Distribution of joint local and total size and of extension for avalanches in the Brownian force model

1.2 MB·

by  Mathieu Delorme

DistributionofjointlocalandtotalsizeandofextensionforavalanchesintheBrownianforcemodel
Mathieu Delorme, Pierre Le Doussal and Kay Jo¨rg Wiese
CNRS-LaboratoiredePhysiqueThe´oriquedel’EcoleNormaleSupe´rieure,24rueLhomond,75005Paris,France
TheBrownianforcemodel(BFM)isamean-fieldmodelforthelocalvelocitiesduringavalanchesinelastic
interfaces of internal space dimension d, driven in a random medium. It is exactly solvable via a non-linear
differentialequation. Westudyavalanchesfollowingakick,i.e.astepinthedrivingforce. Wefirstrecallthe
calculationofthedistributionsoftheglobalsize(totalsweptarea)andofthelocaljumpsizeforanarbitrarykick
amplitude.Weextendthiscalculationtothejointdensityoflocalandglobalsizeswithinasingleavalanche,in
thelimitofaninfinitesimalkick.Whentheinterfaceisdrivenbyasinglepointwefindnewexponentsτ =5/3
0
6 andτ = 7/4,dependingonwhethertheforceorthedisplacementisimposed. Weshowthattheextensionof
1 a“singleavalanche”alongoneinternaldirection(i.e.thetotallengthind = 1)isfiniteandwecalculateits
0 distribution,followingeitheralocaloraglobalkick. InallcasesitexhibitsadivergenceP((cid:96))∼(cid:96)−3atsmall
2 (cid:96).Mostofourresultsaretestedinanumericalsimulationindimensiond=1.

n
a
I. INTRODUCTION linearinstantonequationandperformingLaplaceinversions,
J
whichisnotalwaysaneasytask.Globalavalancheproperties,
9
such as the probability distribution function (PDF) of global
1 In many physical systems the motion is not smooth, but
proceedsbyavalanches. Thisjerkymotioniscorrelatedover sizeS,ofduration,andofvelocityhavebeenobtainedforar-
] bitrary driving. Detailed space time properties however are
n a broad range of space and time scales. Examples are mag-
more difficult. In Ref. [13] a finite wave-vector observable
n netic interfaces, fluid contact lines, crack fronts in fracture,
was calculated, demonstrating an asymetry in the temporal
- strike-slip faults in geophysics and many more [1–3]. These
s shape. Although the distribution of local avalanche sizes S
i systemshavebeendescribedusingthemodelofanelasticin- r
d terfaceslowlydriveninarandommedium. Thismodelisim- has been obtained in some instances, this is not the case for
. thedistributionofthespatialextension(cid:96)ofanavalanche,i.e.
t portantforavalanches, bothconceptuallyandinapplications
a the range of points which move during an avalanche, an im-
[4–6]. The full model of an interface of internal dimension
m portantobservableaccessibleinexperiments. Notethateven
d, inpresenceofrealisticshort-rangeddisorderisdifficultto
- the fact that an avalanche has a finite extent, instead of an
d treatanalytically,andrequiresmethodssuchastheFunctional
exponentially decaying tail in its spatial extension is a non-
n RenormalizationGroup(FRG)[7–14].
trivialresult,whichuptonowwasonlyprovenforverylarge
o A simpler version of the model, the so-called Brownian
avalanchesintheBFM[19].
c
force model (BFM) introduced in [10–13] is very interest-
[
ing in several respects. First it is exactly solvable, and sev-

1 eralavalancheobservableshavebeencalculated,asdiscussed
v below. Second, it was shown [11, 13] to be the appropriate x
0
mean-fieldtheoryforthespace-timestatisticsofthevelocity
4 u
9 field in a single avalanche for d-dimensional interfaces close xt=0 u
4 to the depinning transition for d ≥ duc with duc = 4 for xt=∞
0 shortrangedelasticityandduc =2forlong-rangedelasticity.
. Remarkably,whenconsideringthedynamicsofthecenterof
1
massoftheinterface, itreproducestheresultsofthesimpler
0
6 ABBM model, a toy model for a single degree of freedom r S
1 (particle), introduced long ago on a phenomenological basis r
S
: todescribeBarkhausenexperiments(magneticnoise)[15,16] l
v
and much studied since [1, 12, 17]. Last but not least, the
i
X BFM is the starting point for a calculation of avalanche ob-
r servables beyond mean-field, using the FRG in a systematic
a
expansionind −d[11,13].
uc
ThekeypropertywhichmakestheBFM(andtheABBM)
model solvable is that the disorder is taken to be a Brown-
ian random force landscape. Since it can be shown that un-
der monotonous forward driving the interface always moves
forward(Middleton’stheorem[18]),theresultingequationof
u
motion for the velocity field is Markovian, and amenable to
exactmethods.
Despite being exactly solvable, the explicit calculation of
avalanche observables in the BFM requires solving a non- FIG.1.Anavalancheind=1.
2
Drivingprotocol Observable Exponent II. AVALANCHEOBSERVABLESOFTHEBFM
anyforcekick globalsizeS τ =3/2
A. TheBrownianForceModel
uniformforcekick localsizeS τ =4/3
0 0
uniformforcekick S atfixedS τ =2/3
0 0
Inthispaper,westudytheBrownianForceModel(BFM)in
localizedforcekick localsizeS τ =5/3
0 0 spacedimensiond,definedasthestochasticdifferentialequa-
localdisplacementimposed globalsizeS τ =7/4
tion(intheItosense):
anyforcekick extension(cid:96) κ=3
(cid:112)
η∂ u˙ =∇2u˙ + 2σu˙ ξ +m2(w˙ −u˙ ). (1)
t xt x xt xt xt xt xt
TABLEI.Summaryofsmall-scaleexponentsfordifferentdistribu-
tionsintheBrownian-ForceModel,dependingontheobservableand This equation models the overdamped time evolution, with
thedrivingprotocol. friction η, of the velocity field u˙ ≥ 0 of an interface with
xt
internalcoordinatex∈Rd;thespace-timedependenceisde-
notedbyindicesu˙(x,t) ≡ u˙ . Itisthesumofthreecontri-
xt
butions:
Theaimofthispaperistocalculatefurtherobservablesfor
• short-rangedelasticinteractions,
the BFM which contain information about local properties,
such as the joint density of global and local avalanches, and • stochastic contributions from a disordered medium,
thedistributionofextensions. Weconsidervariousprotocols, where ξ is a unit Gaussian white noise (both in x and
where the interface is either driven uniformly in space or t):
at a single point; in the latter case we identify new critical
ξ ξ =δd(x−x(cid:48))δ(t−t(cid:48)), (2)
exponents. We study avalanches following a kick, i.e. a step xt x(cid:48)t(cid:48)
inthedrivingforce.
• aconfiningquadraticpotentialofcurvaturem,centered
atw ,actingasadriving.
xt
The article is structured as follows: In section II we re-
Thedrivingvelocityischosenpositive,w˙ ≥ 0,anecessary
callthedefinitionoftheBFMandofthemainavalancheob- xt
condition for the model to be well defined, as it implies that
servables, together with the general method to obtain them
u˙ ≥0atallt>0ifu˙ ≥0.
fromtheinstantonequation. SectionIIIstartsbyrecallingthe xt xt=0
Equation(1),takenhereasadefinition,canalsobederived
calculationofthedistributionsoftheglobalsize(totalswept
fromtheequationofmotionofanelasticinterface,parameter-
area)S andofthelocaljumpsizeS ofanavalanche,foran
r izedbyapositionfield(displacementfield)u inaquenched
arbitrarykickamplitude. InSectionIIICweextendthiscal- xt
randomforcefieldF(u,x),
culationtothejointdensityρ(S ,S)oflocalandglobalsize
r
forsingleavalanches,i.e.inthelimitofaninfinitesimalkick. η∂ u =∇2u +F(u ,x)+m2(w −u ). (3)
t xt x xt xt xt xt
InSectionIVwestudythecaseofaninterfacedrivenatasin-
glepoint. Whentheforceatthispointisimposed, wefinda The random force field is a collection of independent one-
new exponent τ = 5/3 for the PDF of the local jump S at sidedBrownianmotionsintheudirectionwithcorrelator
0 0
that point. When the local displacement is imposed, we find
anewexponentτ = 7/4forthePDFoftheglobalsizeS. In F(u,x)F(u(cid:48),x(cid:48))=2σδd(x−x(cid:48))min(u,u(cid:48)). (4)
SectionVweshowthattheextension(cid:96)ofasingleavalanche
Taking the temporal derivative ∂ of Eq. (3), and assuming
alongoneinternaldirection(i.e.thetotallengthind = 1)is t
forward motion of the interface, one obtains Eq. (1) for the
finite;wecalculateitsdistribution,followingeitheralocalor
velocityvariable∂ u ≡u˙ (weuseindifferently∂ oradot
aglobalkick. InallcasesitexhibitsadivergenceP((cid:96))∼(cid:96)−3 t xt xt t
todenotetimederivatives). Thefactthattheequationforthe
atsmall(cid:96),withthesameprefactor.Alltheseexponentscanbe
velocityisMarkovianevenforaquencheddisorderisremark-
foundinTableI.Finally,inSectionVIwestudytheposition
ableandresultsfromthepropertiesoftheBrownianmotion.
oftheinterface,whichisanon-stationaryprocess.Weexplain
Details of the correspondence are given in [12, 13] where
how the Larkin and BFM roughness exponents emerge from
subtleaspectsofthepositiontheory,anditslinkstothemean-
the dynamics. Most of our results are tested in a numerical
field theory of realistic models of interfaces in short-ranged
simulationoftheequationofmotionind=1.
disorderviatheFunctionalRenormalisationGroup(FRG)are
discussed. In the last section of this paper we will mention
ThetechnicalpartsofthecalculationsarepresentedinAp-
somepropertiesofthepositiontheoryoftheBrownianforce
pendices A to J, together with general material about Airy,
model.
WeierstrassandEllipticfunctions. Ashortpresentationofthe
numericalmethodsisalsoincluded.
Finally note a complementary recent study of the BFM, B. Avalanchesobservablesandscaling
where the joint PDF of the local avalanche size at all points
wasobtained. Fromthat,thespatialshapeofanavalanchein The BFM (1) allows to study the statistics of avalanches
thelimitoflargeaspectratioS/(cid:96)4wasderived[19]. asthedynamicalresponseoftheinterfacetoachangeinthe
3
driving.Weconsidersolutionsof(1)asaresponsetoadriving x
oftheform
(cid:90) r1 Sr = 0
w˙ =δw δ(t) , δw ≥0 , δw =L−d δw >0. (5) 1
xt x x x
x
Theinitialconditionis
u˙ =0. (6)
xt=0
l Supp
Thissolutiondescribesanavalanchewhichstartsattimet=0
and ends when u˙ = 0 for all x. The time at which the
xt
avalancheends,alsocalledavalancheduration,wasstudiedin r S > 0
2 r
[20]anditsdistributiongiveninvarioussituations. 2
Within the description (3), i.e. in the position theory, it
u y
correspondstoaninterfacepinned,i.e.atrestinametastable
state at t < 0, it is submitted at t = 0 to a jump in the
total applied force m2δw. More precisely, the center of the
confining potential jumps at t = 0 from w (where it was FIG.2.Anavalancheind=2;thetransversedirectionisorthogonal
x
for t < 0) to w = w + δw (where it stays for all to the plane of the figure and the colored zone corresponds to the
xt=0+ x x
supportoftheavalanche.
t>0). Asaconsequence,theinterfacemovesforward(since
δw ≥ 0) up to a new metastable state. This is represented
x
in figure 1, where u is the initial metastable state and
xt=0 avalanchesofad-dimensionalinterfaceisdonewiththe
u isthenewmetastablestateattheendoftheavalanche.
xt=∞
definition
In fact, as we will see from the distribution of avalanche
durations, the new metastable state is reached almost surely (cid:90) ∞
inafinitetime. Fordetailsonthesemetastablestatesandthe (cid:96)= drθ(Sr >0), (9)
system’spreparationsee[12,13]. −∞
whereθistheHeavisidefunction. Notethatevenfora
Wenowdiscusstheavalancheobservablesatthecenterof d-dimensionalinterface,theextension(cid:96)isaunidimen-
this paper. They can be computed from the solution of (1) sionalobservable(cf. figure2).
given(5)and(6); theyarerepresentedinfigure1foramore
visualdefinitioninthecased=1. Notethat
(cid:92)
• Globalsizeoftheavalanche: S >0 ⇔ Supp {r}×Rd−1 (cid:54)=∅ (10)
r
(cid:90) (cid:90) ∞
S = u˙ dt. (7) whereSuppdenotesallthepointsoftheinterfacemovingdur-
tx
x∈Rd 0 inganavalanche(i.e.itssupport).
Weusenaturalscales(orunits)toswitchtodimensionless
This is the total area swept by the interface during the
expressions,bothforthe(localandglobal)avalanchesizeS ,
avalanche. m
asforthetimeτ expressedas
m
• Localsizeoftheavalanche: σ η
S = , τ = . (11)
(cid:90) (cid:90) ∞ m m4 m m2
S =m−1 u˙ dt. (8)
r tx
x∈{r}×Rd−1 0 The extension, a length in the internal direction of the inter-
face,isexpressedinunitsofm−1.Thisisequivalenttosetting
This is the size of the avalanche localized on a hy- m=σ =η =1. Allexpressionsbelow,exceptexplicitmen-
perplane, where one of the internal coordinates is r; tion,areexpressedintheseunits.
the factor m−1 allows to express S and Sr using the While Sm is the large-size cutoff for avalanches, there is
same units (see below). In d = 1 this yields Sr = genericallyalsoasmall-scalecutoff. AsintheBFMthedis-
m−1(cid:82)0∞u˙trdt, i.e.thetransversaljumpatthepointr orderisscale-invariant(bycontrastwithmorerealisticmod-
of the interface. For d > 1 the variable r is still one- elswithshort-rangedsmoothdisorder), itistheincrementin
dimensional, andSr thetotaldisplacementinahyper- thedrivingδw whichsetsthesmall-scalecutoffforthelocal
planeoftheinterface. andglobalsizeofavalanches. Theyscaleasmin(S) ∼ δw2
(globalsize)andmin(S )∼δw3(localsize).
r
• Avalancheextension:
Masslesslimit
Ford = 1,theextension(denoted(cid:96))ofanavalancheis
the lenght of the part of the interface which (strictly) There are cases of interest where the mass m → 0. This
moves during the avalanche. The generalisation to canbedefinedfromtheequationsofmotion(1)and(3)with
4
thechanges
103 Theoretical distribution Pδw(S)
m2(w −u )→f Numerical simulation
xt xt xt (12) 102
m2(w˙ −u˙ )→f˙ .
xt xt xt
101
Inthatcaseitisnaturaltoconsiderdrivingwithagivenforce 100
f , rather than by a parabola. The definition of the observ-
abxtles is the same except that the factor of m−1 is not added 10-1
inthedefinition(8). Tobringσandηtounity,wethendefine 10-2
timeinunitsofη anddisplacementsuinunitsofσ. There-
sults will still have an unfixed dimension of length. In some 10-3
of them, the system size L leads to dimensionless quantities 10-4
(it also acts as a cutoff for large sizes, although we will not
usethisexplicitly). 10-5
10-3 10-2 10-1 100 101
C. Generatingfunctionsandinstantonequation FIG.3. Greenhistogram: globalavalanche-sizedistributionfroma
directnumericalsimulationofadiscretizedversionofEq.(1)with
parameters : N = 1024,m = 0.01,df = m2δw = 1 and dt =
To compute the distribution of the observables presented
0.05.Redline:theoreticalresultgiveninEq.(16).Fordetailsabout
above, we use a result from [12, 13] which allows us to ex- thesimulationseeappendixH.
press the average over the disorder of generating functions
(Laplacetransforms)ofu˙ ,solutionof(1). Indimensionless
xt
units,thisresultreads III. DISTRIBUTIONOFAVALANCHESIZE
(cid:28) (cid:18)(cid:90) (cid:19)(cid:29)
G[λxt]= exp λxtu˙xt =e(cid:82)xtw˙xtu˜xt . (13) A. Globalsize
xt
Asdefinedin(7)theglobalsizeofanavalancheisthetotal
Here(cid:104)···(cid:105)denotestheaverageoverdisorder. u˜isasolution areasweptbytheinterface.ItsPDFwascalculatedin[11–13]
ofthedifferentialequation(calledinstantonequation) andreads,indimensionlessunits,
∂x2u˜+∂tu˜−u˜+u˜2 =−λxt. (14) Pδw(S)= 2√δπwˆS32e−(S−4δSwˆ)2 . (16)
Sinceavalancheobservablesthatweconsiderareintegralsof Hereδwˆ = Ldδw. Thisresultdoesnotdependonthespatial
the velocity field over all times (c.f. observable definitions formofthedriving(itcanbelocalized,uniform,oranything
above),thesourcesλ weneedin(13)aretimeindependent. inbetween),aslongasitisappliedasaforceontheinterface.
xt
Thusweonlyneedtosolvethespacedependent,buttimein- Driving by imposing a specific displacement at one point of
dependent,instantonequation theinterfaceisanotherinterestingcasethatleadstoadifferent
behavior,seeSectionIVB.
u˜(cid:48)(cid:48)−u˜ +u˜2 =−λ . (15) Wecantestthisagainstadirectnumericalsimulationofthe
x x x x
equation of motion (1). There is excellent agreement over 5
The prime denotes derivative w.r.t x. In the massless case decades,withnofittingparameter,seeFig.3.
discussed above, the term −u˜ is absent, all other terms are Avalancheshavethepropertyofinfinitedivisiblity,i.e.they
x
identical. areaLevyprocess. Thiscanbewrittenasanequalityindis-
tribution,i.e.forprobabilities,
The global avalanche size implies a uniform source in the
instanton equation: λx = λ, while the local size implies a P ∗P =d P . (17)
localizedsourceλ = λδ1(x). Toobtaininformationonthe δw1 δw2 δw1+δw2
x
extension of avalanches, we need to consider a source local- Itimpliesthatwecanextractfromtheprobabilitydistribution
ized at two different points in space, λ = λ δ(x − r ) + (16)thesingleavalanchedensityperunitδw thatwedenote
x 1 1
λ δ(x−r ). ρ(S)andwhichisdefinedas
2 2
This instanton approach, which derives from the Martin-
P (S) (cid:39) δwρ(S). (18)
δw
Siggia-Roseformulationof(1),allowsustocomputeexactly δwˆ(cid:28)1
disorder averaged observables for any form of driving, by
This avalanche density contains the same information as the
solving a “simple” ordinary differential equation, which de-
fulldistribution(16);itsexpressionis
pendsontheobservablewewanttocompute,i.e.onλ ,but
xt
not on the form of the driving w˙xt. For a derivation of (13) ρ(S)= √Ld e−S4 ∼S−τ . (19)
and(14)wereferto[12]. 2 πS32
5
Itisproportionaltothesystemvolumesinceavalanchesoccur
anywhere along the interface. It defines the avalanche expo- 106 Theoretical distribution Pδw(S0)
nentτ = 3 fortheBFM.DuetothedivergencewhenS →0 105 Numerical simulation
2
itisnotnormalizable(itisnotaPDF),butastheinterfacefol- 104
lows on average the confining parabola, it has the following 103
property
102
(cid:90) ∞ 101
dSSρ(S)=Ld. (20)
100
0
10-1
In this picture, typical, i.e. almost all avalanches are of van-
10-2
ishingsize,S ≈ 0,ormorepreciselyS ≤ δwˆ2,butmoments
10-3
ofavalanchesaredominatedbynon-typicallargeavalanches
(oforderS ). 10-4
m
10-5
10-7 10-6 10-5 10-4 10-3 10-2 10-1 100 101
B. Localsize
FIG.4. Greenhistogramm: Localavalanche-sizedistributionfrom
WenowinvestigatethedistributionoflocalsizeS asde-
r adirectnumericalsimulationofadiscretizedversionof(1)withpa-
finedinEq.(8). Wehavetospecifytheformofthekick;we rametersN = 1024,m = 0.01,df = m2δw = 1,anddt = 0.05.
startwithoneuniform(inx):δwx =δwforallx∈R. Inthis Redline: thetheoreticalresultgiveninEq.(21). Fordetailsabout
casethesystemistranslationnalyinvariant,andwecanchoose thesimulationseeappendixH.
r =0,asanylocalsizewillhavethesamedistribution.
The distribution of S is obtained by solving Eq. (15)
0
with the source λx = λδ(x), and then computing the C. Jointglobalandlocalsize
inverse Laplace transform with respect to λ of G(λ) =
exp(δw(cid:82) u˜λ), where u˜λ is the instanton solution (depend-
x Wenowextendtheseresultswithanewcalculationofthe
ingonλ). Thishasbeendonein[13];thefinalresultis
jointdensityoflocalandglobalsizes. ConsiderP (S ,S),
δw 0
P (S )= 2×313e6δwˆδwˆAi(cid:32)(cid:18) 3 (cid:19)13 (S +2δwˆ)(cid:33) tuhneifjoorimntkPiDckFδowf.loFcoarlasribzietrSar0yaδnwdgitlodboaelssnizoetaSd,mfoitllaowsiimngplae
δw 0 S43 S0 0 explicit form (see Appendix D). We thus again consider the
0 (21)
2Ld−1 (cid:18)2S (cid:19) “single avalanche” limit δw → 0. It defines the joint den-
(cid:39)δwˆ(cid:28)1 δw πS0 K13 √30 . csiatlycuρl(aSte,.SE0q),uviviaalePnδtwly(So0n,eSc)an(cid:39)coδnwsiρd(eSr0th,eSc)o,nwdhitiicohnwalepnroobw-
abilityP (S |S)ofthelocalsize,giventhattheglobalsize
Hereδwˆ =Ld−1δw,AiistheAiryfunction,andKtheBessel δw 0
isS. Inthelimitδw →0thesetwoobjectsarerelatedby
function. WeusethatAi(x)= 1(cid:112)xK (2x3/2)forx>0.
π 3 1/3 3
This distribution has again the property of infinite divisibil-
ρ(S ,S)
ity, whichisfarfromobviousonthefinalresultsbut, canbe P0+(S0|S)= ρ(0S) , (23)
checkednumerically.
The small-δw limit defines the density per unit δw of the
where ρ(S) is given in Eq. (19); the two factors of δw can-
localsizesofa“singleavalanche”,whichisgivenby
cel.ForsimplicitywediscusstheresultforP (S |S).While
0+ 0
2Ld−1 (cid:18)2S (cid:19) bothρ(S)andρ(S0,S)arenotprobabilities,i.e.theycannot
ρ(S )= K √0 benormalizedtoone,wewillshowthattheconditionalprob-
0 πS0 31√ 3 (22) abilityP0+(S0|S)iswell-defined,andnormalizedtounity.
(cid:39) Ld−1 63Γ(1/3) ∼S−τφ . AnaturaldecompositionofthisconditionalPDFis
S0(cid:28)1 πS 4/3 0
0 (cid:18) (cid:90) (cid:19)
P (S |S)=Pˆ (S |S)+δ(S ) 1− Pˆ (u|S) .
Itssmall-sizebehaviordefinesthelocalsizeexponentτ = 4 0+ 0 0+ 0 0 0+
φ 3 u>0
fortheBFM. (24)
Thedistribution(21),orthedensity(22),canbecompared The first term is the smooth part defined for S > 0 which
0
totheresultsofdirectnumericalsimulationsoftheBFM,and comes from the avalanches containing the point r = 0. The
theagreementisverygoodover7decades,withoutanyfitting second term arises from all avalanches which do not contain
parameter,c.f. Fig.4. thepointr = 0. Thistermcontainsasubstractionsothatthe
Another interesting property is that the tail of large local total probability is normalized to unity, (cid:82) P (S |S) = 1,
sizesbehavesasρ(S0) (cid:39)S0(cid:29)1 S0−3/2e−2S0/√3,i.e.withthe asitshouldbe. S0 0+ 0
samepower-lawexponentinthepre-exponentialfactorasthe Thesmoothpartiscalculatedusingtheinstanton-equation
globalsize. approach. ThedetailsaregivenAppendixD.Thefinalresult
6
Explicitly
105
√
104 P˜ (S |S)= 4 πe−32α3 (cid:104)αAi(cid:0)α2(cid:1)−Ai(cid:48)(cid:0)α2(cid:1)(cid:105), (31)
103 0+ 0 313Γ(cid:0)14(cid:1)S023S14
with α defined in Eq. (26). It is now normalized to unity,
102 (cid:82) P˜ (S |S) = 1. One sees that the typical local size
S0>0 0+ 0
101 scalesasS0 ∼ S3/4. Computingthefirstm√omentwefindits
conditionalaveragetobe(cid:104)S (cid:105) = π S3/4. ItsPDF
100 0 S,S0>0 3Γ(1/4)
hastwolimitingbehaviors,
1100--21 NNNThuuuemmmoeeererrriiitcccicaaaalll lddd diiisssistttrrrtiiirbbbibuuuutttiiitoooionnnn www (iiiδtttwhhh =SSS0>>>+000)...001011 P˜0+(S0|S)(cid:39)Γe(−541)2SSS30434√ forS0 (cid:29)S34 (32)
π
10-6 10-5 10-4 10-3 10-2 10-1 100 323Γ(31)Γ(54)S023S41 forS0 (cid:28)S34 .
FIG.5. Distributionofα,definedinEq.(26),fromnumericalsim-
The first one shows that the probability of avalanches which
ulations (N = 1024,m = 0.02,δw = 10,dt = 0.01). This is
comparedtothetheoreticalprediction(33). Keepingonlylarge-size are “peaked” at r = 0 decays very fast. The second shows
avalanches,thisconverges(withoutanyadjustableparameter)tothe an integrable divergence at small S0 with an exponent 2/3.
δw=0+result. Comparing, for instance, with the behavior of the local size
density (22), we see that conditioning on S yields a rather
differentbehaviorandexponent.
takesthescalingform
It is interesting to note that changing variables in Eq. (31)
Pˆ0+(S0|S)= L1 4×S023323e−23α3(cid:104)αAi(cid:0)α2(cid:1)−Ai(cid:48)(cid:0)α2(cid:1)(cid:105) (25) fromP˜S0(αto|Sα),d=efi√ne3dπein−(322α63),(cid:104)αgivAeis(cid:0)α2(cid:1)−Ai(cid:48)(cid:0)α2(cid:1)(cid:105), (33)
with 0+ Γ(cid:0)14(cid:1)α34
α:= 323S043 . (26) wcahlliychasisintodwoeisnndoetpreenqduainretoafnSy,caonndditthiuosnneiansgie.rFtiogtuerset5nusmhoewris-
S
the agreement of these predictions with numerical simula-
Thefactor1/Lisnaturalsinceonlyafractionoforder1/Lof tions,inthelimitoflargeS whichisequivalenttoδw = 0+
avalanchescontainsthepointr =0. Aswritten, thissmooth asusedinthetheoreticalderivation.
partisnotnormalized. Itsintegralisequaltotheprobabilityp
thatthepointS hasmoved(i.e.S >0)duringanavalanche,
0 0
forwhichwefind D. Scalingexponents
p:=(cid:90) ∞dS0Pˆ0+(S0|S)= SL14 3Γ√(cid:0)π14(cid:1) . (27) Let us now discuss the various exponents obtained until
0 now. Theyareconsistentwiththeusualscalingargumentsfor
Thescalingofthisprobabilitywithsizeshowsthatinasingle interfaces. Ifanavalanchehasanextensionoforder(cid:96)(inthe
avalanche only a finite portion of the interface is moving. If codirectionofthehyperplaneoverwhichthelocalsizeiscal-
weassumestatisticaltranslationalinvariancewededucethat culated), thetransversedisplacementscalesasu ∼ (cid:96)ζ. Here
theroughnessexponentζ fortheBFMwithSRelasticityis
p=(cid:104)(cid:96)(cid:105) /L, (28)
S
ζ =4−d. (34)
where(cid:96)istheextensiondefinedin(9),and(cid:104)(cid:96)(cid:105) itsmeanvalue BFM
S
conditionedtotheglobalsizeS. Hencewededucethat
The avalanche exponent for the global size follows the
3Γ(cid:0)1(cid:1) Narayan-Fisher(NF)prediction[8]
(cid:104)(cid:96)(cid:105)S = √π4 S14 . (29) 2 BFM 3
τ =2− −−→ . (35)
d+ζ 2
InthefollowingsectionswewillinfactcalculatethePDFof
theextension(cid:96).
The global size then scales as S ∼ (cid:96)d+ζ, since all d internal
Bydividingbyp,wecannowdefineagenuinenormalized
directions are equivalent, and the transverse response scales
PDFforS ,P˜ (S |S),conditionedtobothSandS >0,so
0 0+ 0 0 1
thatthedecomposition(24)becomes withtheroughnessexponentζ. Inturnthisgives(cid:96) ∼ Sd+ζ.
IntheBFMwithSRelasticitythisleadsto(cid:96)∼S1/4asfound
P (S |S)=pP˜ (S |S)+δ(S )(1−p). (30) above.
0+ 0 0+ 0 0
7
Similarly,thelocalsize,definedhereastheavalanchesize B. Imposeddisplacementatapoint
inside a dφ-dimensionel subspace, is S0 ∼ (cid:96)dφ+ζ , leading
to a generalized NF value τφ = 2− dφ2+ζ. In the BFM we We analyze the problem in the massless case. To impose
havefocusedonthecasedφ = d−1(i.e. thesubspaceisan the displacement at point x = 0 we replace in the equation
hyperplane), hence dφ +ζ = 3 and the local size exponent of motion (1) and (3), m2 → m2δ(x). Hence there is no
becomesτ =4/3. ItalsoimpliesS ∼(cid:96)3,henceS ∼S3/4 global mass, but a local one to drive the interface at a point.
φ 0 0
asfoundabove. Toimposethedisplacement,weconsiderthelimitm2 →∞.
Inthatlimitu =w ,andthelocalsizeoftheavalanche
x=0,t 0,t
S isequaltoδw .
0 0
IV. DRIVINGATAPOINT:AVALANCHESIZES WhilethelocalsizeS isfixedbythedriving,wecancalcu-
0
latethedistributionofglobalsizes. ItisobtainedinAppendix
Here we briefly study avalanche sizes for an interface EusinganinstantonequationwithaDiracmassterm. Itcan
driven only in a small region of space, e.g. at a point. There bemappedontothesameinstantonequationasstudiedforthe
aretwomaincases: jointPDFoflocalandglobalsizes. TheLaplace-transformof
the result for the PDF is given in Eq. (E6). Its small-driving
• the local force on the point is imposed, which in our
limit,i.e.thedensity,is
framework means to consider a local kick δw =
x √
δwδ(x). In the massless setting it amounts to use 3
f =δfδ(x), ρ(S)= ∼S−τloc.driv. (40)
x Γ(1/4)S7/4
• thedisplacementu ofonepointoftheinterfaceis
x=0,t withadistinctexponent
imposed.
7
As we now see this leads to different universality classes τloc.driv. = 4 . (41)
andexponents.
V. DISTRIBUTIONOFAVALANCHEEXTENSIONS
A. Imposedlocalforce
In this section we study the distribution of avalanche ex-
Consideranavalanchefollowingalocalkickatx = 0,i.e.
tensions. IntheBFMtheycanbecalculatedanalytically. We
δw =δw δ(x).
x 0 startbyrecallingstandardscalingarguments.
In the BFM the distribution of the global size of an
avalanche does not depend on whether the kick is local in
space or not. One still obtains [13] the global-size distribu- A. Scalingargumentsforthedistributionofextensions
(cid:82)
tionasgiveninEq.(16)withδwˆ = δw =δw .
x x 0
Thedistributionofthelocalsizeatthepointofthekick is
Asmentionedinthelastsection,weexpectthattheglobal
more interesting. The calculation is performed in Appendix
size S and the extension (cid:96) of avalanches are related by the
C2. Forsimplicitywerestricttod = 1,thegeneralcasecan
scalingrelation
beobtainedasabovebyinsertingfactorsofLd−1.Thefullre-
sultforthePDF,P (S ),isgivenin(C7)andisbulky.Inthe S ∼(cid:96)d+ζ (42)
δw0 0
limit δw → 0 it simplifies. Noting P (S ) (cid:39) δw ρ(S ),
0 δw0 0 0 0 intheregionofsmallavalanchesS (cid:28) S (indimensionfull
thecorrespondinglocal-sizedensitybecomes m
units). Fromthedefinitionoftheavalanche-sizeexponent
1 (cid:16) (cid:17)
ρ(S )=− Ai(cid:48) 31/3S2/3 . (36)
0 31/3S5/3 0 P(S)∼S−τ (43)
0
At small S , or equivalently in the massless limit at fixed andusingthechangeofvariablesP(S)dS =P((cid:96))d(cid:96)wefind
0
δf =m2δw ,itdivergesas
0 0 P((cid:96))∼(cid:96)−κ with κ=1+(τ −1)(d+ζ). (44)
S−5/3
ρ(S ) (cid:39) 0 ∼S−τ0,loc.driv. . (37) Usingthevalueforτ fromtheNFrelation(35)weobtain
0 S0(cid:28)1 32/3Γ(1/3) 0
2
Thisleadstoanewavalancheexponent τ =2− . (45)
d+ζ
5
τ0,loc.driv. = 3 . (38) ForSRelasticity,thisyields
Thecutoffatsmallsizeisgivenbythedriving,S ∼ δw3/2. κ=d+ζ−1. (46)
0 0
AtlargeS thePDFiscutbythescaleS ≡1anddecaysas
0 m ThepredictionfortheBFMisthatζ =4−dandτ =
BFM BFM
S−3/2 √ 3/2,whichleadsto
ρ(S ) (cid:39) √0 e−2S0/ 3. (39)
0 S0(cid:29)1 2 π31/4 κBFM =3 (47)
8
inalldimensions. Wewillnowcheckthispredictionfromthe slower). Thentheintegralin(49)isnotconvergent, equalto
scalingrelationswithexactcalculationsontheBFMmodelin −∞,whichimplies
d=1.
P (S =0,S =0)=0.
δwx r1 r2
B. Instantonequationfortwolocalsizes Thismeansthattheavalanchecontainssurelyatleastoneof
thepointsr orr .
1 2
(ii)-Ifδw vanishesfastenough,forexampleifδw islo-
If we want to investigate the joint distribution of two lo- x x
calisedawayfromx=±r (e.gδw =δwδ(x−y)forsome
cal sizes at points r and r , we need to solve the instanton 1,2 x
1 2 y ∈R\{r ,r }),theprobablity(49)becomesnontrivial.
equationwithtwolocalsources, 1 2
u˜(cid:48)(cid:48)−u˜ +u˜2 =−λ δ(x−r )−λ δ(x−r ). (48)
x x x 1 1 2 2
C. Avalancheextensionwithalocalkick
Thissolutionisdifficulttoobtainforgeneralvaluesofλ and
1
λ . Neverthelessλ →−∞isaninterestingsolvablelimit, Wenowconsideralocalkickcenteredatx = 0,i.e.w =
2 1,2 x
and sufficient to compute the extension distribution. Let us δw δ(x). Iffurther0<r <r ,then
0 1 2
denotebyu˜ (x)asolutionofEq.(48)withr <r inthis
limit λ1,2 →r1,r−2∞. It allows to express the pro1babil2ity that Pδw0(Sr1 =0,Sr2 =0)=Pδw0(Sr1 =0) . (54)
two local sizes in an avalanche following an arbitrary kick
δwxequal0, Thiscomesfromthefactthatintheintervalx∈[−∞,r1],the
solution u˜ (x) is identical to the instanton solution with
r1,r2
Pδwx(Sr1 =0,Sr2 =0) onlyoneinfinitesourceatr1(inotherword,itdoesnot“feel”
(cid:18)(cid:90) (cid:19) (49) thesourceinr2). Thisshowsforinstancethatthesupportof
=exp δw u˜ (x) . the avalanche is larger or equal than the set of points where
x r1,r2
x∈Rd thedrivingisnon-zero.
Thispropertyalsoshowsthatavalanchesareconnected,i.e.
We further restrict for simplicity to the massless case, i.e.
it is impossible to draw a plane where the interface did not
withoutthelineartermu˜ inEq.(48). Oneeasilyseesfrom
x
movebetweentwomovingpartsoftheinterface. Asafunc-
thelatterequationthatu˜ takesthescalingform
r1,r2 tionofr (whichisone-dimensional), thesupport(i.e.theset
(cid:18) (cid:19) of points where S > 0) of an avalanche following a local
1 2x−r −r r
u˜r1,r2(x)= (r −r )2 f 2(r −1 r )2 . (50) kick at x = 0 must be an interval. Since this interval con-
1 2 2 1 tains x = 0 we will write it as [−(cid:96) ,(cid:96) ] with (cid:96) > 0 and
1 2 1
(cid:96) >0. Thisallowstodefinetheextensionofanavalancheas
Thefunctionf(x)issolutionof 2
(cid:96)=(cid:96) +(cid:96) .
1 2
f(cid:48)(cid:48)(x)+f(x)2 =0. (51) TocalculatethejointPDFof(cid:96)1 and(cid:96)2 forakickatx = 0
we consider (49) with r = −x < 0 < r = x . Us-
1 1 2 2
Itdivergesatx = ±1,vanishesatx → ±∞andisnegative ingthepreviousresultsabouttheinstantonequationwithtwo
2
everywhere: f(x) ≤ 0. Asδw ≥ 0,thelatterisanecessary sources, and the fact that the interface model is translation-
x
conditions.t.theprobability(49)isboundedbyone. aly invariant, we obtain the joint cumulative distribution for
Intheintervalx∈]−1,1[,thescalingfunctionf(x)canbe (cid:96)1 >0and(cid:96)2 >0:
2 2
expressedintermsoftheWeierstrassP-function,see(I15),
F (x ,x ):=P ((cid:96) <x ,(cid:96) <x ) . (55)
δw0 1 2 δw0 1 1 2 2
(cid:18) 1 Γ(1/3)18(cid:19)
f(x)=−6P x+ 2;g2 =0;g3 = (2π)6 . (52) Itcan,foranyx1,x2 > 0,beexpressedintermsofthefunc-
tionf obtainedintheprecedingsection,
The value of g > 0 is consistent with the required period
3 F (x ,x )=P (S =0,S =0)
2Ω=1,see(I12). NotefromAppendixIthatthereisanother δw0 1 2 δw0 r1 r2
(cid:16) √ (cid:17)6 (cid:18)(cid:90) (cid:19)
solutionoftheform(52)withg = − 2 π Γ(1/3) < 0 =exp δw δ(x)u˜ (x)
3 413Γ(5/6) x 0 −x1,x2 . (56)
Fwohric|xh|v≥iol1a/te2s,tthheefcuonncdtiiotinonf(fx()xr)ea≤ds0, hence is discarded. =eδw0(x1+1x2)2f(cid:16)−2(xx21−+xx12)(cid:17)
6 Since the argument of f is within the interval ] − 1,1[ we
f(x)=− . (53) 2 2
(|x|−1/2)2 mustusetheexpression(52).
Fromthisonecanobtainseveralresults. Firsttakingx →
2
One property of the solution u˜r1,r2(x) is that it diverges as ∞oneobtainsthePDFof(cid:96)1alone,
∼(x−r )−2whenx≈r . Therearethustwocases:
1,2 1,2
or(vi)an-isthheesdtroiovinslgowδwlyxniesarnothni-szeprooinatt(oe.nge. oonfltyhelisneeaprolyintosr, Pδw((cid:96)1)= 12(cid:96)δ3we−δw(cid:96)621 . (57)
1
9
100
Small scale power law
101 Large scale asymptotic
80 Theoretical density distribution
Numerical simulation
100
60
10-1
40 10-2
20 10-3
10-4
(cid:45)0.4 (cid:45)0.2 0.2 0.4
10-5
FIG. 6. Decay amplitude R(k) as a function of the aspect ratio k 100 101
involvedinthejointdensityof(cid:96)andk,anddefinedinEqs.(60)and
(61).
FIG.7. Thedistributionofextensionsρ((cid:96)),asobtainedfromtheel-
lipticintegrals(F7)and(F8)(blackline).The(straight)greendotted
lineisthesmall-(cid:96)asymptotics(69),whereasthe(curved)reddotted
Asimilarresultholdsfor(cid:96) .
2 lineisthelarge-(cid:96)asymptotics(71).Thenumericalsimulation(green
In principle, one can now obtain the distribution of
histogram)iscutatsmallscaleduetodiscretizationeffects.
avalanchesextensions
(cid:90) ∞ (cid:90) ∞
P ((cid:96))= d(cid:96) d(cid:96) δ((cid:96)−(cid:96) −(cid:96) )∂ ∂ F ((cid:96) ,(cid:96) )
δw0 1 2 1 2 (cid:96)1 (cid:96)2 δw0 1 2 However,inthelimitofasmallδwwhichisalsothelimitofa
0 0
(58) “singleavalanche”,wecanrecovertheresultforthedistribu-
Ithasarathercomplicatedexpression. Letusdefineinaddi- tion of extensions. This is consistent with the idea that “sin-
tiontothetotallength,theaspectratio gleavalanches”donotdependonthewaytheyaretriggered.
These calculations allow to obtain the extension distribution
(cid:96) −(cid:96) 1 1
k = 1 2 , − <k < . (59) withoutsolvingexplicitlytheinstantonequation. (Theuseof
2((cid:96)1+(cid:96)2) 2 2 ellipticintegralsisinfactequivalenttotheuseofWeierstrass
functionsassolutionsoftheinstantonequation,c.f.Appendix
Using a change of variables, we obtain the joint density of
I).
totalextensionandaspectratiointhelimitδw →0,
0
Wenowfocusonthefollowingratioofgeneratingfunctions
1 R(k)
ρ((cid:96),k):= lim P ((cid:96),k)= , (60)
δw0→0δw0 δw0 (cid:96)3 (cid:104)eλ1s0+λ2sr(cid:105) (65)
(cid:18) 1(cid:19) (cid:104)eλ1s0(cid:105)(cid:104)eλ2sr(cid:105)
R(k):=6f(k)+6kf(cid:48)(k)+ k2− f(cid:48)(cid:48)(k). (61)
4
in the limit λ ,λ → −∞. It compares the probability that
1 2
The function f(x) was defined in Eq. (52). While the prob- bothlocalsizess := S ands := S aresimultaneously0
0 0 r r
ability as a function of (cid:96) decays as (cid:96)−3, the dependence totheproductofthetwoprobabiltiesthateachoneis0.
on the aspect ratio is more complicated and plotted in fig- Wecanexpressthisratio,usingtheinstanton-equationap-
ure 6. Note that in this expression f(k) can be replaced by proach,as
f (k):=f(k)+ 6 + 6 ,whichisaregularfunc-
reg (k+1)2 (k−1)2
tionofk,vanishingatk2=±1. In2tegrationoverkgives lim (cid:104)eλ1s0+λ2sr(cid:105)
B 2 λ1,λ2→−∞(cid:104)eλ1s0(cid:105)(cid:104)eλ2sr(cid:105) (66)
ρ((cid:96))= with (62) (cid:18)(cid:90) (cid:104) (cid:105)(cid:19)
(cid:96)3 =exp δw u˜ (x)−u˜ (x)−u˜ (x−r)
x r ∞ ∞
(cid:90) 1/2 √ x
B =24+2 f (k)=8 3π. (63)
reg whereu˜ := u˜ . Wedenotebyu˜ := u˜ ,
−1/2 r r1=0,r2=r ∞ r1=0,r2=∞
thesolutionoftheinstantonequationwithonesourceatr =0
andtheotheroneatinfinity. Itisthesameasthesolutionfor
D. Avalancheextensionwithauniformkick only one source in r = 0. The above expression is valid for
anyformofdrivingδw .
x
If a kick extends over the whole system, as e.g. a uniform Wecannowspecifytothecaseofsmallanduniformdriv-
kick δwx = δw, the avalanche will have almost surely an ingδwx =δw;thequantityofinterestisthen
infiniteextensionasthelocalsizeisnon-zeroeverywhere,
(cid:90)
P (S =0)=0 forany r ∈R. (64) Z(r)= u˜r(x)−u˜∞(x)−u˜∞(x−r). (67)
δw r x
10
Whileu˜ (x)isnotintegrable,Z(r)iswelldefinedasthetwo Here the disorder has the correlations of independent one-
r
u˜ terms cancel precisely the two non-integrable poles lo- sidedBrownianmotion
∞
catedatx=0andx=r.
Using that u˜ is a solution of Eq. (48), we can obtain an F(u,x)F(u(cid:48),x(cid:48))=2σδd(x−x(cid:48))min(u,u(cid:48)). (73)
r
expressionofZ(r)asanellipticintegral,seeAppendixFfor
details of the calculation. The formulas written there are for Considertheinitialconditionu = 0. Wecanthencom-
xt=0
themassivecase,butonlyallowtogetanimplicitexpression putethecorrelationfunctionoftheposition
for Z(r). They however allow us to extract the small-scale
behavioroftheavalanche-extensiondistribution(equivalently (cid:90) t
themasslesslimit). Forsmallr,thebehaviorofZ(r)is uxt = u˙xsds
0
√
4 3π
Z(r)(cid:39) . (68) for a uniform driving wt = vtθ(t), starting at t = 0. The
r calculationissketchedinAppendixJ.Indimensionlessunits
andinFourierspace,theresultreads
To understand the connection with the avalanche extension,
whaevneesepdectoifigeedttbhaeckkitcoktthoebinetuenrpifroetramti,otnheoftw(6o5a).veNraogwesthoafttwhee (cid:104)u u (cid:105)c =v(cid:34)2q2(t−1)+2t−5 − 4e−(q2+1)t
denominator are independant of r, and act only as a normal- qt −qt (q2+1)3 q2(q2+1)3
iiszatthieonprcoobnasbtailnittys.tThahtebnouthmseraatnodr,sintahreelsiimmiutlotafnλe1o,u2s→lye−q∞ual, 4e−t e−2(q2+1)t (cid:35)
0 r + + . (74)
to0. Derivingthistwotimesw.r.t. r (whichletsthedenom- q2(2q2+1) (q2+1)3(2q2+1)
inatorinvariant)givestheprobabilitythattheavalanchestart
in x = 0 and end in x = r. Dividing by δw and taking the At large times, the displacement correlations behave as
limit1 δw → 0 , we obtain the extension density in the limit (restoringunits)
ofasingleavalancheas
2σvt
1 (cid:104)u u (cid:105)c (cid:39) . (75)
ρ((cid:96))= δw∂r2eδwZ(r)|δw=0+,r=(cid:96) (69) qt −qt t→∞ (q2+m2)2
(cid:12)
=∂2 Z˜(r)(cid:12) (cid:39)B˜(cid:96)−3 when (cid:96)→0 The q dependence is similar to the so-called Larkin random-
r (cid:12)
r=(cid:96) force model [21], but with a time-dependent amplitude, i.e.
with the effective disorder is growing with time, which is natural
√ given the correlations (73). The correlation of the position
B˜ =8 3π. (70) thusremainsnon-stationaryatalltimes2.
FromEq.(75)oneobtainsthecorrelationsofthedisplace-
Werecoverherethe(cid:96)−3 divergenceforsmall(cid:96)oftheexten-
mentinrealspace,stillinthelarge-tlimit
sion of avalanches. Note that this calculation gives exactly
thesameprefactorasinEq.(62),whichconfirmsthatweare (cid:90) ddq 1
studyingthesameobject,namelya“singleavalanche”. (uxt−u0t)2(cid:39)2vt (2π)d(q2+m2)2(1−cosqx)
Finally, in themassive case, one canalso compute thetail
oftheextensiondistribution,resultinginto(seeAppendixF) ∼vt×x2ζL (76)
ρ((cid:96))(cid:39)72(cid:96)e−(cid:96)when(cid:96)→∞. (71) with ζL = (4−d)/2 the Larkin roughness exponent. Note
that the average displacement is u = vt − 1−e−m2t (see
xt m2
AppendixJ).HenceweseethattheBFMroughnessscaling
VI. NON-STATIONNARYDYNAMICSINTHEBFM u ∼ x4−d isdimensionallyconsistentwiththecorrelationat
largetimes,
The easiest way to construct a position theory equiva-
lent to the BFM model define in Eq. (1) is to consider the (u −u )2 (cid:39)2u x4−d. (77)
xt 0t xt
non-stationnary evolution of an elastic line in some specific
quencheddisorder, This result, ζ = 4−d = ε, is in agreement with the FRG
approch: the position theory of the BFM model is an exact
η∂ u =∇2u +F (u ,x)+m2(w −u ). (72)
t xt x xt xt xt xt fixedpointfortheflowequationoftheFRGwitharoughness
exponentζ =ε,asdiscussedin[10,12].
1Notethatthedenominatorscanthenbesettounity.Thereisnoambiguity
sincethecalculationcouldbeperformedfirstatfinitebutlargeλi, and
settingδwtozeroaftertakingthederivativeanddividingbyδw,andonly 2NotethattherearestationaryversionsoftheBFM,whichwewillnotdis-
attheendtakingthelimitofinfiniteλi. cusshere,seediscussionsine.g.[11–13].