performedin different geometries. Moreover, we showed evidence thatsuggests that the classical claim of the existence of a criticaloutlet size beyond which no jamming occurs—i.e., J be-comes a step function—could be invalid in a thermodynamicsense. Comparison between experiment and recent numericalsimulations @16# could help to gather more understanding.Finally, other interesting aspects that have not been consid-ered, such as the influence of the roughness of the beads andtheir packing fraction, also merit further investigation.The Spanish Ministry of Science and Technology ~ProjectBFM2002-00414!, the Government of Navarre, and the Uni-versity of Navarre have supported this work. L.A.P. ac-knowledges support from Fundacio´n Antorchas ~Argentina!and the British Council. We thank Dr. G. C. Barker for hishelpful comments, and Dr. O. Dauchot for discussing hisideas with us.FIG. 4. Mean avalanche size S as a function of pR : experimen-tal data ~points! and model ~solid line!. For the experimental data,pR has been estimated as Nb /(Nb1N j), where Nb is the total num-ber of fallen beads and N j the total number of jammings, for a givenorifice size. Inset: the same data, plot in logarithmic scale as afunction of 12pR .@1# A.J. Liu and S.R. Nagel, Nature ~London! 396, 21 ~1998!.@2# A. Drescher, A.J. Waters, and C.A. Rhoades, Powder Technol.84, 177 ~1995!.@3# M.E. Cates, J.P. Wittmer, J.-P. Bouchaud, and P. Claudin, Phys.Rev. Lett. 81, 1841 ~1998!.@4# D. Hirshfeld and D.C. Rapaport, Eur. Phys. J. E 4, 193 ~2001!.1-3RAPID COMMUNICATIONSZURIGUEL et al. PHYSICAL REVIEW E 68, 030301~R! ~2003!@5# We have done this by performing experiments in several silosof different diameters, namely, 30, 80, 120, and 150 mm, withbeads of 2 and 3 mm in diameter. The results from the threelarger silos are indistinguishable; not so for the smaller silo.@6# Z. Zhong, J.Y. Ooi, and J.M. Rooter, Eng. Struct. 23, 756~2001!.@7# J. Duran, Sables, Poudres et Grains ~Eyrolles, Paris, 1997!.@8# E.R. Nowak, J.B. Knight, E. Ben-Naim, H.M. Jaeger, and S.R.Nagel, Phys. Rev. E 57, 1971 ~1998!,@9# We set the pressure of the air jet to the minimum needed tobreak the arches, which is about 2 bar. We checked that apressure of up to 7 bar does not affect the results ~the expo-03030nential tail of the histogram remains unchanged!.@10# K. To, P.-Y. Lai, and H.K. Pak, Phys. Rev. Lett. 86, 71 ~2001!.@11# P. Bak, C. Tang, and K. Wiesenfeld, Phys. Rev. A 38, 364~1988!.@12# V. Frette et al., Nature ~London! 379, 49 ~1996!.@13# S.S. Manna and H.J. Herrmann, Eur. Phys. J. E 1, 341 ~2000!.@14# D. Stauffer and A. Aharony, Introduction to PercolationTheory ~Taylor and Francis, London, 1992!.@15# M. Me´zard, Physica A 306, 25 ~2002!.@16# L.A. Pugnaloni, G.C. Barker, and A. Mehta, Adv. ComplexSyst. 4, 289 ~2001!.1-4

Jamming during the discharge of grains from a silo described as a percolating transition

Universidad de Navarra

Deposito Adademico Digital Universidad De Navarra

RAPID COMMUNICATIONS
PHYSICAL REVIEW E 68, 030301~R! ~2003!Jamming during the discharge of grains from a silo described as a percolating transition
Iker Zuriguel,1 Luis A. Pugnaloni,2 Angel Garcimartı´n,1,* and Diego Maza1
1Departamento de Fı´sica, Facultad de Ciencias, Universidad de Navarra, E-31080 Pamplona, Spain
2Procter Department of Food Science, University of Leeds, Leeds LS2 9JT, United Kingdom
~Received 23 December 2002; published 9 September 2003!
We have looked into an experiment that has been termed the ‘‘canonical example’’ of jamming: granular
material, clogging the outlet of a container as it is discharged by gravity. We present quantitative data of such
an experiment. The experimental control parameter is the ratio between the radius of the orifice and the radius
of the beads. As this parameter is increased, the jamming probability decreases. However, in the range of
parameters explored, no evidence of criticality—in the sense of a jamming probability that becomes infinitely
small for a finite radius—has been found. We draw instead a comparison with a simple model that captures the
main features of the phenomenon, namely, percolation in one dimension. The model gives indeed a phase
transition, albeit a special one.
DOI: 10.1103/PhysRevE.68.030301 PACS number~s!: 45.70.Ht, 45.70.MgThe suggestive idea that jamming is at the origin of a new
state of matter, amenable to be described in a thermodynamic
formalism, has been recently proposed @1#. Our experiment
is akin to the familiar experience of shaking a saltcellar; salt
grains plug the outlet of the container due to the formation of
arches. Shaking or tapping is needed to break up the arch and
restore the flow after each jam. A similar situation is found in
many industrial applications, such as dosage hoppers, and in
various transport phenomena. Ad hoc laws have been put
forward based on experience and observation @2#, but the
physics of arching is still poorly understood. A new vision of
jamming has recently been put forward, relying on the con-
cept of fragile matter, which encompasses not only granular
materials but foams, unstructured glasses, and other systems
@3#. The common feature they share is that they support cer-
tain stresses, called compatible, developed as a response to
the external forces, but are unstable against incompatible
stresses.
We have devised a simple arrangement to conduct a quan-
titative study of the jamming that halts a granular flow. Our
assembly consists of a scaled cylindrical silo with a circular
opening at the base. The bin is filled with monodisperse glass
spheres ~radius standard deviation is about 1%!. Beads of
different sizes were used in separate trials, but always with a
diameter larger than 1 mm in order to reduce the relative
importance of disturbances—such as humidity or electro-
static interactions—against gravity. The diameter of the bin
is larger than 30 bead diameters; in this way, the finite size of
the system can be neglected, as reported in the literature @4#
and checked specifically for our setup @5#. The silo is always
filled with the same procedure ~a distributed filling @6#, pour-
ing the grains rapidly in the silo! in order to obtain a repeat-
able compaction of the material ~a volume fraction of about
0.59! from run to run. Besides, the pressure at the base of a
silo is known to be almost constant as long as the height of
the material above it exceeds a certain level @7#. Indeed, the
*Author to whom correspondence should be addressed. Email ad-
dress: angel@fisica.unav.es1063-651X/2003/68~3!/030301~4!/$20.00 68 0303results of our experiment are repeatable—irrespective of the
amount of material in the silo—provided that the height
reached by the beads is larger than twice the diameter of the
silo. We refill the silo from time to time so that this condition
is met. We have used a transparent glass silo in order to
check that a mass flow develops inside it @6#; a funnel was
therefore not formed during the experiments.
When the experiment begins, beads pour freely from the
outlet due to gravity. Soon thereafter, however, if the diam-
eter of the orifice does not exceed a few bead diameters, an
arch is formed arresting the flow. An electronic balance be-
neath the silo measures the number of grains s which fall
between two successive jams. The size of the avalanche is
stored in a computer and then the arch at the outlet is de-
stroyed so as to trigger another avalanche. This is accom-
plished by means of a jet of compressed air aimed at the
orifice from beneath the silo. In this way, the compaction of
the granular material remains approximately constant during
an experimental run—shaking would instead pack the grains
together more densely @8#. Further information on the setup
will be provided elsewhere. We made sure that variables
such as the pressure of compressed air or the material from
which the bin is made have a negligible effect on the data
that we are presenting @9#. The parameter that we control is
the ratio between the diameter of the outlet orifice and the
diameter of the beads, called R in the following. Both the
size of the orifice and the size of the beads may be changed;
only the ratio R is relevant.
For a fixed value of R, the data collected after a large
number of avalanches—several thousands typically—are
best viewed in the form of a histogram. In Fig. 1, nR(s),
which is the number of avalanches consisting of s grains for
a given R, is presented for R53 in a semilogarithmic plot.
From the histograms of all the orifice sizes studied (1,R
,4.5) it is possible to define the jamming probability
J(R ,N) as the probability of finding an avalanche of size less
than or equal to N grains for a given R. That is to say,
J(R ,N) is the probability that an orifice of relative size R
gets obstructed at least once before more than N grains fall
through it. Then J can be evaluated as©2003 The American Physical Society01-1
RAPID COMMUNICATIONS
ZURIGUEL et al. PHYSICAL REVIEW E 68, 030301~R! ~2003!J~R ,N !5
(
s50
N
nR~s !
(
s50
‘
nR~s !
. ~1!
We have obtained J for different values of R from the
experiment; the results are shown in Fig. 2. The curves in the
graph show that the bigger the size of the orifice, the smaller
the probability that it gets blocked before an avalanche larger
than N grains falls. Also, J grows with N for fixed R. These
data are consistent with those found in a two dimensional
hopper @10#, obtained with a fixed number of grains, much
smaller than in our experiment.
One could conceivably speculate from Fig. 2 that J would
perhaps tend to a step function as N→‘ . The question is
whether for that limit the jamming probability is one below a
certain size Rc , while it is zero above Rc . In other words,
whether there is a finite value of R beyond which the orifice
would not jam at all for an infinitely large silo. We can assess
this by evaluating the size of the orifice for which the jam-
ming probability is one half—let us call it R0(N). This can
be easily obtained from the fit of J(R ,N) ~see Fig. 2!. The
particular fitting function for J chosen to obtain R0(N) is
almost irrelevant because the slope of J is quite large at R
5R0. In Fig. 3~a! we plot R0 versus N in semilogarithmic
scale. It is unclear whether R0 saturates. Several functions
can be fit reasonably well to R0(N) @see Fig. 3~a!#, some of
which saturate for N→‘ , while others do not. Values of N
much bigger than 105 are experimentally inaccessible in our
laboratory. Therefore we cannot conclude whether or not
there is a critical value for R0. The family of curves used to
fit J also depends on a , which is related to the slope of J at
FIG. 1. Histogram for the number of grains that flow between
two successive jammings. Data correspond to R53 ~the beads have
a diameter of 2 mm, and the circular orifice is 6 mm wide!. More
than 4000 events have been recorded. The line is a linear fit. Inset:
first return map, i.e., the avalanche size nt vs the next avalanche
size nt11. Note that t is just a correlative index ordering the se-
quence of avalanches.03030R0. The inverse of a is the width of the transition region
from J51 to J50. The values for 1/a are plotted in Fig.
3~b!. It decreases with N but it is not possible to ascertain if
it cancels out for a finite value of N, as would be the case for
a step function. Both parameters, R0 and a , are not related in
a simple way.
These results can be understood in the framework of a
one-dimensional percolation model. Let us call pR the prob-
ability that one grain gets past the outlet of size R without
blocking it. Assuming that these events are independent, an
FIG. 2. Jamming probability J as a function of R. Each series
corresponds to a fixed N, i.e., the probability of the orifice getting
jammed before N grains pass through it. Data taken for several bead
diameters are displayed. Successive series, from left to right, corre-
spond to N53, 10, 30, 100, 300, 1000, 3000, 10 000, and 30 000
beads. The fit corresponds to a hyperbolic tangent with two adjust-
able parameters: J(R ,N)5$12tanh@a(R2R0)#%/2. The two pa-
rameters of this family of curves, a and R0, are shown in Fig. 3.
Error bars are statistical.
FIG. 3. ~a! R0 vs N. Note the logarithmic scale on the horizontal
axis. ~b! 1/a as a function of N in semilogarithmic scale. Both R0
and a were obtained from the fits of Fig. 2.1-2
RAPID COMMUNICATIONS
JAMMING DURING THE DISCHARGE OF GRAINS FROM . . . PHYSICAL REVIEW E 68, 030301~R! ~2003!avalanche of size s would consist of one jamming event, s
nonjamming events, and another jamming event. The prob-
ability of this sequence is
nR~s !5~12pR!2pR
s
. ~2!
This also holds if one considers the grains falling in clusters,
or groups, rather than one at a time. In this case, pR is the
probability that a group of grains passes the outlet without
jamming, i.e., without forming an arch. If outgoing groups
have k grains in average, the probability of finding an ava-
lanche of s grains would be nR(s)5(12pR)2pRs/k . From Eq.
2 one obtains log(nR)52log(12pR)1s log(pR), so log(nR) has
a linear dependence on s, exactly as observed experimentally.
The value of pR can be determined from the histogram. The
exponential dependence of nR(s) on s precludes the interpre-
tation of these data in the framework of self-organized criti-
cality ~SOC! @11#, because there exists a characteristic ava-
lanche size manifested in the exponential distribution of
events. Conversely, experiments with rough grains in differ-
ent experimental conditions @12# and numerical simulations
in a two-dimensional silo @13# yield a power law
distribution—asignature of SOC.
Incidentally, the exponential distribution is a compelling
evidence that arch formation is an uncorrelated process.
From the temporal point of view, this is also revealed by the
autocorrelation function of the series of avalanches and by
the first return map ~see Fig. 1, inset!, which show no sign of
memory between consecutive avalanches.
From the avalanche size distribution given by Eq. ~2!, the
mean size avalanche S can be explicitly calculated, yielding
S5
(
s50
‘
s2nR~s !
(
s50
‘
snR~s !
5
11pR
12pR
. ~3!
The mean avalanche sizes obtained from our data are
shown along with the above expression in Fig. 4. Indeed the
histogram given by Eq. ~2! and the mean avalanche size ~3!
are those of a one-dimensional percolation model @14#,
which leads to a second order phase transition, but only at
pR51. This is marked by a divergence to infinity in the
mean avalanche size S, which is a measure of the suscepti-
bility of the system. Let us remark that the limit pR51 is
unattainable, so such a phase transition would never be ob-
served. In practice, however, pR could be made high enough
~if the radius of the outlet orifice is big enough!, so that
jamming would seldom be observed.
We can put forward the idea that the jamming transition
bears an analogy with the glass transition. By increasing R
one selects larger bridges only, so 1/R could be thought of as03030the ‘‘temperature.’’ In the glass transition, the characteristic
viscous time is certainly infinite for zero temperature; for a
finite temperature, it can be extremely large but a power law
divergence is not found @15#. In the same sense, the mean
avalanche size ~or the susceptibility, for that matter! is infi-
nite for 1/R→0, and could be very large for finite values, but
there would not exist a critical exponent for S.
In this paper, we have shown that jamming events during
the discharge of a silo of glass beads are uncorrelated and
show no evidence of self-organized criticality, in contrast
with other avalanche studies @12,13# which were performed
in different geometries. Moreover, we showed evidence that
suggests that the classical claim of the existence of a critical
outlet size beyond which no jamming occurs—i.e., J be-
comes a step function—could be invalid in a thermodynamic
sense. Comparison between experiment and recent numerical
simulations @16# could help to gather more understanding.
Finally, other interesting aspects that have not been consid-
ered, such as the influence of the roughness of the beads and
their packing fraction, also merit further investigation.
The Spanish Ministry of Science and Technology ~Project
BFM2002-00414!, the Government of Navarre, and the Uni-
versity of Navarre have supported this work. L.A.P. ac-
knowledges support from Fundacio´n Antorchas ~Argentina!
and the British Council. We thank Dr. G. C. Barker for his
helpful comments, and Dr. O. Dauchot for discussing his
ideas with us.
FIG. 4. Mean avalanche size S as a function of pR : experimen-
tal data ~points! and model ~solid line!. For the experimental data,
pR has been estimated as Nb /(Nb1N j), where Nb is the total num-
ber of fallen beads and N j the total number of jammings, for a given
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Jamming during the discharge of grains from a silo described as a percolating transition

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