Table Of ContentSuper-diffusion around the rigidity transition:
L´evy and the Lilliputians
F. Lechenault
CEA Saclay/SPCSI, Grp. Complex Systems & Fracture, F-91191 Gif-sur-Yvette, France
R. Candelier
LPS, Ecole Normale Sup´erieure, URA D 1306, 24 rue Lhomond 75005 Paris, France.
O. Dauchot
0 CEA Saclay/SPEC, URA2464, L’Orme des Merisiers, 91 191 Gif-sur-Yvette, France
1
0 J.-P. Bouchaud
2
Capital Fund Management, 6-8 Bd Haussmann, 75009 Paris, France
n
a G. Biroli
J
CEA Saclay/IPhT, UMR2306, L’Orme des Merisiers, 91 191 Gif-sur-Yvette, France
1 (Dated: January 13, 2010)
1
By analyzing the displacement statistics of an assembly of horizontally vibrated bidisperse fric-
] tionalgrainsinthevicinityofthejammingtransitionexperimentallystudiedbefore[1],weestablish
t
f that their superdiffusivemotion is a genuine L´evy flight,but with ‘jump’ size very small compared
o tothediameterofthegrains. Thevibrationinducesabroaddistributionofjumpsthatarerandom
s in time,butcorrelated in space, and that can beinterpretedas micro-crack eventsat all scales. As
.
t the volume fraction departs from the critical jamming density, this distribution is truncated at a
a
smallerandsmallerjumpsize,inducingacrossovertowardsstandarddiffusivemotionatlongtimes.
m
This interpretation contrasts with the idea of temporally persistent, spatially correlated currents
- and raises new issues regarding the analysis of thedynamics in terms of vibrational modes.
d
n
PACSnumbers:
o
c
[
INTRODUCTION i.e. faster than the familiar diffusive √τ law, while al-
1 ways remaining small compared to the diameter of the
v grains! It was also found that dynamical heterogeneities
5 As the volume fraction of hard grains is increased be- reach a maximum when τ = τ∗(φ), precisely when the
6 yonda certainpoint, the systemjams andis able to sus-
superdiffusive characterof the motion is strongest[2], in
7
tain mechanical stresses. This rigidity/jamming transi-
1 agreementwiththe generalboundonthedynamicalsus-
tion has recently been experimentally investigated [1, 2]
. ceptibility established in [3, 4]. This superdiffusion was
1 in an assembly of horizontally vibrated bi-disperse hard
interpretedasthe existenceoflargescaleconvectivecur-
0
disks, using a quench protocol that produces very dense
0 rents, which were tentatively associated to the extended
configurations, with packing fractions beyond the glass
1 soft modesthatappearwhenthesystemlosesoracquires
v: density φg, suchthat the structural relaxationtime τα is rigidity at φJ [7], and along which the system should
much larger than the experimental time scale. There is
i fail. A similar interpretation of the dynamics of partic-
X a density range φ <φ < φ where the strong vibration
g a ulate systems close to the glass or jamming transitions
r can still induce micro-rearrangements through collective
was promoted in [8]. The intuitive idea is that energy
a
contact slips that lead to partial exploration of the por-
barriers that the system has to cross to equilibrate are
tionofphasespacerestrictedtoaparticularfrozenstruc-
essentially in the directions (in phase space) defined by
ture. For φ <φ<φ 0.842, the system is frozen but
g J ≈ thesoftmodes. Withinthispicture,theharmonicmotion
not rigid; the system can only sustain an external stress
of the particles around a metastable configurations can
for φ larger than φ , which appears as a genuine critical
J be usedto guessthe structure of the anharmonic barrier
point where a dynamical correlation length and a cor-
crossing events. This is certainly reasonable when the
relation time simultaneously diverge, showing that the
barriersaresmall,sothatthe topofthe barrierisstillin
dynamics occurs by involving progressively more collec-
a quasi-harmonic regime.
tive rearrangements.
The primary aim of the present paper is to revisit the
One of the most surprising result of [1] was the dis- above interpretation for our system of hard-disks with
covery of a superdiffusive regime in the vicinity of φ . friction, in the light of a deeper analysis of the statistics
J
In a time range that diverges when φ φ , the typical ofdisplacements. Tooursurprise,wefindthatthedistri-
J
→
displacement of the grains grows as τν, with ν > 1/2, bution of rescaled displacements is, in the superdiffusive
2
FIG. 1: Left: Sketch of the experimental setup. Right: Pic- FIG. 2: Left: Typical trajectories of a grain during a 10000
ture of the grains together with an illustration of the relax- cycle acquisition. The packing fractions are φ = 0.8402
ation patternofthefieldqi at τ∗ (seetextbelowfordetails). (red), 0.8417 (orange), 0.8422 (yellow), 0.8430 (green). φJ
would stand between the orange and yellow trajectories.
Right: Fraction of broken neighborhood relations (links in
regime, accurately givenby an isotropic L´evy stable dis- the Vorono¨ı tessellation sense), as a function of time for all
packingfractions.
tribution L . Furthermore, the exponent µ characteriz-
µ
ing this L´evystabledistributionis equalto 1/ν,whereν
is the superdiffusion exponent, precisely what one would
described in detail in [1], and summarized in Fig. 1.
expect if the motion of the grains was a L´evy flight,
The stroboscopic motion of a set of 8500 brass cylin-
i.e. a sum of uncorrelated individual displacements with
ders (“grains”) is recorded by a digital video camera.
a power-law tail distribution of sizes such that the vari-
The cylinders have diameters d = 4 0.01mm and
1
ance of the distribution diverges [5, 6]. This divergence ±
d = 5 0.01mm and are laid out on a horizontal glass
2
only occurs at φ = φ , but is truncated away from the ±
J plate that harmonically oscillates in one direction at a
critical point, which explains why the motion reverts to
frequency of 10 Hz and with a peak-to-peak amplitude
normal diffusion at very large times. This finding shows
of 10 mm. The cell has width L 100 d , and its
small
that the rearrangementscorresponding to the maximum ≈
lengthcanbeadjustedbyalateralmobilewallcontrolled
of dynamical correlations cannot be thought of as large
by a µm accuracy translation stage, which allows us to
scalecurrentsthatremaincoherentoveralongtimescale
vary the packing fraction of the grains assembly by tiny
τ∗. Superdiffusionisnotinducedbylong-rangetemporal
amounts (δφ/φ 510−4). The position of the grains is
correlations of the velocity field, as was surmised in [1]. ∼
trackedwithinanaccuracyof2.10−3d . Inthefollowing,
1
Quite on the contrary, the total displacement on scale
lengths are measuredin d units andtime in cycle units.
τ∗ is made of a large number of temporally incoherent 1
jumps withabroaddistributionofjump sizes. Asis well
known,a L´evyflightis, overany time τ, dominatedby a STATISTICAL CHARACTERIZATION OF THE
handfulofparticularlylargeevents. Hence,theonlypre- MOTION OF GRAINS
dictability of the total displacement between t = 0 and
t = τ∗ on the basis of the motion in any early period,
Fig. 2-left shows the trajectories of a single grain dur-
say between t = 0 τ∗/10, comes from the presence of
ing the whole experimentalwindow for packingfractions
→
one of these large jumps in [0,τ∗/10]. The situation for
sitting on both sides of the transition φ . As expected,
J
frictional grains might therefore be quite different from
the typicaldisplacements are much largerat lowerpack-
what happens in supercooled liquids or frictionless hard
ing fractions. The so-called caging dynamics appears
spheres, when soft modes should indeed play an impor-
clearlyonthetrajectoriesobtainedatlargerpackingfrac-
tant role.
tions: thegrainsseemtostayconfinedaroundfixedposi-
Ontheotherhand,theveryfactthatindividualjumps tions for a longtime andthen hoparoundfromone cage
are broadly distributed in such a dense system where to another over longer time scales. The striking feature
grainscanhardlymovemeansthattheselargejumpsare of these curves is the remarkably small amplitude of the
necessarily collective, with a direct relation between the motion that we alluded to above. Even at the lowest
sizeofthejumpandthenumberofparticlesinvolvedthat packing fractions, the grains do not move much further
we discuss below. The physical connection between su- thanafractionofa diameteroverthe totalcourseofone
perdiffusionanddynamicalheterogeneitiesinthissystem experimental run. Another evidence of the absence of
becomes quite transparent. Note however that although structural relaxation is given by the low percentage of
wespeakabout“broaddistributions”and“largejumps”, neighboring links (in the Vorono¨ı sense) that are broken
one should bear in mind that all of this takes place on during the 104 cycles of an experimentalrun (see Fig. 2-
minuscule displacement scales, a few 10−2 of the grains right). Thisfractiongoesfromaround5%fortheloosest
diameter! OurL´evyflightsarethereforeLilliputianwalks packing fraction to around 0.2% for the densest. This
with a diverging second moment... confirmsthatfordensitiesaroundφ ,thesystemisdeep
J
The experimentalset-up and the quench protocolsare in the glass phase where structural relaxation is absent,
3
sented in Fig. 3, see [1]. Note the very small values of
σ (τ) d at all timescales, which is a further indica-
φ 1
≪
tion that the packing indeed remains in a given struc-
tural arrangement. At low packing fractions φ<φ and
J
at small τ, σ (τ) displays a sub-diffusive behavior. At
φ
longertime,diffusivemotionisrecovered. Asthepacking
fraction is increased, the typical lag at which this cross-
overoccursbecomeslargerand,atfirstsight,σ (τ) does
φ
notseemtoexhibitanyspecialfeatureforφ=φ (corre-
J
sponding to the bold line in Fig.3). Above φ , an inter-
J
mediate plateau appears before diffusion resumes. How-
ever, a closer inspection of σ (τ) reveals an intriguing
φ
behavior,thatappearsmoreclearlyonthelocallogarith-
mic slope ν = ∂lnσ (τ)/∂ln(τ) shown in Fig. 3. When
φ
ν = 1, the motion is diffusive, whereas at small times,
2
ν < 1 indicates sub-diffusive behavior. At intermedi-
2
ate packing fractions, instead of reaching 1 from below,
2
ν overshoots and reaches values 0.65 before reverting
≈
to 1 from above at long times. Physically, this means
2
that after the sub-diffusive regime commonly observed
in glassy systems, the particles become super-diffusive
FIG. 3: Top: Average displacement σφ(τ) ≡ h|rti|ii,t (left); atintermediatetimes beforeeventuallyenteringthelong
and local slope ν = ∂lnσ (τ)/∂lnτ (right) as functions of
φ
time diffusive regime. At higher packing fractions, this
the lag τ for all studied values of φ. The dotted dark line
corresponds to φJ. Bottom left: Characteristic time scales unusualintermediatesuperdiffusiondisappears: oneonly
extractedfromtheanalysisofν(τ)includingτ∗,whichwillbe observesthestandardcrossoverbetweenaplateauregime
introducedanddiscussedinthesectiondevotedtodynamical at early times and diffusion at long times. In order
heterogeneities. Bottom left: Pdf of the displacements ri, to characterize these different regimes, we define three
Pτ(r), normalized by σφ(τ), for different values of the lag characteristic times: τ (φ) as the lag at which ν(τ) first
1
time τ at φ=0.8446 (right). A Gaussian distribution (black
reaches1/2,correspondingtothebeginningofthesuper-
dotted curve) is shown for comparison.
diffusive regime, τ (φ) when ν(τ) reaches a maximum
sD
ν∗(φ) (peak of super-diffusive regime), and τ (φ) where
D
i.e. in a very different regime from the one studied in ν(τ)hasaninflectionpoint,beyondwhichthesystemap-
proaches the diffusive regime. These characteristic time
[9–11].
We will denote by Rt =(xt,yt) the vector position of scalesareplottedas afunctionofthe packingfractionin
i i i
the bottom-left panel of Fig. 3. Whereas τ does not ex-
grainiattimetinthecenter-of-massframe. Laggeddis- 1
placements are defined as rt(τ) = Rt+τ Rt for grain i hibit any special feature across φJ, both τsD and τD are
i i − i strongly peaked at φ . We have also shown on the same
betweentimetandt+τ. Wehavecheckedindetailsthat J
the statistics of rt along the x and the y axes are identi- graphthe time scale τ∗ where dynamicalheterogeneities
i
are strongest (see [1] and below). As noted in the intro-
cal,ormorepreciselythatthemotionisisotropic,inspite
duction,τ∗ andτ areveryclosetooneanotherandwe
of the strongly anisotropic nature of the external drive. sD
will identify these two time scales in the following.
Thisinitselfisanontrivialobservation,thatshowsthat
therandomstructureofthepackingisenoughtoconvert Wenowturnto theprobabilitydistributionofthe dis-
a directional large scale forcing into an isotropic noise placements rit(τ). We have represented on the bottom-
on small scales. From now on, we will thus focus on right panel of Fig. 3 the distributions accumulated for
the totaldisplacementrt = rt . We measuretypicaldis- all grains and instants for the packing fractions closest
placementsforadensityiφan|di|lagτ asthemeanabsolute to φJ and several values of lag-time τ. The horizontal
displacement, defined as: axis has been normalized by the root mean square dis-
placements, and a unit Gaussian is also plotted for com-
σ (τ) rt(τ) (1) parison. We find that the tails of the distributions are
φ ≡ | i | i,t
muchfatterthantheGaussian,unveilingtheexistenceof
(cid:10) (cid:11)
where the time average is performedover 10,000cy- extremely large displacements (compared to typical val-
t
h·i
cles. We have chosen to estimate σ (τ) by the mean ues)inthe regionoftime lagscorrespondingtosuperdif-
φ
absolute value deviationinsteadof the rootmeansquare fusion. As we will discuss below, these fat tails tend
displacementbecause–asweshallsee–thelaterdiverges to disappear when one leaves the superdiffusion regime,
when approaching the transition. i.e. when φ φc and/or lnτ/τ∗ become large.
| − | | |
Theevolutionofσ (τ)withthepackingfractionispre- Before characterizing more precisely these probability
φ
4
distributions, we would like to mention that we have re-
moved “rattling events” from our analysis, i.e. events
where particles make an anomalously large back and
forthmotion,ofamplitudelargerthan(say)0.1d . Some
1
oftheseeventsareduetothesamegrainrattlingduringa
largefractionoftheexperimentalrun,whileotherevents
arelocalizedintimeforagivengrain,andcouldactually
well be part of the extreme tail of the distribution seen
in Fig. 3. Our point here is that the observed fat-tails
arenotanartifactduetoafew“loose”grains–eachand
everyoneofthe grainsseemto contributeatone pointor
another in its history to these fat tails.
Aneatwaytocharacterizetheprobabilitydistribution
of the displacements is to study the generating function
of the squared displacements:
F(λ,τ)≡he−λrit2(τ)ii,t = ∞ (−kλ!)khrit2k(τ)i. (2)
k=0
X
As a benchmark, one can easily compute (λ,τ) in the
F
case of isotropic Gaussian diffusion. One obtains:
FIG. 4: Top: Fτ(x) as a function of −log10(x) for τ =
∗
1 τ = 373 (dots, left panel) and 1/Fτ(x)−1 as a function of
(λ,τ) = ˜ (x)= , (3) x for different values of τ (dots, right panel), from 1 (blue)
F G FG 1+x to 104 (red). For both plots: φ = φJ and the solid line
correspondstotheGaussian fit1/(1+x). Bottom-left: Fitof
where x = 2σ (τ)2λ is the scaled variable. When com- ∗
φ Fτ(x)forφ≈φJ andτ ≈τ usingaL´evydistributionmodel
puting (λ,τ) for the empirical data, one finds, as ex- F(λ,τ) withindexµ=1.6,andcomparisonwithaStudent
F L
pected,systematicdiscrepancieswiththe (λ,τ) . The distributionmodelF(λ,τ) with thesamesmallxbehavior.
F G L
rescaling of the different curves when plotted as a func- Bottom-right: Fittingparameter ε=1−µ/2as afunctionof
tion of x, on the other hand, works very well, as can be φ and τ.
inferred from the scaling of the probability distributions
themselves.
the fit suggests that the distribution of displacements is
The most important finding is that the small x be-
indeed an isotropic L´evy stable distribution L of index
havior of the empirical (λ,τ) is singular: as shown in µ
F µ.
the top-right panel of Fig. 4, 1/ (λ,τ) 1 behaves as
F − Wehavecheckedthatotherdistributionfunctionswith
xα when x 1, with α 0.80. It is easy to show
≪ ≈ power-lawtailsachieveamuchpoorerfittothedata. For
that such a singular behavior is tantamount to the ex-
example, a Student distribution of displacements (which
istence of a power-law tail in the distribution of r(τ),
cannot be obtained as the sum of independent jumps,
P (r) r−1−µ with α = min(1,µ/2). The value of
τ ∼r→∞ contrarily to the L´evy distribution) leads to:
α therefore corresponds to µ 1.6 < 2, which means
≈
that the variance of the distribution of r is formally di- (λ,τ) = ˜ (x) (5)
vergent, or at least dominated by a large physical cut- F S FS
∞ e−z
off rmax. This suggests to fit (λ,τ) over the whole x =1 (C′x)µ/2 dz ,
regime by the appropriate LapFlace transform of a L´evy − Z0 (z+C′x)µ/2
stabledistributionL ,thatreadsforthepresentisotropic
µ where µ < 2 and C′ is another numerical constant. Al-
two-dimensional problem [12] :
though (λ,τ) has the same small x singularity and
F S
∞ the same large x behavior as (λ,τ) , it turns out not
F(λ,τ)L =F˜L(x)= dze−z−C(xz)µ/2 (4) to be possible to adjust the paFrameteLr C′ in such a way
Z0 to adjust simultaneously the small and large x behavior
where µ<2 and C is a numericalconstant. It is easy to of the empirical (λ,τ).
checkthatforsmallx, ˜ (x) 1 CΓ(1+µ)xµ/2. Our What is mostFsignificant, however,is that the value of
FL ≈ − 2
main result is that this function is a very good fit to the the L´evy index µ corresponding to the best fit turns out
empirical data corresponding to φ φ and τ τ∗, for tobeveryclosetotheoneexpectedinthecaseofaL´evy
J
≈ ≈
all values of x, see the bottom-left panel in Fig. 4. The flight where the diffusion exponent ν is given by 1/µ,
optimal value of µ is slightly smaller, µ 1.6, than the since we find ν 0.65 (see Fig. 3) whereas 1/µ 0.625.
≈ ≈ ≈
oneobtainedbyafitofthesmallxregion. Thequalityof Fig. 4 bottom-right gives the value of the fitting pa-
5
rameter ε=1 µ/2 in the plane φ,τ, obtained from the
−
small x behavior of (λ,τ), corresponding to large dis-
F
placements. We see that the effective values of ε (resp.
µ)become closerto 0(resp. 2), correspondingtonormal
diffusion, as when φ φ and/or lnτ/τ∗ increase. We
c
| − | | |
believe that this corresponds to a so-called “truncated”
L´evyflight[13],withacut-offvalueinthedistributionof
elementary jump size that becomes smaller and smaller
asonedepartsfromthe criticalpoint,ratherthanacon-
tinuouslyvaryingexponentµ. Inotherwords,aplausible
scenario that explains the behavior of ε as a function of
φ,τ is that the tail of the elementary jump size distribu-
tion is given by:
FIG. 5: Correlation of the instantaneous (τ = 1) velocity
P1(r)∼ [rr01(+φµ)]µ, (6) vtheecitro′rasm|phrlitit.urdti+ets′ihir,tit|.r(it+bltu′iei,tci(rrceldess)qvuearrseus)satshefucnocrtrioelnastioofnthoef
lag t, for φ≈φJ.
wherethe exponentµ 1.6isindependentofφ, whereas
≈
the typical scale of the jumps r (φ) and the cut-off
0 DYNAMICAL CORRELATIONS
r (φ) both depend on φ. It is reasonable that r (φ)
max max
only diverges at φ = φ , while r (φ) has a regular, de-
J 0
The above analysis focused on single grain statistics
creasingbehaviorasafunctionofφ. Infact,theparame-
and established that the displacement of a grain has
terC appearingintheL´evydistributionaboveisdirectly
proportional to rµ. a L´evy stable distribution in the superdiffusive regime.
0
This means that the motion of each grain can be de-
Assuming that the jumps are independent, the distri-
composedinasuccessionofpower-lawdistributedjumps.
bution of displacements P (r) on a time scale τ is given
τ Butif“large”jumpscanoccurinsuchajammedsystem,
by the τ-th convolution of P (r), which converges to a
1 this necessarily implies that these jumps are correlated
L´evy distribution of order µ when r = and pre-
max ∞ in space. In order to characterize these dynamical cor-
dicts r τν with ν =1/µ (see e.g. [6]). For finite r ,
∝ max relations, we have introduced in [1] the following local
P will be very close to L in the intermediate regime
τ µ correlationfunction:
1 τ τ , where τ (r /r )µ, before crossing
D D max 0
≪ ≪ ∼
over to a diffusive regime at very long times. If r (φ) rt2(τ)
max qt(a,τ) exp i , (7)
behaves as φ φJ −ζ, the diffusion time τD should di- i ≡ − 2a2
| − |
verge as φ φ −ζµ when approaching φ .
J J
| − | whereaisavariablelengthscaleoverwhichweprobethe
In order to check directly whether the jumps are in- motion. Essentially, qt(a,τ)=0 if within the time lag τ
i
deed independent, we have measured the correlation of theparticlehasmovedmorethana,andqt(a,τ)=1oth-
instantaneous (τ = 1) velocities hrti.rit+t′ii,t. As shown erwise: qit(a,τ)isamobilityindicator. Tihe averageover
in Fig. 5, this correlation function indeed decays ex- alliandtdefinesafunctionakintotheself-intermediate
tremelyfastwiththelagt′,excludingthatsuperdiffusion scattering function in liquids:
could be due to long-range correlations in the displace-
ments. However, we discovered that the amplitude of Q¯(a,τ) 1 qt(a,τ) . (8)
the displacements, proportional to the parameter r , re- ≡hN i it
0
i
veal long-range correlations, decaying approximately as X
lnt′. This suggests that some slow evolution takes Notethatitiscloselyrelatedtothepreviouslyintroduced
−place during an experimental run, that affects the value generating function through Q¯(a,τ) = λ= 1 ,τ ,
F 2a2
of r over time. A natural mechanism is through the which we have characterizedin the previous section.
0
(cid:0) (cid:1)
fluctuations ofthe localdensity φ (see below), that feed- The dynamical (four-point) correlation function is de-
back on r . The approximately logarithmic decay of the fined as the spatial correlationof the q field:
0
correlations is interesting in itself and characteristic of
multi-time scale glassy relaxation. These long-ranged G4(R,a,τ)= hδqit(a,τ)δqjt(a,τ)i t;|R~i−R~j|=R (9)
correlationsslowdownbutfortunately do notjeopardize
(cid:12)
the convergence towards a L´evy stable distribution.[17] where δqt(a,τ) = qt(a,τ) Q¯(a,τ(cid:12)) and is plotted
i i −
Therefore, the above interpretation in terms of a jump for three packing fractions on fig. (6)-left. Its sum
dominated superdiffusion, instead of persistent currents, over all R’s defines the so-called dynamical susceptibil-
is warranted. ityχ (a,τ). Throughsimple manipulations,itiseasyto
4
6
and φ ). This immediately leads to a relation between
g
thetypicalsizeofthejumpsaftertimeτ andtherequired
scale of the cooperative motion:
r(τ)
ξ (τ) τν, τ τ∗ (11)
4
∼ c ∝ ≤
This very simple argument predicts that ξ should be
4
102 103 times larger than typical displacements,
∼ −
whichis indeedthe case(seeFig.6). Furthermore,using
τ∗ τ one finds r(τ∗) r , and therefore, using
FIG. 6: Left: Four-point correlator G∗4(R) = the∝trunDcated L´evy flight m∝odmelaxabove, a power-law di-
∗ ∗
G4(R;σφ(τ ),τ ) as a function of R for φ = 0.8402 (red), vergence of ξ (φ) as φ φ −ζ. This divergence might
0.8417 (black), 0.8426 (blue). Right: Dynamical correlation 4 | − J|
change if c strongly depends on the distance φ φ .
as a function of the packing fraction; (Inset: Rescaling of | − J|
∗ ∗ Turning now to higher cumulants of the distribution
log[G4(R)/G4(0)] as a function of pR/ξ4 for 8 densities of Q(a∗,τ∗), one can show that they are related to the
around φJ). From [1].
space-integralofhigherordercorrelationfunctionsofthe
dynamicalactivity. For example, the skewness ς of Q is
6
show thatχ4(a,τ) is relatedto the varianceof Q as (see 1/N2 times the space integral of the 6-point correlation
e.g. [4] for a review): function, defined as:
χ4(a,τ)= N1 δqit(a,τ) 2 . (10) G6(R,R′,a∗,τ∗)= hδqit∗δqjt∗δqkt∗i t;|R~i−R~j|=R;|R~i−R~k|(=1R2′),
* ! + (cid:12)
Xi t where δqt∗ is a shorthand notatio(cid:12)n for δqt(a∗,τ∗).
i i
We haveshownin[1] thatχ4(a,τ) has,for agivenφ, an Similar relations hold for higher moments. The sim-
absolute maximum χ∗4 = χ4(a∗,τ∗), which as expected plest scenario is that all these higher order correlation
sits onthe line correspondingto Q¯(a,τ) 1/2,i.e. such functions are governed by the same dynamical correla-
≈
thathalfoftheparticleshavemovedbymorethana. The tion length ξ , extracted from G (R,a,τ). This is what
4 4
amplitude ofthis maximum,whichcanbe interpretedas happensinthe vicinityofstandardphasetransitions,for
a number of dynamically correlated grains, grows as φ example. If this is the case, and provided that χ∗ ξ2,
4 ∝ 4
approached φJ, indicating that the jump motion of the one can show that the following scaling relations should
grainsbecomesmoreandmorecollectiveasoneentersthe hold:
superdiffusive L´evy regime, as anticipated above. Note
inparticularthatτ∗ behavesinthesamewayasτsD,the ς ς χ∗4, κ κ χ∗4 (13)
6 6,c 8 c,8
time at which the superdiffusion exponent ν reaches its ∼ N ∼ N
r
maximum. We furthermorefound [1]that the four-point
whereς ,κ arerespectivelytheskewnessandkurtosisof
correlation G∗(R) G (R,a∗,τ∗) is a scaling function 6 8
of R/ξ (see4inset≡of F4ig. 6-right), where ξ (φ) is the 1/N iqi (relatedtothe6-and8-pointconnectedcorre-
4 4 lation functions), and ς ,κ the corresponding values
dynamical correlation length such that χ∗ ξ2. P c,6 c,8
4 ∝ 4 at the critical point, i.e. for systems of size N smaller
All these results were reported in [1] and are re-
than the correlation volume χ∗ ξ2. If these scaling
called here for completeness. The behavior of χ∗(φ) 4 ∝ 4
4 relations are valid, the determination of ς ,κ using
c,6 c,8
was furthermore shown in [2] to be well accounted for
the above equations should give the same values for any
by the following upper bound, derived in [3, 4]: χ
(∂Q¯/∂φ)2 φ2 , where φ2 is the variance of the lo4ca≥l φ close to φJ. Unfortunately, our statistics is not suffi-
h ic h ic cient to make definitive statements, althoughthe data is
density fluctuations. Here, we want to present further
indeed compatible with such scalings. The notable fea-
speculations, firston the relationbetween the size of the
ture is that both ς 1 and κ 5 are found to
c,6 c,8
‘micro-jumps’anddynamiccorrelations,andthenonthe ≃ − ≃ −
be negative, although the error bar on both quantities
higher moments of the distribution of Q(a,τ).
is large. Interestingly, if we assume that the displace-
By conservation of the number of particles, the local
ments are perfectly correlated within a correlation blob
change of density δφ is related to the divergence of the
of size ξ , the L´evy flight model with µ=1.6 makes the
displacement field by: δφ/φ = ~ ~r. If we invoke a 4
∇ · following predictions:
kind a Reynolds dilatancy criterion whereby the local
density must fall below some threshold for the system ς 0.12; κ 1.37, (14)
c,6 c,8
to move, the displacement field must be correlated over ≈− ≈−
some length ξ such that ~ ~r r/ξ c, where c is a in qualitative agreement with our data (note however
4 4
∇· ∼ ≈
smallnumber,possibly dependentonφ,andofthe order that there is an unknown proportionality factor in
of 10−3 10−2 (i.e. the relative difference between φ eq.(13). AGaussiandiffusionmodel, ontheotherhand,
J
−
7
predicts ς = 0 and κ = 6/5, corresponding to a model system. We thank him in particular for insisting
c,6 c,8
−
uniform distribution of Q in [0,1]. A kurtosis smaller on the importance of measuring velocity correlations in
than 6/5 means that the distribution of Q tends to be our system.
−
spikedaround0and1. Finally,asφincreasesbeyondφ ,
J
the negative skewness of Q markedly increases. This is
a sign that the dynamics becomes more and more inter-
[1] F. Lechenault, O. Dauchot, G. Biroli, and J.-P.
mittent, “activated”, with a few rare events decorrelat- Bouchaud, Europhys. Lett. 83, 46003 (2008).
ing the systemcompletely, while mosteventsdecorrelate
[2] F. Lechenault, O. Dauchot, G. Biroli, and J.-P.
onlyweakly. SincewefixQ¯(a∗,τ∗)=1/2,thisleadstoa Bouchaud, Europhys. Lett. 83, 46002 (2008).
divergingnegativeskewnessinthelimitwheretheproba- [3] L. Berthier, G. Biroli, J.-P. Bouchaud, L. Cipelletti, D.
bilityofraredecorrelatingeventstendstozero. Theidea El Masri, D. L’Hote, F. Ladieu and M. Pierno, Science
of characterizingthe skewnessof Q might actually be an 310, 1797 (2005)
[4] L. Berthier, G. Biroli, J.-P. Bouchaud, W. Kob, K.
interesting tool to characterize the strength of activated
Miyazaki and D.Reichman,J. Chem. Phys. 126 184503
events in other glassy systems.
(2007).
[5] G. Zaslavsky, U. Frisch, M. Shlesinger (Edts.), L´evy
Flights and Related Topics in Physics, Lecture Notes in
CONCLUSION Physics 450, Springer, New York(1995).
[6] J.-P.Bouchaud,A.Georges,Anomalous diffusionindis-
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ical applications, Phys. Rep. 195, 127 (1990).
sive motion of our frictional grains in the vicinity of the
[7] M.Wyart,S.R.NagelandT.A.Witten,Europhys.Lett.,
jamming transition appears to be a genuine L´evy flight,
72, 486 (2005); M. Wyart,L. E. Silbert, S. R.Nagel, T.
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arXiv:0907.0222
are random in time, but correlated in space, and that
[9] G. Marty,O.Dauchot Subdiffusion and Cage Effect in a
can be interpreted as micro-crack events on all scales.
Sheared Granular Material, Phys. Rev. Lett. 94, 015701
As the volume fraction departs from the critical jam-
(2005). O.Dauchot,G.Marty,G.Biroli Dynamical Het-
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picture severely undermines the usefulness of harmonic Nature Physics 3, 260 (2007).
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tem (although this conclusion might of course not carry
[12] G. Samorodnitsky, M. S. Taqqu, Stable Non-Gaussian
over to frictionless grains, or to thermal systems). The
Random Processes, Chapman & Hall, New York,1994.
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lyze them in the present system, is discussed in [15]. [14] J.-F.Muzy,J.Delour,E.Bacry,Eur. Phys. J. B17,537
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[15] C. Brito, O. Dauchot, G. Biroli, J.-P. Bouchaud, com-
length ξ and the size of the jumps, and the structure
4
panion paper.
of higher order dynamical cumulants. The idea of using
[16] C. Heussinger, L. Berthier, J.-L. Barrat, Superdiffu-
the 6-point skewness as a quantitative measure of the
sive, heterogeneous, and collective particle motion near
importance of activated events in the dynamics of glassy the jammingtransition inathermal disordered materials,
systems seems to us worth pursuing further. arXiv:1001.0914
[17] Technically, however, these logarithmic correlations sug-
gestthatoursuperdiffusionprocessisamultifractalL´evy
Note added and acknowledgment: We want to thank
flight, by analogy with the multifractal random walk in-
L. Berthier for interesting discussions about a preprint
troduced in [14]. We will not dwell on this interesting
of his and collaborators [16] (that appeared in the very
new process, that would deserve a theoretical study of
last stages of the present work), and where effects simi- its own.
lar to those discussed here are observednumerically in a
