Comment on Macrospopic Equation for the Roughness of Growing Interfaces in Quenched Disorde

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Commenton”MacrospopicEquationfor
theRoughnessofGrowingInterfacesin
QuenchedDisorder”

InarecentLetter[1]BraunsteinandBucetaintro-
duceda’macroscopic’equationforthetimeevolutionof
thewidthofinterfacesbelongingtothedirectedperco-
lationdepinning(DPD)universalityclass[2].Fromnu-
mericalsimulationsoftheDPDmodel,theyinferredan
ansatz(Eq.(1)inRef.[1])forthetimederivativeofthe
interfacewidth(calledDSIWinRef.[1])atthedepin-
ningtransition.BraunsteinandBucetafoundthattheir
formulafittedthenumericaldataatthedepinningtra-
sition,forqc=0.539andβ=0.63,withtheappropriate
electionofsomearbitraryconstants.
Herewearguethat,contrarytowhatitisclaimedin
Ref.[1],BraunsteinandBuceta’sformuladoesnotde-
scribethe’macroscopic’behaviouroftheinterface.The
formulaproposedinRef[1]fortheDSIWisanapproxi-
mationtotheveryshorttimesregime(whenlessthanone
layerhasbeencompleted),whichisnotsignificantforthe
descriptionofthesurfacedynamicsatlargescales.We
obtainanaliticallytheshorttimebehaviouroftheDPD
model,whichisvalidforanyqandexplainstheapper-
anceofanexponentialtermintheformulaofRef.[1]for
theDSIW.
LetusconsidertheDPDmodelinasystemofsize
Landadensityqofblockedcells(p=1−qden-
sityoffreecells).Weareinterestedintheveryshort
timesregimewhenthefirstmonolayerstillhasnotbeen
completed,i.e.thenumberofgrowthattemptsNis
N≪L(thiscorrespondstotimest=N/L≪1).Inthis
regime,theprobabilityofhavingacolumniwithheight
hi>min(hi−1,hi+1)+2isnegligibleandthecolumnsare
growingalmostindependently.Thegrowthatthisearly
stagecanbeseenasarandomdeposition(RD)process
[3]inwhicheverycolumngrowsinoneunitwithprob-
abilityp/L.TheshorttimeregimeoftheDPDmodel
isthenlikeRD,whichissolvableexactly,butwiththe
additionalingredientofadensityqofblockedsites.
Onecanseethat,withinthisapproximation,theprob-
abilityofhavingacolumnwithheighthafterNgrowth
attemptsisgivenby

P(N,h)=
(Nsp)
h
r!
e−Ns,(1)

wheres=1/Listheprobabilityofattemptingto
growthacolumnandtheusualapproximationsr(1−
s)N−rN!/[(N−r)!r!]≈(Ns)rexp(−Ns)/r!hasbeen
made.
Fromtheprobability(1),onecancalculatetheinter-
facewidthW2=󰀄h2󰀅−󰀄h󰀅2andthenthetimederivative,
whichleadingtermsare

dW
2
q
+t󰀂,(2)

wheret=Ns=N/Listhetimeintheunitsusedin
Ref.[1].Thisformulagivestheexacttimeevolutionof
dW
2

[1]L.A.BraunsteinandR.C.Buceta,Phys.Rev.Lett.81,
630(1998).
[2]L.-H.TangandH.Leshhorn,Phys.Rev.A45,R8309
(1992);S.V.Buldyrevet.al.,Phys.Rev.A45,R8313
(1992).
[3]A.-L.Barab´asiandH.E.Stanley,FractalConceptsin
SurfaceGrowth(CambridgeUniversityPress,Cambridge,
(1995).

−10.0−5.00.05.010.0
ln (t)

−3.0

−2.0
−1.0
0.0
l
n
(
d
W
2
/
d
t
)
~t
−0.5
~t
0.258

FIG.1.NumericalresultsfortheDPDmodelinasystem
ofsizeL=213forqc=0.539(circles)andq=0.3(squares).
ContinuouslinescorrespondtoEq.(2)andfitthedatafor
t≪1.Forlargertimesourapproximationisnotvalidany
longerandthepowerlawt2β−1takesoverwithβ=0.623and
β=0.3forqc=0.539andq=0.3respectively(dottedlines).