Table Of ContentUniversality in the mean spatial shape of avalanches
Thimoth´ee Thiery and Pierre Le Doussal
CNRS-Laboratoire de Physique Th´eorique de l’Ecole Normale Sup´erieure, 24 rue Lhomond,75231 Cedex 05, Paris, France
Quantifyingtheuniversalityofavalancheobservablesbeyondcriticalexponentsisofcurrentgreat
interest in theory and experiments. Here, we improve the characterization of the spatio-temporal
process inside avalanches in the universality class of the depinning of elastic interfaces in random
media. Surprisingly, at variance with the temporal shape, the spatial shape of avalanches has not
yet been predicted. In part this is due to a lack of an analytically tractable definition: how should
the shapes be centered? Here we introduce such a definition, accessible in experiments, and study
the mean spatial shape of avalanches at fixed size centered around their starting point (seed). We
calculatetheassociateduniversalscalingfunctions,bothinamean-fieldmodelandbeyond. Notably,
6
they are predicted to exhibit a cusp singularity near the seed. The results are in good agreement
1
with a numerical simulation of an elastic line.
0
2
PACSnumbers: 05.40.-a,05.10.Cc,64.60.av,64.60.Ht
n
u
J Numerous slowly driven non-linear systems exhibit
6 motion which is not smooth in time but rather proceeds
discontinuously via jumps extending over a broad range 12.5
] of space and time scales. Developing predictive models
n
ofavalanchemotionandunderstandingtheiruniversality,
n 10.0
- or lack thereof, has emerged as an outstanding challenge
s of modern statistical physics [1]. In condensed matter
i 7.5
d recent developments have led to distinguish two broad
t. classes, depending on the importance of plastic defor-
a mations. In systems such as dislocated solids, metallic 5.0
m
glasses, granular media near jamming, plastic deforma-
- tions play a crucial role and despite recent progresses a
d 2.5
theoretical description is still under construction [2–5].
n
o In many other situations the description by an elastic
0
c interface driven in a disordered medium has proved rel-
[ evant [6–9]. Examples are domain walls in soft magnets
[10, 11], fluid contact lines on rough surfaces [12, 13],
3 FIG. 1: Density plot of the velocity field v(x,t) inside an
v strike-slip faults in geophysics [14], fractures in brittle avalancheofsizeS =1760inthemean-fieldmodel(Brownian
4 materials [15–18] or imbibition fronts [19]. This class Force Model) for d = 1 discretized with N = 128 points.
7 exhibits a dynamical phase transition - the so-called de- Time is given in machine-time unit. Line in red: backward
1 pinning transition - accompanied by collective avalanche path produced by the algorithm used to find the seed of the
0
motion. While the microscopic details of the dynam- avalanche(seetext). Inset: thespatialshapeofthisavalanche
0
ics are specific to each system, the large scale statistical when centered around its starting point.
.
1 properties of the avalanches are believed to be univer-
0
sal. The most studied quantities in this context are the
6
critical exponents characterizing the scale-free probabil- stand whether and how scaling and universality extend
1
ity distribution function (PDF) of avalanche total sizes to v(x,t).
:
v S, P(S) ∼ S−τS and durations T, P(T) ∼ T−τT. They Until now the focus was on the center of mass velocity
Xi are related to the roughness and dynamical exponents, ζ vcm(t) ∼ (cid:82) ddx v(x,t) and the mean temporal shape at
and z, defined at the depinning transition of the inter- fixed duration T, (cid:104)v (t)(cid:105) , where here (cid:104)(cid:105) denotes the
r cm T T
a face, using the scaling relations S ∼ (cid:96)d+ζ and T ∼ (cid:96)z statistical average over all avalanches of fixed duration
with (cid:96) the lateral extension of the avalanche. T. A scaling analysis suggests, through the sum rule
S = (cid:82) dtddxv(x,t), the existence of a scaling function
Recentimprovementsinexperimentaltechniquesallow
studies of avalanches with higher accuracy and to access fdtemp(t) such that (cid:104)vcm(t)(cid:105)T = Tγ−1fdtemp(t/T), where
new,finerquantities,withtheaimofdistinguishingmore γ = (d+ζ)/z. The universality of ftemp(t) was shown
d
efficiently the different universality classes. This notably theoretically and studied experimentally in [20–24]. The
includes the direct imaging of the spatio-temporal pro- beautiful parabola-shape predicted at mean field level,
cessofthevelocityfieldinsideanavalanchev(x,t)where ftemp(t)=t(1−t)(andγ =2),stimulatedtheexcitement
x denotes the internal coordinate of the (d-dimensional) around this observable.
interface and t is the time since the beginning of the Though very interesting, this observable does not con-
avalanche. A question of great interest is to under- tain information on the remarkable spatial structure of
2
avalanche processes (see for illustration Fig. 1). A char-
acterizationofeventhemeanspatialshapeofavalanches
in terms of a simple scaling function is presently lacking.
In this Letter we propose and calculate such a scaling
function. We consider the mean shape of avalanches at
fixed total size S, for which a scaling analysis suggests
(in real or in Fourier space (cid:104)S(q)(cid:105) =(cid:82) ddxeiqx(cid:104)S(x)(cid:105) )
S S
x
(cid:104)S(x)(cid:105)S =S1−d+dζfd( 1 ) ,
Sd+ζ FIG.2: Plotofthemean-fieldresultforthespace-timemean
(cid:104)S(q)(cid:105)S =Sf˜d(qSd+1ζ) , (1) v(4e)lo)caitnydpLrRofielelaisntsiciditeya(nrigahvta,lasneech(e7)i)n. d=1 for SR (left, see
(cid:82)
where S(x) = dtv(x,t) is the “local size” at x, f (x)
d
and f˜(q) are radial scaling functions (hence x and q
d
the scaling ansatz (1) holds and that the scaling func-
as arguments of the scaling functions always denote the
normofthevectorsxandq),normalizedas(cid:82) ddxf (x)= tions contain only one non-universal scale (cid:96)σ (which is
d
f˜(q =0)=1, since S =(cid:82) ddx S(x). Here the local size discussed in details below)
d
at x, S(x) is the local displacement of the interface be-
1 x
tween the beginning and the end of an avalanche at the f (x)= F ( ) , f˜(q)=F˜ ((cid:96) q) , (3)
d (cid:96)d d (cid:96) d d σ
pointx,whilethetotalsizeS istheareasweptbythein- σ σ
terface during the avalanche. Note that these definitions
are not complete: there are various ways of centering an where Fd and F˜d are fully universal and depend only
avalanche. Our proposal is to study the spatial struc- on the space dimension d and the universality class of
turebycentering the avalanches on their starting points. the model (i.e. range of elasticity and disorder). The
Hence in (1) (cid:104)(cid:105) denotes the statistical average over all precise model that is the starting point of our theoreti-
S
avalanches of fixed total size S and starting point x=0. cal analysis(for elasticinterfaceswith short-rangedelas-
We call this procedure the seed-centering which appears ticity) is given in (12). Our conclusions however apply
naturalwhenonethinksofhowanavalancheunfoldsfol- in much greater generality and the details of the model
lowing a branching process (see Fig. 1). Furthermore, it are unimportant (once the range of elasticity and disor-
permits analytical treatment and is thus appropriate to der correlation have been set). Indeed, since the scaling
compare theory and experiments. functions that we compute are universal and entirely de-
termined by the properties of the FRG fixed point for
We first calculate the above scaling functions at the
models in the quenched Edward-Wilkinson universality
level of mean-field. This requires to go beyond the sim-
class, any model in the same universality class leads to
plest mean-field toy model, the ABBM model [25, 26]
the same scaling functions. In the first part of the Let-
which only describes the center of mass motion of the
ter we thus focus on stating our results, and report the
interface. To this aim we consider the Brownian Force
discussion of the model and of the method to the second
Model(BFM),recentlyintroducedastherelevantmean-
part. Foragenericsystem,weexpectscalinganduniver-
field theory to describe spatial correlations [27–30]. For
sality to hold for avalanche of size S in a scaling regime
this model, we even compute the full mean velocity-field
S (cid:28) S (cid:28) S . Note that in (3), the space variable
min max
inside a seed-centered avalanche of given size S which in 1
x is measured in units of Sd+ζ (see (1)). In the original
general obeys the scaling form
units,theuniversalityintheavalancheshapeshouldhold
1
(cid:104)v(x,t)(cid:105)S =Sζd−+ζzF(t/Sd+zζ,x/Sd+1ζ) . (2) for both small and large x (compared to Sd1+ζ) as long
as x (cid:28) x (cid:28) x where x ∼ Sd+ζ . We
More generally, in this Letter we consider elastic in- min max min/max min/max
willstartbydiscussingtheexactresultsobtainedforthe
terfaces in the quenched Edward-Wilkinson universality
BFM (defined below, see (12)). These results are also
class with short ranged disorder. In this context, the
of interests for the SR disorder universality class as the
BFMisaccurateford≥d ,whered istheuppercritical
c c lowest order terms in the (cid:15) expansion (i.e. O((cid:15)0) terms)
dimension of the depinning transition, d = 4 for short-
c of the true universal scaling functions.
range (SR) elasticity and d = 2 for the most common
c
long-range (LR) elasticity. In lower dimensions d < dc, Results within mean-field: The BFM can be studied
correlations play an important role. To take them into analytically in any dimension d. Let us first consider
account and study this more difficult case, we use the the case of SR elasticity. The exponents are τ = 3/2,
S
Functional Renormalization Group (FRG) and calculate τ = z = 2 and ζ = 4−d. The scaling function in (2)
T
the scaling functions fd(x) and f˜d(q) perturbatively in admits a very simple expression:
(cid:15) = d − d, to one-loop, i.e. O((cid:15)) accuracy (see [31–
c
34] for background on FRG, and [27, 30, 35, 36] for its F(t,x)=2te−t2 1 e−x2/(4t) , (4)
application to the study of avalanches). We show that (4πt)d/2
3
which is plotted in Fig. 2. Here we use dimensionless ℱ(�)
units, the original units can be recovered using x→mx, ��� ℱ�=�(�) ℱ�=�(�)
t → t/τm and S → S/Sm where τm = η/m2 and Sm = ��� ��� ����
σ/m4 and the parameters η,m and σ are those in the ��� ��� ����
equationofmotionofthemodel(12). Timeintegrationof
(4)confirmsfortheBFMthegeneralscalinglaw(1)and ��� ��� ����
(3) with FdMF(x) = (cid:82)0+∞dtF(t,x) and (cid:96)σ = σ−1/4. The ��� ��� ����
result is simplest in Fourier space and does not depend
� � �
on the dimension: � � � � � � � � � � � � � � �
√
F˜dMF(q)=F˜MF(q)=1− π2q2eq44erfc(cid:18)q22(cid:19) , (5) lFeIvGel.f3o:r t(hceoluorniovnerlisnael)s.caAlinngalyfutinccatliornesFu˜lts aitnMFoFurainerdsOpa(c(cid:15)e)
d=1
(Left) and F in real space for d = 1 (Middle) and d = 2
d
whereerfc(z)= √2 (cid:82)+∞e−t2. Inrealspace, FMF(x)de- (Right)forSRelasticity. Blacklines: tree/mean-fieldresults.
π z d Dotted blue lines: universal corrections, δF˜ (q) (left, O((cid:15))
pends on the dimension and can be expressed using hy- 1
correction in Fourier space in d = 1), δF (x) (middle) and
pergeometric functions [37] with FMF(0) = 2−dπ1−d2 . δF (x) (right). Red-dashed lines: O((cid:15)) est1imate obtained by
d≤4 Γ(d)sin(πd) 2
BothF˜MF(q)andFMF (x)areplottedinblack4inFig4.3. simply adding the corrections to the MF value. Red lines:
d=1,2 improved O((cid:15)) estimate, which, through a re-exponentiation
AfundamentalpropertyofF˜MF(q)isthatitpossessesan procedure, takes properly into account the modification of
algebraic tail F˜MF(q) ∼ q−4 at large q, which generates exponents(10)and(11)(see[37]). Notethatthecuspatthe
a non-analytic term ∼|x|4−d in the small x expansion of origin of the avalanche shape at O((cid:15)) is not obvious in this
FMF(x)aroundtheorigin. Itsbehavioratlargexiseval- plot since the non-analyticity is rather small, but it can be
d
uated using a saddle-point on (4), leading to a stretched emphasized using a log-log scale (and measured in numerics,
see Fig. 5).
exponential decay with a d-independent exponent 4/3:
FdMF(x)(cid:39)x→∞ 2−d/√2π321−d2 x2−3de−3x44/3 . (6) FdMF(x)+δFd(x)+O((cid:15)2). From the ILT expression (9)
we obtain the following analytical properties of the O((cid:15))
corrections:
TheseresultseasilyextendtoLRelasticity,inwhichcase
1) Its large q expansion is δF˜ (q)(cid:39) (cid:15)8log(q)−γE−8,
z =1, ζ =2−d and the mean shape in Fourier space is d q(cid:29)1 9q4
obtained replacing q2 → q in (5). Let us also give here interpreted as a change in the tail exponent η˜d:
thespatiotemporalshape(2)fortheexperimentallymost
4(cid:15)
relevant case of d=1, with F˜d(q)(cid:39)q(cid:29)1 A˜dq−η˜d , η˜d =4− 9 +O((cid:15)2) , (10)
2t2e−t2
F(t,x)= . (7) with a universal prefactor A˜ = 2(1−(2+ γE)2(cid:15)). In
π(x2+t2) d 4 9
realspacethisimplies,intheexpansionofF (x)atsmall
d
x, a non-analytic term ∼ |x|ηd with ηd = η˜d−d = 59(cid:15) +
Results beyond mean-field for SR elasticity: For real- O((cid:15)2). RestoringtheS dependencefrom(1)thisleadsto
i(cid:15)st=ic4S−Rddiesxoprdaenrs,iotnh.e IBtFisMmiosstthcelesatralrytiinmgppleominetnitnedthine (cid:104)S(q)(cid:105)S ∼q→+∞ S1−dη˜+dζq−η˜d and the non-analytic part
Fourier space, since the mean-field result for F˜d(q) does (cid:104)S(x)(cid:105)nS.a ∼x→0 S1−dη˜+dζ|x|ηd. Note that in the BFM the
not depend on d: value η˜ = 4 = d+ζ implies that the large q behavior
d
of (cid:104)S(q)(cid:105) does not depend on S. This may seem nat-
S
F˜dSR(q)=F˜MF(q)+δF˜d(q)+O((cid:15)2) , (8) ural: in the BFM the small scales do not know about
the total size of the avalanche. A generalization of this
with δF˜d(q)=(cid:15)F˜(1)(q). Here F˜(1)(q)=(cid:82)C 2diµπeµH˜(µ,q) propertytotheSRdisordercasewouldsuggesttheguess
isobtainedasanInverseLaplaceTransform(ILT)µ→1: η˜guess =d+ζ. Ourresultexplicitlyshowsthatthisprop-
d
4√π(cid:20)2−3γ 1 4√µ erty fails with η˜d >d+ζ. Hence in the SR disorder case
H˜(µ,q)= E √ − √ (9) the large avalanches tend to be more smooth than small
9 8 q2+2 µ (q2+2 µ)2 avalanches. Notethatthepredictedvalueofη issmaller
d
(cid:18) q2+9õ (cid:32) q (cid:33) 3 (cid:19)(cid:21) than 2 in all physical dimension: this non-analytic term
× (cid:112) √ sinh−1 (cid:112) √ −1+ ln(4µ) should actually dominate the behavior of Fd(x) around 0
q q2+8 µ 2 2 µ 16
(andthusleadtoacuspsingularity). Apossibleinterpre-
tation of this cusp singularity is that around 0 the mean
where γ is Euler’s Gamma constant (see [37] for the
E shape of avalanches F (x) is dominated by avalanches
d
choice of C). We then define the correction to the mean
whose largest local size is at their seed. This could cor-
shape in real space as the d-dimensional Fourier trans-
respond to the fact that such avalanches occur as a con-
form δFd(x) = (cid:82) (2dπdq)de−iqxδF˜d(q). Hence, FdSR(x) = sequence of large fluctuations of the disorder that would
4
pinaspecificpointoftheinterfaceforalongtime. These of the first two moments of the avalanche size distri-
wouldresultinconfigurationsoftheinterfacewithasin- bution, S = (cid:104)S2(cid:105)/(2(cid:104)S(cid:105)), which can be measured in
m
gle point well behind the rest of the interface. The de- numerics and experiments. Here (cid:104)(cid:105) denotes the aver-
pinning of such a point would then trigger an avalanche age with respect to the avalanche size distribution. In
that is peaked around its seed [38]. cases where the numerical or experimental setup cor-
2)Atlargex,weobtainthatthestretchedexponential responds to our model (as in our simulations, see be-
decay exponent of the mean shape is modified from its low), this prediction for (cid:96) allows unambiguous compar-
σ
MF behavior δMF =4/3: ison between our results and the data. In cases where
(cid:96) cannot be predicted, some scale-independent features
σ
4 2
F (x)∼e−Cxδ , δ = + (cid:15)+O((cid:15)2) , (11) of the mean-shape still allow comparison with the ex-
d 3 27 periments. This includes the tail exponent of F˜ (q) in
d
√ (10), the small and large distance behavior of Fd(x) in
withauniversalprefactorC = 34+(7363−1)29(cid:15). Remark- (11), and the universal ratios c = (cid:82)ddx|x|2pFd(x) . In
ably, using ζ =(cid:15)/3+O((cid:15)2), this agrees to O((cid:15)) with the p ((cid:82)ddx|x|pFd(x))2
conjecture δ = d+ζ that we justify in [37]. d = 1, (c1,c2) (cid:39) (1.6944,3.8197) for the BFM while
d+ζ−1 (c ,c ) (cid:39) (1.641±0.001,3.43±0.02) for SR disorder to
Furthermore, the ILT expression (9) is easily calcu- 1 2
O((cid:15)).
lated numerically. The corrections δF˜ (q) and δF (x)
d d
areshowninFig.3,togetherwiththeresultingestimates
for the functions FSR(x) and F˜SR(q). S-dζ+ζ SxSd+1ζ S-dζ+ζ SxSd+1ζ
d d S S
0.5
Model and method: For SR elasticity, the equation of
motion for the interface position u(x,t) (denoted uxt) is 0.3 0.4
η∂ u =∇2u −m2(u −w )+F(u ,x) , (12) 0.2 0.3
t xt x xt xt t xt
0.2
where η is the friction, m is a mass cutoff which sup- 0.1
0.1
presses fluctuations beyond the length (cid:96) = 1/m and
m
m2w is the driving force. In the BFM, the random pin- x x
t 1 2 3 4 5 1 2 3 4 5
ning force F(u,x) is an independent Brownian motion
in u for each x with (F(u,x)−F(u(cid:48),x))2 = 2σ|u−u(cid:48)|. FIG. 4: (color online). Plain lines: rescaled mean shapes of
For the SR disorder universality class, the second cu- avalanches at fixed size S from the simulation of the BFM
mulant is F(u,x)F(u(cid:48),x(cid:48)) = δd(x−x(cid:48))∆ (u−u(cid:48)) with model (left) and of the model with SR disorder (right), in
0
∆ (u) a fast decaying function. Eq. (12) is ana- d=1, for S =10 (left only, blue), S =50 (right only, blue),
0
lyzed using the dynamical field theory and the FRG S =102(red),S =103(green),S =104(purple)andS =105
[37]. This leads to an expression for (cid:104)S(x)(cid:105) as an (leftonly,yellow). Dashedblacklines: theoreticalMFresult.
S
ILT: (cid:104)S(y)(cid:105) ∼LT−1 ((cid:104)u˜1 (cid:105) )/ρ(S) where ρ(S) is the Red dashed line: O((cid:15)) result. No fitting parameter.
S µ→S x=0 ξ
avalanche-size density (previously computed to O((cid:15)) ac-
curacy in [30, 36]) and u˜1 is the O(λ) term taken at
x=0 Numerical simulations. AconvenientchoiceofSRdis-
x=0ofthesolutionu˜ ofthefollowingdifferentialequa-
x order, amenabletoMarkovianevolution, istheGaussian
tion (here in dimensionless units):
disorder F(u,x) with ”Ornstein-Uhlenbeck” (OU) cor-
relator ∆ (u) = σδue−|u|/δu. It is defined by two cou-
−µ+λδ(x−y)+(u˜ )2+∇2u˜ −(1+ξ )u˜ =0 ,(13) 0
x x x x x pledequationsforthevelocityv ≡v(x,t)andtheforce
xt
√ F(x,t) (the first one being the time-derivative of (12)):
where ξ is a white-noise of order (cid:15) and (cid:104).(cid:105) denotes
x ξ
the average over it. For the BFM, the result is thus ob-
η∂ v =∇2v +m2(w˙ −v )+∂ F(x,t) ,
tainedsettingξ →0above. AtO((cid:15))fortheSRdisorder t xt xt t xt t
x √ v
universality class, it is thus sufficient to solve (13) per- ∂ F(x,t)= 2σv χ − xtF(x,t) , (14)
t xt xt δu
turbatively to second order in ξ . Here the fact that we
x
are looking at the local size of avalanches at x = y and with χ a centered Gaussian white noise χ χ =
xt xt x(cid:48)t(cid:48)
whose seed is centered at x=0 is encoded in (13) as the δd(x−x(cid:48))δ(t−t(cid:48)) and initial condition v =F(x,t=
xt=0
factthatwearecomputingthevalueatx=0(seedposi- 0)=0. Inthestationaryregime,thismodelisequivalent
tion)ofthesolutionof(13)withadeltasourceλδ(x−y) [39,40]toEq. (12)withu˙ =v andF(x,t)=F(u ,x)
xt xt xt
(local size position). The seed centering therefore allows and initial condition u = 0. When 1/δu = 0 this
xt=0
analyticaltreatmentherebecauseu˜x=0 onlycontainsthe model becomes equivalent to the BFM. We discretize
contribution of avalanches starting at 0 (see [37]). Using time in units dt and space with periodic boundary con-
anothertypeofspatialcenteringdoesnotallowasimilar ditions along x. To measure quasi-static avalanches,
simple treatment. we apply a succession of kicks of sizes δw: we impose
In our model (12), the non-universal scale (cid:96)σ in (3) vxt =(m2/η)δw at t=0+ (beginning of the avalanche),
is m−1S−1/(d+ζ) where S is defined from the ratio iterate (14) and wait for the interface to stop before
m m
5
applying a new kick [37]. To identify the seed of each discretized with N =L points and a mass m=10/L. In
avalanche, we record the velocity v(x,t) for the n =103 Fig.4weshowourresultsforthemean-shapefordifferent
t
first time-steps of the avalanche. We find the position valuesof S andcomparewithourtheoreticalpredictions
x (n ) of maximum velocity at t = n dt (or at the using the predicted value of (cid:96) (deduced from the mea-
max t nt t σ
end of the avalanche if it has stopped before), and then surement of S ), hence with no fitting parameter. The
m
successively identify at each time step t <t the posi- results for the BFM are excellent. For the model with
n nt
tionx (n)definedastheneighborof x (n+1)with SR disorder, the improvement brought by the O((cid:15)) cor-
max max
the largest velocity at time t . x (n=1) is identified rection is substantial. If one instead uses a measurement
n max
astheseedoftheavalanche. Thesizeofthekicksischo- of (cid:96) by e.g. setting the value of the shape at the ori-
σ
sensmallenoughsothattheprobabilitytotriggerseveral gin, the agreement with the SR disorder model is, to the
macroscopicandoverlappingavalanchesisnegligible(see naked eye, almost perfect. We also measure properties
[37] for details). independent of the value of (cid:96) : (i) in Fig.5 the small x
σ
and large q behaviors (ii) the universal ratios c . We ob-
˜ p
ℱd=1(q) tain (c1,c2) (cid:39) (1.699±0.003,3.83±0.05) for the BFM
ℱd=1(0)-ℱd=1(x) 0.100 and (c1,c2) (cid:39) (1.612±0.004,3.16±0.03) for the model
withSRdisorder(error-barsare3sigmaestimates). The
0.1 above predictions are in perfect agreement for the BFM,
0.010
and our O((cid:15)) corrections go in the right direction for the
0.01
SR disorder case.
0.001 0.001 To conclude, we introduced an original way of char-
acterizing the mean shape of an avalanche by centering
0.0001
around its seed. We obtained theoretical predictions for
x q
0.1 1 2 4 6 8 10 this observable and confronted them to numerical sim-
ulations. We also proposed a protocol to measure it.
FIG. 5: (color online). Left: (resp. Right:) Log-Log plot of We hope that this work stimulates measurements of this
F (0)−F (x) (resp. F˜ (q)) numerically obtained in quantity in numerical setups and imaging experiments.
d=1 d=1 d=1
the BFM model (blue) and in the model with SR disorder
(red). Dotted lines: guide lines for the BFM result x2 (left)
and 1/q4 (right). Dashed lines: x1.5 (left) and 1/q2.5 (right).
These results are consistent with (i) the exact result η˜ =
d=1 Acknowledgments
4 for the BFM (ii) η˜ (cid:39) 2.5 for the SR disorder model
d=1
(in between the guess η˜guess = d+ζ (cid:39) 2.25 and our O((cid:15))
d=1
prediction η˜d=1 (cid:39)8/3(cid:39)2.66). We thank Matthieu Wyart for interesting discussions.
WeacknowledgesupportfromPSLgrantANR-10-IDEX-
In dimension d = 1 we use a system of size L = 2048 0001-02-PSL.
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Supplemental Material
We give here a derivation of the results presented in the main text of the letter and details on the numerical
simulations.
Dynamical Field Theory Setting
Here we first introduce the formalism used to derive the results presented in the letter.
Equation of motion and dynamical action
Aswritteninthemaintext,weconsidertheequationofmotionfortheover-dampeddynamicofanelasticinterface
of internal dimension d in a quenched random force field and driven by a parabolic well of position w
xt
η∂ u =∇2u −m2(u −w )+F(u ,x) (15)
t xt x xt xt xt xt
where x∈Rd, t∈R, u ∈R (the space-time dependence is indicated by subscripts). The elastic-coefficient as been
xt
set to unity by a choice of units. In this formulation, the driving force of the parabolic well is f = m2(w −u ).
xt xt xt
The pinning force F(u,x) is chosen centered, Gaussian with second cumulant F(u,x)F(u(cid:48),x(cid:48))=δd(x−x(cid:48))∆ (u−u(cid:48))
0
(the overline denotes the average over disorder) where ∆ (u) is a short-ranged function. Higher cumulant can also
0
exist (i.e. non Gaussian force, and are taken into account in the FRG treatment). Note that here we have written
the case of short-ranged (SR) elasticity with an elastic term of the form ∇2u . Other elastic kernels can also be
x xt
considered, by changing
(cid:90)
∇2u −m2u → g−1u (16)
x xt xt xx(cid:48) x(cid:48)t
x(cid:48)
where g−1 is a translationally invariant (g−1 = g−1 ) elastic kernel. In particular, we will consider the following
xx(cid:48) xx(cid:48) x−x(cid:48)
kernel (here written in Fourier space) (g−1 =(cid:82) eiqxg−1, here and throughout the rest of the Supplemental Material
q x x
(cid:82) =(cid:82) ddx and (cid:82) =(cid:82) ddq )
x x∈Rd q q∈Rd (2π)d
(cid:112)
g−1 = µ2+q2 (17)
q
which is known to be relevant in the description of standard long-ranged (LR) elasticity. In this situation, the
√
parameter µ is related to the mass m as m= µ. In most of the following, we will deal with the SR elasticity case,
7
and explicitly mention when we consider the LR one. Introducing a response field u˜ , the generating function of the
xt
velocity field G[λxt]=e(cid:82)xtλxtu˙xt is computed using the dynamical action formalism for the velocity theory, that is for
the time-derivative of (15) [42, 43]:
(cid:90)
G[λxt]= D[u˜]D[u˙]e(cid:82)xtλxtu˙xt+m2(cid:82)xtu˜xtw˙xt−S0−Sdis
(cid:90) 1(cid:90)
S = u˜ (η∂ −∇2+m2)u˙ , S =− u˜ u˜ ∂ ∂ ∆ (u −u ) (18)
0 xt t xt dis 2 xt xt(cid:48) t t(cid:48) 0 xt xt(cid:48)
xt xtt(cid:48)
The renormalized field theory
As discussed in [30], in the limit of small m, and in the quasi-static limit w˙ = v → 0+, universal quantities
xt
associated to the motion inside a single avalanche can be computed in an expansion in (cid:15) = 4−d using an effective
action identical to (18) with the replacement ∆ (u) → ∆(u) = ∆(0)−σ|u|−4π2αm4−du2 +O((cid:15)2), where σ and
0
α=O((cid:15)) are renormalized quantities. σ is a non-universal parameter whose value is related to the two first moments
oftheavalanchesizedistributionthroughtheexactrelation2σ/m4 =(cid:104)S2(cid:105)/(cid:104)S(cid:105). Ontheotherhandαisdimensionless
and universal at the FRG fixed point with value α = −2(cid:15)/9+O((cid:15)2). In terms of the action, this replacement reads
S →Seff =S +δ S with
dis dis tree 1−loop
(cid:90) (cid:90)
S =−σ u˜2 u˙ , δ S =−4π2αm4−d u˜ u˙ u˜ u˙ (19)
tree xt xt 1−loop xt xt xt(cid:48) xt(cid:48)
xt xtt(cid:48)
At lowest order in (cid:15), the action is Seff = S . Using the renormalized value of σ, it gives the exact result
dis tree
for universal quantities in d > 4. In any dimension, this tree/mean-field theory also corresponds to an interface
slowly driven in a Brownian force landscape: for each x, F(u,x) is a Brownian in u independent of the others with
(F(u(cid:48),x)−F(u,x))2 =2σ|u(cid:48)−u|. This is the Brownian Force Model (BFM). The O((cid:15)) corrections around the BFM
are easily computed using the fact that δ S can also be taken into account by introducing a fictitious Gaussian
1−loop
centered white noise ξ with correlations (cid:104)ξ ξ (cid:105) =8π2αm4−dδd(x−x(cid:48)) through the identity
xt x x(cid:48) ξ
e−S0−Sdeifsf =(cid:104)e−(cid:82)xtu˜xt(η∂t−∇2+m2+ξx)u˙xt−Stree(cid:105)ξ (20)
where(cid:104)(cid:105) denotestheaverageoverξ. One-loopobservablesarethusrewrittenasaveragedtreeobservablesinatheory
ξ √
withspace-dependentmassm2 →m2+ξ . Sinceξ =O( (cid:15)),theeffectofξ canbetakenintoaccountpertubatively
x x x
up to order O(ξ2).
x
Avalanches observables
Avalanches in non-stationary driving
Letusfirstintroduceouravalancheobservablesinanon-stationarysetting. Wereferthereaderto[28–30]formore
details on this procedure. We first prepare the interface is in its quasi-static stationary state w˙ ∼v =0+, then turn
xt
the driving off: w˙ = 0 and finally wait for the interface to stop at some metastable position. Supposing we are in
xt
such a state at t=0, we apply to the interface a step in the driving force localized at x=t=0, f˙ =m2δwδ(x)δ(t)
xt
(local kick) and let it evolve. Information about the resulting motion of the interface is encoded in the generating
functional G[λxt]=e(cid:82)x,t>0λxtu˙xt. Remarkably, since the action (19) (written at one-loop in terms of ξx (20)) is linear
in u˙ , the evaluation of G[λ ] through the path-integral formalism simplifies. The integration on the velocity field
xt xt
u˙ leads to a delta functional and to the result:
xt
G[λxt]=(cid:104)em2δwu˜λx=,ξt=0(cid:105)ξ (21)
where u˜λ,ξ is the solution of the so-called instanton equation:
xt
∂ u˜ +∇2u˜ −(1+ξ )u˜ +u˜2 +λ =0 (22)
t xt xt x xt xt xt
here written in dimensionless units using the variables u˜ = m2uˆ˜ , x = xˆ/m, t = η tˆ, λ = m4λˆ , and omitting
x σ xˆ m2 xt σ xˆtˆ
the hats in what follows, to lighten notations. The boundary conditions is u˜ = 0 for t = +∞. Here we will only
xt
be interested in single avalanche, defined as the response of the interface to an infinitesimal step in the force. We
introduce the generating functional Z[λ ] as (expanding (21) in δw):
xt
e(cid:82)x,t>0λxtu˙xt −1=δwZ[λxt]+O(δw2)
Z[λ ]=m2(cid:104)u˜λ,ξ (cid:105) (23)
xt x=t=0 ξ
8
In the above expansion, the δw factor just accounts for the probability to trigger an avalanche at t = x = 0.
Introducing ρ [u˙ ], the density of velocity field u˙ inside an avalanche that starts at t=x=0, we write
t=x=0 xt tx
Z[λxt]=(cid:90) D[u˙](cid:16)e(cid:82)xtλxtu˙xt −1(cid:17)ρt=x=0[u˙xt] , (24)
where here this equation can actually be viewed as a definition of the density ρ . The fact that these definitions
t=x=0
indeedcorrespondtowhatisusuallymeantbyavalanchesinthequasi-staticlimitisdiscussedbelow. Thisformulation
is up to now completely general. Let us now focus on two types of sources: λ1 = (−µ+λδ(x−y)δ(t−s))θ(t) and
xt
λ2 = (−µ+λδ(x−y))θ(t) (θ(.) denotes the Heaviside theta function). In both cases, the µ variable probes the
xt (cid:82)
total size of the avalanche S = u˙ . In the first case, λ probes the local velocity at t = s and x = y during
x,t>0 xt
(cid:82)
the avalanche. In the second case, λ probes the local size of the avalanche at x = y, S = u˙ . We write the
y t>0 yt
associated generating function Z(1)[λ1 ] = Z(1)(µ,λ,y,s) and Z(2)[λ2 ] = Z(2)(µ,λ,y). These are obtained through
xt xt
the formula (23) by solving (22) which leads to
(cid:90) (cid:90)
Z(1)(µ,λ,y,s)= dSdu˙ e−µS+λu˙ysρ(1) (S,u˙ ) , Z(2)(µ,λ,y)= dSdS e−µS+λSyρ(2) (S,S ), (25)
ys t=x=0 ys y t=x=0 y
where ρ(1) (S,u˙ ) (resp. ρ(2) (S,u˙ )) is the joint density of total size S and velocity field u˙ (resp. of total
t=x=0 ys t=x=0 ys ys
size S and local size S ) for avalanches starting at t=x=0. In practice we will only be interested in computing the
y
mean velocity-field inside avalanche of total size S, (cid:104)u˙ (cid:105) (resp. the mean local size inside avalanche of total size S,
ys S
(cid:104)S (cid:105) ). These are computed as
y S
LT−1 ∂ Z(1)| LT−1 ∂ Z(2)| (cid:90) ∞
(cid:104)u˙ (cid:105) = µ→S λ λ=0 , (cid:104)S (cid:105) = µ→S λ λ=0 = ds(cid:104)u˙ (cid:105) (26)
ys S ρ(S)/Ld y S ρ(S)/Ld ys S
s=0
whereLT−1 denotestheInverseLaplaceTransform(ILT)operationLT−1 = 1 (cid:82) dµeµS withappropriatecontour
µ→S µ→S 2iπ C
of integration, and we have introduced ρ(S) the density of avalanches of total size S, previously computed up to one-
loop in [27, 30, 36] (ρ(S)/Ld =(cid:82) du˙ ρ(1) (S,u˙ )=(cid:82) dS ρ(2) (S,S ) is the density of avalanches of total size S
ys t=x=0 ys y t=x=0 y
starting at x=0). For the observables we are interested in, we will thus only need to solve (22) at first order in λ.
Link with the stationary driving
Letusnowpresentherehowtheprecedentapproachislinkedtoavalanchesoccurringinthequasi-staticstationary
state of the interface dynamic w˙ =v →0+. We introduce ρ the mean density of avalanche per unit of driving and
xt 0
p[u˙ ]the(functional)probabilityofvelocityfieldu˙ insideanavalanche. Atfirstorderin v, thegeneratingfunction
tx tx
G[λxt]=e(cid:82)xtλxtu˙xt can be written as
G[λxt]=(1−ρ0vT)+ρ0vT (cid:90) D[u˙]e(cid:82)xtλxtu˙xtp[u˙xt]+O(v2)=1+vT (cid:90) D[u˙](cid:16)e(cid:82)xtλxtu˙xt −1(cid:17)ρ[u˙xt]+O(v2) (27)
where we reintroduced ρ[u˙ ] = ρ p[u˙ ] the density of velocity field u˙ inside an avalanche. The equation (27) can
xt 0 tx tx
be seen as a definition of what is meant by avalanches in the quasi-static setting. The time scale T that appears in
(27) should be much larger than the time-scale of avalanche motion (to allow the avalanche to terminate) and much
smaller than the typical waiting time between avalanches. This only works if λ is also non-zero in a time window
xt
smaller than T: this ensures that the measurement made on the velocity-field is also inside a single-avalanche. On
the other hand, the small velocity expansion made directly on the action (18) and compared to (27) gives
G[λxt]=1+v(cid:104)m2(cid:90) u˜xt(cid:105)λxt −→(cid:90) D[u˙](cid:16)e(cid:82)xtλxtu˙xt −1(cid:17)ρ[u˙xt]=(cid:90) mT2(cid:104)u˜xt(cid:105)λxt, (28)
xt xt
where here the average (cid:104),(cid:105) refers to the average with respect to the dynamical action (18) with source λ . In the
λxt xt
right of (28), the integral over time and space originates from the fact that we have consider the effect of avalanches
starting at any point of the interface, and at any time in the time-window T. From a field-theory point of view, it is
then natural to interpret m2(cid:104)u˜ (cid:105) as the contribution from avalanches starting at t=x=0 (diagrams entering
x=t=0 λxt
into (cid:104)u˜ (cid:105) can only have a first non-zero u˙ at x = 0). Furthermore, this is supported by the non-stationary
x=t=0 λxt xt
setting in which this interpretation is immediate. In the quasi-static setting we can only a priori consider sources λ
xt
non-zero in time windows smaller than T to make sure that only one avalanche is taken into account. However, from
a practical point of view, when T >> τ where τ is the typical time scale of avalanches, both descriptions give
m m
exactly the same result as detailed in [30, 41].
9
Calculation in the BFM
Mean-velocity field inside an avalanche in the BFM
Here we present the calculations leading to the resuts Eq.(4) and Eq.(7) of the letter for the mean-velocity field
inside avalanche of total size S in the BFM (cid:104)u˙ (cid:105) (denoted v(y,s) in the main text with y =x and s=t). We have
ys S
to solve to first order in λ the instanton equation
∂ u˜ +∇2u˜ −u˜ +u˜2 −µ+λδ(x−y)δ(t−s)=0 . (29)
t xt xt xt xt
Notethathere,indimensionlessunits,timeandavalanchesizearemeasuredintermsofthenaturalunitsofavalanches
motion τ =η/m2 and S =σ/m4. The perturbative solution is u˜ =u˜0 +u˜1 λ+O(λ2) with
m m xt xt xt
u˜0x =Z(µ)= 21(cid:0)1−κ2(µ)(cid:1) , κ(µ)=(1+4µ)14 , u˜1qt =−(cid:90) t e(q2+κ2(µ))(t−t(cid:48))+iqyδ(t(cid:48)−s)dt(cid:48) (30)
t(cid:48)=+∞
here written in Fourier space for the O(λ) part: u˜1 =(cid:82) eiqxu˜1 . This immediately gives
qt x xt
(cid:90)
u˜1 = eiqy−(q2+κ2(µ))s (31)
t=x=0
q
Using the tree result for the avalanche size density ρMF(S) = √Ld e−S/4 we obtain the mean velocity field inside
2 πS3/2
a single avalanche using (26) as
<u˙ys >S=2√πS3/2eS/4LTµ−→1S(cid:90) eiqy−(q2+√1+4µ)s =2se−sS2 (cid:90) eiqy−q2s =2se−s2/S(4πs1)d/2e−y2/(4s) (32)
q q
In the notation of the main text, we thus obtain (4) that we recall here
1
<v(x,t)>S=S2−4dF(t/S1/2,x/S1/4) , F(t,x)=2te−t2(4πt)d/2e−x2/(4t) (33)
Extension to LR elasticity
Following the same computation, one obtains for the case of the BFM with long-ranged elasticity (with the kernel
(17))
(cid:90) √
u˜ = eiqy−( 1+q2−1+κ2(µ))s (34)
t=x=0
q
And thus
(cid:90) √
<u˙ys >S=2se−sS2 eiqy−( 1+q2−1)s (35)
q
Note that here, the spatio-temporal shape does not satisfy the expected scaling form (2), < u˙ > =
ys S
S2−d2−1F(s/S12,y/S12) for all S. This should not be surprising, it is known that the present theory describes scale-
invariant avalanches only for S (cid:28) S (here S = 1 in dimensionless units is the large scale cutoff S mentioned
m m max
in the main text, and note that in our theory the low-scale cutoff on the scaling regime S also mentioned in the
min
main text can effectively be taken to 0 for shape observables). The fact that the scaling hypothesis for the mean
velocity field holds ∀S in the BFM with short-ranged elasticity is the true surprise. Scaling in the long-ranged model
is restored at small S and here
(cid:90)
F(s,y)= lim Sd−21 <u˙ 1 1 >S=2se−s2 eiqy−|q|s (36)
S→0 S2y,S2s q
Evaluating this integral in dimension 1 immediately leads to the result (7).
The mean shape of avalanches in the BFM: results in Fourier space
We now derive the result Eq.(5) of the letter. Using (31), we immediately obtain the mean-shape of avalanche in
Fourier space in the BFM as
√
F˜MF(q)=(cid:90) ∞ 2se−s2−q2s =1− πq2eq44erfc(cid:18)q2(cid:19) (37)
2 2
s=0
10
i.e. the result (5) of the main text. Note that here avalanche sizes have been expressed in units of S = σ/m4 and
m
distances in units of 1/m. Hence the non-universal scale (cid:96) of the main text is indeed (cid:96) = 1S−1/4 =σ−1/4. Let us
σ σ m m
give here the large and small momenta behavior of F˜MF(q):
2 12 120 1
F˜MF(q)= − + +O( ) (38)
q(cid:29)1 q4 q8 q12 q16
√ √
F˜MF(q)= 1− πq2 + q4 − πq6 +O(cid:0)q8(cid:1) (39)
q(cid:28)1 2 2 8
Extension to LR elasticity
Wenowcomputethemeanshapeinrealspace. InparticularweobtaintheresultEq.(6)oftheletter. Theextension
of the precedent results to the case of LR elasticity is straightforward. As written in the main text and following
the formula (36), the mean-shape in Fourier space in the scaling regime for LR elasticity is simply obtained from the
precedent results by changing q2 →|q|:
√
F˜MF,LR(q)=F˜MF( q). (40)
In particular it now has an algebraic tail at large q with exponent 1/q2, F˜MF,LR(q)(cid:39) 2 .
q(cid:29)1 q2
The mean shape of avalanches in the BFM: results in real space
In real space, FMF(x) is most simply obtained by integration of (33):
d
FdMF(x)= (4π2)d/2 (cid:90) +∞dtt1−d/2e−t2−x42t (41)
0
This integral can be expressed either as the sum of three series:
FdMF(x)=π1−d2 (cid:88)∞ (−1)p2−4p[sinapdπx4p− 4acpo+s21dπx4p+2+ sinbpdπx4−d+4p] (42)
p=0 4 4 2
2−d 2−3
a = , b = (43)
p (2p)!Γ(cid:0)d +p(cid:1) p p!Γ(cid:0)−d +2p+3(cid:1)
4 2
or, equivalently, as the sum of three generalized hypergeometric functions (corresponding term by term to the series):
(cid:18)23−dcsc(cid:0)πd(cid:1) F (cid:16);1,d;−x4(cid:17) 21−dx2sec(cid:0)πd(cid:1) F (cid:16);3,d + 1;−x4(cid:17)
FdMF(x)= 81π1−d2 4 Γ0(cid:0)d2(cid:1) 2 4 64 − 4 Γ0(cid:0)d2+2(cid:1)2 4 2 64 (44)
4 4
x4−dcsc(cid:0)πd(cid:1) F (cid:16);3 − d,2− d;−x4(cid:17)(cid:19)
2 0 2 2 4 4 64
+
Γ(cid:0)3− d(cid:1)
2
The expressions (42) and (44) are adequate for d = 1,3. For d = 2,4 one must first take the limit d → 2,4 before
evaluating. This is easy to do with mathematica, and we give here only the two leading terms at small x:
FMF(x)= √1 − x2(−4log(x)−3γE +2+log(16)) +O(cid:0)x3(cid:1) (45)
2 4 π 16π
FMF(x)= −4log(x)−3γE +log(16) + x2 +O(cid:0)x3(cid:1) (46)
4 16π2 32π3/2
For d<4 the value at zero is finite:
FMF(0)= 2−dπ1−d2 (47)
d Γ(cid:0)d(cid:1)sin(cid:0)πd(cid:1)
4 4
FMF(0)≈0.345684 , FMF(0)≈0.141047 , FMF(0)≈0.0813891 (48)
1 2 4
and FMF(0) diverges as 1 as d → 4− (it has a minimum near d = 3.2). For d > 4 it diverges near zero as
d 4π2(cid:15)
FMF(x)(cid:39) π1−d2 csc(π2d)x4−d. The large distance behavior is easily obtained from the saddle-point method on (41). It
1 8Γ(3−d)
2
yields a stretched exponential decay at large x with exponent 4/3, independent of d:
FdMF(x)(cid:39) 2−d/√2π12−d2 x2−3de−3x44/3 (49)
3
