IaroslavIspolatov
DepartmentofPhysics,McGillUniversity,3600rueUniversity,Montr´eal,Qu´ebec,H3A2T8,Canada
and
ChemistryDepartment,BakerLaboratory,CornellUniversity,Ithaca,NY14853,USA
(February1,2008)Persistenceincoarsening1Dspinsystemswithapowerlawinteractionr−1−σisconsidered.Numericalstudiesindicatethatforsufficientlylargevaluesoftheinteractionexponentσ(σ≥1/2inoursimulations),persistencedecaysasanalgebraicfunctionofthelengthscaleL,P(L)∼L−θ.ThePersistenceexponentθisfoundtobeindependentontheforceexponentσandclosetoits
¯=0.17507588....Forsmallervaluesoftheforceexponentvaluefortheextremal(σ→∞)model,θ
(σ<1/2),finitesizeeffectspreventthesystemfromreachingtheasymptoticregime.Scalingargumentssuggestthatinordertoavoidsignificantboundaryeffectsforsmallσ,thesystemsizeshouldgrowas[O(1/σ)]1/σ.
PACSnumbers:02.50.Ga,05.70.Ln,05.40.+j
arXiv:cond-mat/9811196v1 [cond-mat.stat-mech] 13 Nov 1998Coarseningdynamicsofone-dimensionalsystemswithapower-lawV(r)∼r−σ−1interactionbetweenspinshasrecentlybeenstudiedbyLeeandCardy[1],andRuten-bergandBray[2].Ithadbeenestablishedthatafterquenchingfromahigh-temperaturedisorderedphasetoT=0thesesystemsdevelopadomainstructurecharac-terizedbyasinglelengthscaleL(t).Anaiveargument
˙∼L−σbasedonthelawofmotionfordomainwalls,L
(whereL−σisatypicalforcebetweendomainwalls),pro-ducesanasymptoticallycorrecttimedependenceofL,
L(t)∼t1/1+σ.
(1)
Otherpropertiesofthissystem,includingcorrelationfunctionsanddomainsizedistribution,havebeenstudiedin[2]aswell.
Inthispaperweshalllookatanotherfacetof1Dphase-orderingsystemswithapower-lawinteraction:whatfractionPofspinshaveneverchangedsignuptothetimet?Or,equivalently,whatfractionofthespacehasneverbeencrossedbyadomainwall?Suchapropertyofcoarseningsystemsisusuallycalledpersistenceandhasrecentlybecomeamajorsubjectofresearchinstatisticalphysics[3–7].Letusbrieflyreviewsomeknownresultsinthisfieldrelevanttoourproblem.In[3]theexactsolutionwasfoundforpersistenceinanorderingsystemwithextremaldynamics.Theextremaldynamicslimitisreachedbyformallysettingσ→∞,whichmeansthatinteractionsbecomeinfinitelyshort-range.Inthislimit,coarseningproceedsbyconsecutiveshrinkinganddisap-pearanceofthecurrentsmallestdomainsinthesystem,whileotherdomainboundariesremainvirtuallymotion-less.Itwasestablishedin[3]thatpersistenceatastageofevolutionwhentheaveragedomainsizeisLispropor-¯¯=0.17507...isthetionaltoLθ,wheretheexponentθ
solutionofanimplicitintegralequation.
In[4]persistenceexponentshavebeencalculatedforcoarsening1DPottsmodelswithGlauberdynamics.Forthe2-statePotts(Ising)model,persistencedecaysast−θ,θ=3/8,orintermsoftheaveragedomainsizeL,P(L)∼L−3/4.
1
Thefollowingconclusioncanbedrawnfromacompar-isonofpersistenceexponentsforextremalandGlauberdynamics.Extremaldynamicsismoreefficientinpre-servingpersistence,sincethemotionofdomainwallsisalwaysdirectedtowardstheirultimateannihilationpart-ners,whileinthecaseofGlauberdynamics,domainwallsperformrandomwalksandsweepthroughalargeramountofspace,whichotherwisecouldhaveremained
¯setsapersistent.Theextremaldynamicsexponentθ
lowerboundonpersistenceexponentsforsystemswithafiniteforceexponentσ.Itiseasytovisualizeascenariowhenadomainwallfirstmovesawayfromitsultimateannihilationpartner,andthen,afterthestrongerforcesourcedisappears,itturnsback.Sucheventsresultinspinflipsonpartsofthelinethatbelongtoasurviv-ingdomainandwouldhavebeenleftuntouchedintheextremaldynamicscase.
Theresultspresentedbelowsuggestthatthislower
¯=0.17507...isinfacttheexactvalueoftheboundaryθ
persistentexponentforarbitraryσ>0.
Letusformallyintroduceourmodel:Weconsidercoarseningofthe1D2-statespinsystemwithalong-rangeferromagneticHamiltonian:
H=
−4
(xi−xj)
σ+1.
(2)
Afterquenchingfromahigh-temperaturerandomphasetoT=0,coarseningdynamicsforthissystemisdeter-minedbythemotionofdomainwalls,governedbytheLangevinequation.Thevelocityofawallisequaltothesumofpairwiseforcesfromotherwalls,withwallsofthesamesignsrepellingandwallsoftheoppositesignsattractingeachother:
dri
|ri−rj|
σ.
(4)
Whentheadjacentwallsmeet,theyannihilate.Aswementioned,thedegreeofcoarseningisuniquelycharac-terizedbyatypicaldomainsizeL(t)∼t1/(1+σ).WemeasurethefractionofspaceP(L)thathasneverbeencrossedbyasingledomainwallasafunctionofthislengthscaleL(t).Weperformmoleculardynamicssimu-lationsofthemodelforthefollowingvaluesoftheforceexponent:σ=3/2,5/4,1,3/4,1/2,1/4.EachrunstartswithasystemconsistingofN0=100000domainwalls;resultsforeachσareaveragedover20initialconfigura-tions.Openboundaryconditionswithnoreplicasaddedtotheboundariesareused.Tospeeduptheevaluationofforces,a1Dmultipoleexpansionhasbeenperformed,andtermsofuptoquadrupoleorderweretakenintoac-count[8].
1
)L(Pσ=1/4σ=1/2σ=3/4σ=1σ=5/4σ=3/2L**−0.175
0.110
0
10
1
10
2
103
10
4
10
5
L
Fig.1.Thelog-logplotofpersistenceP(L)vs.averagedo-mainsizeLforvariousforceexponentsσ.Thestraightline
correspondstoP(L)∼L−¯
θ.
TheresultsforpersistenceasafunctionoftheaveragedomainlengthLarepresentedinlog-logforminFig.1.Exceptforsmallforceexponents(σ=1/4andlaterevo-lutionstagesforσ=1/2)allthecurvescollapseatalinewithaslope≈−0.175,whichcorrespondstoσ=∞extremalmodel.
StatisticalerrorbarsareshowninFig.2forasinglesetofdata(σ=5/4).
Oursimulationssuggeststhatscalingofpersistence,correspondingtoσ=∞,isvalidforallothernotverysmallσ.Thefollowingasymptoticargumenthelpstoun-derstandwhythisisso.Atanycurrentmomentoftime,persistentspinsaremostlycontainedinthedomainsthatwereexpandingatalmostallpreviousstagesofcoarsen-ing;i.e.thesedomainswerelargerthantheaverageatthosestages.Ifoneofsuchlargedomainsissurroundedbytwosmallneighbors,itwouldmostprobablygrowout-wards,andnospinflips,additionaltothoseinevitablycausedbydirectedcoarseningitself,wouldhappen.Thesituationmaybedifferentiftwoorthreebigdomainsareadjacenttoeachother:theirdomainwallsmaywonder
andgetinsidetheterritoryofthefuturesurvivor,causingsomeexcessivespinflips.
−0.2
]−0.4
)L(P[gl−0.6
−0.8
σ=5/4
−1.0
0.0
1.0
2.0
3.04.05.0
lg(L)
Fig.2.Thelog-logplotofpersistenceP(L)vs.averagedo-mainsizeLforceexponentsσ=5/4withstatisticalerror
bars.ThestraightlinecorrespondstoP(L)∼L−¯
θ.
Wecanestimatethecharacteristicscaleofsucha
persistence-loosingevent.Atypicaldistance∆LthatawallofdomainofsizeL0,surroundedbyagroupofdomainsofsimilarsizes,travelsduringtimetis
∆L∼L0−(L(1+σ)
0
−t)1/1+σ≈L(t)[
L(t)
L0
≫1,hencethenumberofspinflipsaddi-tionaltothosepresentinextremaldynamicscoarseningscenariobecomesnegligible.Anotherconclusionthatfol-lowsfromEq.(5)isthatforsmallσ,thecrossovertimeto
P(L)∼Lθ¯
scalingmustbelargersincethesystemmustdevelopstructurethatincludessufficientlylargedomains.However,besideslongintitialtransitionaltimes,thereisanotherreasonforthebreakdownofscalingforsmallσthatweobservedinoursimulations.Letusfirstconsidertheoppositetoσ=∞caseofσ=0.Inthislimitforces,aredistance-independent,anddomainwalldynamics(3)isdescribedbytheequation
dri
domainlengthandv=1isthevelocityofdomainwalls.Foranexponentialdistributionofinitialdomainsizes,W(L0)=exp(−L0),persistencecanbeexpressedas
∞∞
W(x)dx+2vtW(x)dx˜(t)=0(7)P
2Systemswithfewparticlesandsmallσcoarsenalmostaccordingtotheσ=0scenario:particlesacrossthewholesystemfeelthepresenceoftheboundary.Oddandeven-numberwallstendtomovepredominantlytotheleftandright,respectively,independentoftheposi-tionoftheirnearestneighbors.
¯
ToprobewhetherthedeviationfromP(L)∼Lθscal-inginpersistencebehavioriscausedbyσ=0finitesizeeffects,wedothefollowingmeasurements.First,forthesystemofthesameinitialsize(N=105)weplottheaveragedomainlengthL(t)asafunctionoftimeandcompareittotheL∼t1/1+σprediction.
thatoursystemisneverinthetrueinfinite-sizeregimeforσ=1/4,finitesizeeffectsarebecomingevidentforσ=1/2,evenatearlystagesofevolution,andonlyforσ≥3/4theboundaryeffectscouldbeneglected.
0.8
σ=1/4σ=1/2σ=3/4σ=1σ=5/4σ=3/2
0.6
B(L)0.4
0.2
0010
10
1
10
210L
3
10410
5
10
4
σ=1/4σ=1/2σ=3/4σ=1σ=5/4σ=3/2
Fig.4.NumberofdomainwallsB(L)thatmoveoppositetothedirectionprescribedbyboundaryeffectsvs.averagedomainlengthLforvariousforceexponentsσ.
10
2
Finally,wepresentaroughestimateofhowbigasys-temshouldbeforaparticularvalueofσ≪1toavoidsignificantfinite-sizeeffects.WeevaluateaforceF1−2,exertedonatestdomainwallbyadipolepairofneigh-boringdomainwalls,
F1−2≈(
1
2L
σ
L(t)10
0
−327)=(
1
101010
t
Fig.3.Thelog-logplotofaveragedomainsizeL(t)vs.timeforvariousforceexponentsσ.Straightlinescorrespondtoscalingpredictions,L(t)∼t1/1+σ.
σ
NL
).(9)
ResultsforthissimulationarepresentedinFig.3.Onecanseethatthesystemwithσ=1/4isneverinscalingregime(1),systemwithσ=1/2behavesaccordingto(1)onlyuptosomeintermediatestageofevolution.Forallotherforceexponentsσ>1/2,foracertainperiodofevolutionaftershorttransitionaltime,typicaldomainsizesscaleaccordingto(1).Anothercheckofwhetherasystemfeelsthepresenceoftheboundariesandthere-forecrossesovertotheσ=0coarseningregime,istomeasuredirectlythefractionofdomainwallsB(L)thatmoveoppositetothedirectionprescribedbyboundaryeffects.InFig.4weplotthefractionofeven-numberdo-mainwallsmovingtotherightandodd-numberdomainwallsmovingtotheleftforthesystemsinitiallyconsist-ingofthesamenumberofdomains,N=105.Forfiniteσ>0andatrulyinfinitesystemthisfractionshouldbeequalto1/2,forσ=0itshouldbe0.Weobserve
3
HereLandNarethetypicaldomainlengthandthenumberofdomainsinthesystem.Theboundaryeffectsbecomesignificantwhentheseforcesareofthesameor-der.Henceforparticularσ,theminimumnumberofparticlestoavoidfinitesizeeffectsNminis
Nmin≈2(
1
σ
(10)
Wehaveobserved(seeFig.4)thatforσ=1/2,Nmin≈105.Assumingthefollowingparameterizationofminimal
1/σ
sizeofthesystemforsmallσ,Nmin=(constant/σ)andfittingittoNmin(σ=1/2)=105,weobtainNmin(σ=1/4)≈1.6×1011.Thisiswellbeyondthelimitsofcomputationalpoweravailabletous.
Insummary,wepresentednumericalevidenceandascalingargumentsuggestingtheuniversalityofpersis-¯=0.17507588...,tentexponentforextremalmodel,θ
formodelswitharbitraryforceexponentsσ>0.Wefoundthatadeviationfromscalingforpersistence,thathappensforsmallσ,isaccompaniedbyasimilarde-viationfromscalingforatypicaldomainsizeL(t)andiscausedbyfinitesizeeffectsthatcausecrossovertoaσ=0coarseningscenario.Weestimatedthatinordertoavoidboundaryeffects,thesystemsizeshouldgrowas[O(1/σ)]1σ.Apossibleextensionofthisworkisforhigherdimensionalsystems,thoughthedualitybetweendomainwallsandspindynamicsthatwasextensivelyusedforthiswork,maynotbesostraightforwardtoapply.
TheauthorwouldliketothankP.Krapivsky,A.Rutenberg,A.Hare,andR.Hillforinterestingdiscus-sions.
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