Effects of mixing and stirring on the critical behavior

91 ]chem-tats.tam-dnoc[ 1v4344060/tam-dnco:viXraNVAntonov,1MichalHnatich2,3,4andJuhaHonkonen5,6

1

DepartmentofTheoreticalPhysics,St.PetersburgUniversity,Uljanovskaja1,St.Petersburg,Petrodvorez,198504Russia2

InstituteofExperimentalPhysics,SlovakAcademyofSciences,Watsonova47,04011Koˇsice,Slovakia3

FacultyofCivilEngineering,TechnicalUniversity,Vysokoˇskolsk´a4,04353Koˇsice,Slovakia4

N.N.BogoliubovLaboratoryofTheoreticalPhysics,JointInstituteforNuclearResearch,141980Dubna,MoscowRegion,Russia5

DivisionofTheoreticalPhysics,DepartmentofPhysics,FIN-00014UniversityofHelsinki,Finland6

DepartmentofMilitaryTechnology,NationalDefenceCollege,FIN-00861Helsinki,Finland

E-mail:juha.honkonen@helsinki.fi

Abstract.Stochasticdynamicsofanonconservedscalarorderparameternearitscriticalpoint,subjecttorandomstirringandmixing,isstudiedusingthefieldtheoreticrenormalizationgroup.ThestirringandmixingaremodelledbyarandomexternalGaussiannoisewiththecorrelationfunction∝δ(t−t′)k4−d−yandthedivergence-free(duetoincompressibility)velocityfield,governedbythestochasticNavier–StokesequationwitharandomGaussianforcewiththecorrelationfunction

∝δ(t−t′)k4−d−y′

.Dependingontherelationsbetweentheexponentsyandy′andthespacedimensionalityd,themodelrevealsseveraltypesofscalingregimes.Someofthemarewellknown(modelAofequilibriumcriticaldynamicsandlinearpassivescalarfieldadvectedbyarandomturbulentflow),buttherearethreenewnonequilibriumregimes(universalityclasses)associatedwithnewnontrivialfixedpointsoftherenormalizationgroupequations.Thecorrespondingcriticaldimensionsarecalculatedinthetwo-loopapproximation(secondorderofthetripleexpansioniny,y′andε=4−d).

PACSnumbers:.75.+g,05.10.Cc,.60.Ht,05.40−a

Effectsofmixingandstirringonthecriticalbehaviour1.Introduction

2

Overthepastthreedecades,increasingattentionhasbeenattractedbythedynamicsofphaseordering—thegrowthoforderthroughdomaincoarsening(spinodaldecomposition),whenasystem(e.g.aferromagnetorabinaryalloy)isquenchedfromitshigh-temperaturehomogeneousphaseintothelow-temperaturemulti-phasecoexistenceregion;see[1]–[18]andreferencestherein.

Muchinterestwasfocusedonthelatestagesofthecoarseningprocess,whensomekindofaself-similar(scaling)regimedevelopswithapparentlyuniversalexponents—thefeaturesnormallyassociatedwiththecriticalbehaviour.Thatregimeisbynowratherwellunderstood;seeRef.[1]andthereviewscitedthe.Phenomenologicalapproaches,renormalizationgroup(RG)techniques,exactlysolublemodelsandnumericalsimulationsshowthatthecharacteristicdomainsizeincreasesasapoweroftime,L(t)∼tα,wherethegrowthexponentαdependsontheglobalcharacteristicsofthesystem(conservingornonconservingdynamics,scalarorvectororderparameter,dimensionalityofspace)butnotonitsdetailedstructure(likethevaluesofthecouplingconstants).Therefore,inrecentyearsattentionhasbeendirectedtosystemssubjectedtoexternalstirring,likebinarymixturesunderimposedshearfloworotherkindsofdeterministicorrandom(e.g.turbulent)velocityfields;see[3]–[15]andreferencestherein.

Numericalexperimentsandtheoreticalanalysis(e.g.thelinearstabilityanalysisofthecorrespondingdynamicequations)ofbinaryalloyssubjectedtostatisticallyisotropicandhomogeneousrandomvelocityensemblesofverydifferentkindsalsosuggestthat,atleastclosetothecriticalpointandundervigorousstirring,thedomaingrowthis“arrested”andanewdynamicalnonequilibriumsteadystateemerges,whichischaracterizedbyacontinuousformationandruptureoffinite-sizedomains[3,9,16,18].

Emergenceofthenonequilibriumsteadystatesappearsratheragenericandrobustphenomenon,beingobservedintwo-dimensionalnumericalsimulationsforpassive[9,16,18]andactive[18]orderparameterssubjectedtoarandomGaussianvelocityfieldwithfinitecorrelationlengthandtime[9]andvariouskindsofregularandchaoticcellularflows[16,18].Thequestionswhichnaturallyarisewithinthiscontext,andwhichwillbeaddressedinthepresentpaper,arethefollowing:Dothosesteadystatesrevealsomekindsofself-similarbehaviour?Dothecorrespondingcorrelationandstructurefunctionsexhibitpowerlaws?Ifyes,dothosestatesbelongtotheuniversalityclassesknownforthemodelsofequilibriumcriticaldynamics[19,20],ordotheyrepresentnewtypesofscalingbehaviour?Arethereanycrossoverdimensionsforthenewscalingregimes?Isitpossibletoestablishtheexistenceofthesescalingregimesonthebasisofmicroscopicmodels,andtocalculatethecorrespondingexponentsinconsistentapproximationsor,better,withinregularperturbationexpansions?Towhatextentthisbehaviourisuniversal?Whataretheparametersthescalingdimensionsdependon?

Wewillconsiderthedynamicsofascalar(one-component)passive(nofeedbackonthevelocityfield)nonconservingorderparameterϕ(x)≡ϕ(t,x)governedbythe

Effectsofmixingandstirringonthecriticalbehaviourstochasticequation

σ0∇tϕ=∂2ϕ−V′(ϕ)+f,

∇t=∂t+vi∂i,

3

(1)

whereσ0>0isthereciprocalofthekineticcoefficientandthepotentialV(ϕ)willbechosenasinthewell-knownmodelsofcriticaldynamics[19,20,21,22].However,incontrasttothelatter,thestirringnoisef(x)andthevelocityvi(x)arenotchosensuchthatthesteadystateofthesystemisinequilibrium,or,inotherwords,itsequal-timecorrelationfunctionsarenotdescribedbytheLandau–GinzburgHamiltonian.Namely,thetransverse(divergence-free,duetotheincompressibilitycondition∂ivi=0)velocityfieldsatisfiestheNavier–Stokesequationwitharandomdrivingforce

∇tvi=ν0∂2vi−∂iP+fi,

(2)

wherePandfiarethepressureandthetransverserandomforceperunitmass(allthesequantitiesdependonx).

Therandomsourcesf(x)andfi(x)maintainthesteadystateofthesystemandmodeltheeffectsofexternalstirringand/orshakingandinitialand/orboundaryconditions.Theuseofsuchrandomstirringtermsisacommonplaceinthestatisticaltheoryofturbulence[23,24,25,26]andothernonequilibriumphenomena[23,27]:itallowsonetodoawaywiththedetailsofthegeometryofthesystemandtoconsiderahomogeneousandisotropicproblemintheinfinitespace.Letusspecifytheirstatisticalproperties.

InmodelsofequilibriumcriticaldynamicstheformofsuchcorrelatorsforLangevinequations(likee.g.(2)withoutthevelocity)isuniquelydeterminedbytherequirementthatthedynamicsandstaticsbemutuallyconsistent(thatis,theequal-timecorrelationsofthedynamicalproblembegivenbytheLandau–GinzburgHamiltonian);see[19,20,21].Suchargumentsdonotapplytoournon-equilibriummodel.LikeintheRGtheoryofturbulence,thecorrelatorswillbechosenonthebasisofbothphysicalandtechnicalarguments.

Considerfordefinitenessthecorrelationfunction

Dϕ(x,x′)≡󰀈f(x)f(x′)󰀉=δ(t−t′)D(r),

r=|x−x′|,

(3)

ofthesourcefieldofthestochasticequation(1).ThefunctionD(r)dependsonlyonr=|x−x′|,itsFouriertransformbeingD(k).Thephysicalargumentsarethatthenoisesmodeltheinjectionofenergytothesystemowingtointeractionwiththelarge-scalestirring.ThusforrealisticcasethedominantcontributiontothecorrelatorsD(k)mustcomefromsmallmomentak∼m,wherem=1/Listhereciprocaloftheintegral(external)scaleL(thesizeofthesystemorastirringdevice).IdealizedinjectionbyinfinitelylargemodescorrespondstoD(k)∝δ(k).Ontheotherhand,fortheuseofthestandardRGtechniqueitisimportantthatthefunctionD(k)haveapower-lawbehaviouratlargek.ThisconditionissatisfiedifD(k)ischosenintheform[24]D(k)=D0h(m/k)k4−d−y,whereD0>0isanamplitudefactor,disthespacedimensionandtheexponentyplaysthepartanalogoustothatplayedby4−dintheRGtheoryofcriticalbehaviour.Thefunctionh(m/k)withh(0)=1providestheIRregularization.

Effectsofmixingandstirringonthecriticalbehaviour4

Itsspecificformisunessential;wewillusethesharpcutoffh(m/k)=θ(m−k)withtheHeavisidestepfunctiontosimplifythepracticalcalculation(inthecalculationsinthespiritofdimensionalregularizationonecouldsimplyseth=1).

Thelarge-scaleforcingisreproducedinthelimity→4,asfollowsfromthewell-knownpower-lawrepresentationofthed-dimensionalδfunction,

1

δ(k)=lim

y→4

Effectsofmixingandstirringonthecriticalbehaviour5

withcriticalphenomena,however,theapplicationoftheRGtothegrowthproblemssuffersfromthelackofan(obvious)smallparameter(analogoustoε=4−dfortheLandau–Ginzburgmodelandcorrespondingdynamicalmodels),theproblemalsoencounteredforthestochasticBurgersandKardar–Parisi–Zhangmodels[23,27].Tocircumventthisobstacle,in[1,5,6,7,8],theRGwasusedintheformofblock-spintransformationsperformedusingnumericalMonteCarlosimulations,proposedearlierin[31].Anotherpossibility,exploredin[8](seealsodiscussioninthereviewpaper[1]),wastoassumetheexistenceoftheRGsymmetryandanappropriatestrong-couplingfixedpoint,andthentousespecificfeaturesoftheconserveddynamics(absenceofrenormalizationofthetransportcoefficient,wellknownforthemodelBofequilibriumdynamics[19,20,21])toderivesomeexactrelationsbetweenthecriticalexponents.Tocompletethatanalysis,however,oneshouldtakesomeexponentsfromtheexperimentorderivethemusingadditionalphenomenologicalconsiderations[1,8].

Theplanofthepaperisasfollows.Webeginwiththeanalysisofthemodelwithoutvelocity,whichappearsnontrivialandrevealsanewtypeofscalingbehaviour.Insection2wepresentthefieldtheoreticformulationofthemodelanditsrenormalization.Afteranappropriateextension,themodelbecomesmultiplicativelyrenormalizable,andthedifferentialRGequationscanbederivedinastandardfashion(section3).ThefixedpointsandtheirregionsofIRstabilityareanalyzedinsection4.Itisshownthatasystematicperturbationexpansioninthetwoparameters,yandε=4−d,canbeconstructedforthecoordinatesofthefixedpointsandcriticaldimensions,withtheadditionalassumptionthaty−ε=O(ε2).OneofthetwonontrivialfixedpointscorrespondstothewellknownmodelAofequilibriumcriticaldynamics,whiletheotherrepresentsanewnonequilibriumuniversalityclass;thecorrespondingcriticaldimensionsarecalculatedinthetwo-loopapproximation(section5).Insection6,thefullmodelwiththevelocityfield,governedbythestochasticNavier–Stokesequation,isstudied.Twoadditionalnonequilibriumscalingregimes(universalityclasses)areidentified;thecorrespondingdimensionsarefoundtothesecondorderofthetripleexpansioniny,y′andε=4−d.Section7isreservedforabriefconclusion.Someinterestingdetailsofthetwo-loopcalculationaregiveninAppendixA.

Thefield-theoreticrenormalizationgroupwasearlierappliedtotheproblemoftheeffectsofturbulenceonthecriticalbehaviourofbinarymixturesin[17].Themodelstudiedinthatworkwaslessrealisticthanthepresentoneintworespects:turbulencewasmodelledbyaGaussiantime-decorrelatedstatisticalensembleandthenoisewastakentobepurelythermal.Ontheotherhand,ourmodelislessrealisticinthesensethattheorderparameterhereisnotconserved.Nevertheless,themainqualitativeconclusiondrawnfromthetwocasesisthesame:theinstabilityoftheequilibriumfixedpointandtheexistenceofanewnon-equilibriumcriticalregimewasestablished.Thuswemayconcludethatsuchaphenomenonappearsquiterobustandinsensitivetothedetailsofthemodel.

Effectsofmixingandstirringonthecriticalbehaviour

2.Themodelwithoutconvection:Fieldtheoreticformulationandrenormalization

6

Itisinstructivetobegintheanalysiswiththemodelwithnoconvectiontermin(1),whichalreadyexhibitsanonequilibriumscalingregimeandinvolvessomeinterestingformalsubtleties.Thedynamicalequationfortheorderparameterϕ(x)≡ϕ(t,x)thenbecomes

σ0∂tϕ=∂2ϕ−V′(ϕ)+f.

Correlatoroftherandomnoisef(x)willbetakenintheform

Dϕ(x,x′)≡󰀈f(x)f(x′)󰀉=δ(t−t′)D(r),

withsomefunctionD(r)dependingonlyonr=|x−x′|.ThechoiceD(r)=2σ0δ(x−x′)correspondstothewell-knownmodelAofcriticaldynamics,whichdescribeskineticpropertiesoftheequilibriumcriticalstate[19,20,21,22];theprobabilitydistributionfunctionofitsequal-timecorrelatorsisthengivenbyexp(−H(ϕ))whereH(ϕ)=−ϕ∂2ϕ+V(ϕ)withtheimpliedintegrationoverxistheHamiltonianforthetime-independentfieldϕ(x).

Weassumethatthemodelisnearitscriticalpoint,and,inthespiritoftheLandautheory,retaininV(ϕ)onlythefirsttermsoftheTaylorexpansion:V(ϕ)=τ0ϕ2/2+λ0ϕ4/24,whereτ0isthedeviationofthetemperaturefromitscriticalvalue.ThefunctionD(r)in(6),however,willbechoseninthepower-likeformD(r)∝r−4+y,whichinthemomentumrepresentationgives

D(k)=D0k4−d−y,

(7)

r=|x−x′|,

(6)(5)

wherek=|k|isthewavenumber,yisanarbitraryparameterandD0>0anamplitudefactor.TheIRcutoffatk=misimplied.

Accordingtothegeneraltheorem(seee.g.[21,22]),stochasticproblem(5),(6)isequivalenttothefieldtheoreticmodelofthedoubledsetoffieldsΦ≡{ϕ′,ϕ}withactionfunctional

󰀃󰀄′′′23

S(Φ)=ϕDϕϕ/2+ϕ−σ0∂tϕ+∂ϕ−τ0ϕ−λ0ϕ/6,(8)

withDϕfrom(6)andimpliedintegrationsovertheargumentx={t,x}.Formulation(8)

meansthatstatisticalaveragesofrandomquantitiesintheoriginalstochasticproblemcanberepresentedasfunctionalaverageswiththeweightexpS(Φ).Themodel(8)correspondstoastandardFeynmandiagrammatictechniquewithtwobarepropagators(linesinthediagrams)󰀈ϕϕ󰀉0and󰀈ϕϕ′󰀉0(theirexplicitformisgiveninAppendixA)andthevertexϕ′ϕ3.

Theanalysisofultraviolet(UV)divergencesisbasedontheanalysisofcanonicaldimensions.Dynamicalmodelsofthetype(8),incontrasttostaticmodels,havetwoscales,i.e.,thecanonicaldimensionofsomequantityF(afieldoraparameterintheactionfunctional)isdescribedbytwonumbers,themomentumdimensiondkFandthekω−dF

[T]−dF,whereLfrequencydimensiondωF.Theyaredeterminedsuchthat[F]∼[L]

Effectsofmixingandstirringonthecriticalbehaviour7

isthelengthscaleandTisthetimescale.Thedimensionsarefoundfromtheobvious

kωωkkωω

normalizationconditionsdkk=−dx=1,dk=dx=0,dω=dt=0,dω=−dt=1,andfromtherequirementthateachtermoftheactionfunctionalbedimensionless(withrespecttothemomentumandfrequencydimensionsseparately).Then,basedondkF

ωkω

anddF,onecanintroducethetotalcanonicaldimensiondF=dF+2dF(inthefreetheory,∂t∝∂2),whichplaysinthetheoryofrenormalizationofdynamicalmodelsthesameroleastheconventional(momentum)dimensiondoesinstaticproblems,seee.g.[21].Theresultingcanonicaldimensionsaregivenintable1,includingthedimensionsoftheparameterswhichwillappearlateron(renormalizedparametersandtheothers).Itiseasilycheckedthattheroleofthecouplingconstant(expansionparameterintheordinaryperturbationtheory)inmodel(8)withcorrelator(6)isplayedbythecombinationλ0D0/σ0.Fromtable1itfollowsthatthisconstanthasthedimensionΛywithsomemomentumscaleΛ.Thusthecasey<0correspondstotheGaussianIRbehaviour(perturbationtheoryworksintheIRrange),y=0isthelogarithmicvalue,andfory≥0theRGsummationisneeded.TheUVdivergenceshavetheformofthepolesinyinthecorrelationfunctionsofthefieldsΦ≡{ϕ′,ϕ}.

Thetotalcanonicaldimensionofanarbitrary1-irreduciblecorrelationfunction

ω

Γ=󰀈Φ···Φ󰀉isgivenbytherelationdΓ=dkΓ+2dΓ=d+2−NΦdΦ,whereNΦ={Nϕ,Nϕ′}arethenumbersofcorrespondingfieldsenteringintothefunctionΓ,andthesummationoveralltypesofthefieldsisimplied.ThetotaldimensiondΓinthelogarithmictheory(thatis,aty=0)istheformalindexoftheUVdivergence.SuperficialUVdivergences,whoseremovalrequirescounterterms,canbepresentonlyinthosefunctionsΓforwhichdΓisanon-negativeinteger.Straightforwardanalysisshowsthatforalld>4,superficialUVdivergencescanbepresentonlyinthe1-irreduciblefunctions󰀈ϕ′ϕ󰀉withthecountertermsϕ′∂tϕ,ϕ′∂2ϕ,τ0ϕ′ϕ,and󰀈ϕ′ϕ3󰀉withthecountertermϕ′ϕ3.Suchtermsarepresentintheaction(8),sothatthemodelismultiplicativelyrenormalizable.

However,ford≤4anewdivergenceappearsinthefunction󰀈ϕ′ϕ′󰀉(ford=4andsmally,thenoisecorrelatorbecomesalmostpolynomialink—namelyconstant—andlocalinxrepresentation).Itwouldbeerroneoustotrytoeliminatethisdivergencebyrenormalizingthenonlocalnoiseterm(asthoroughlydiscussedinRef.[32]forthestochasticNavier–Stokesequation).Wethereforemustaddthelocalcountertermoftheformϕ′ϕ′.Soweareforcedtoconsiderthevariablespacedimensionand,ford≤4,toextendtheoriginalmodel(toincludethelocalterminthenoisecorrelatorfromtheverybeginning)inordertohavemultiplicativerenormalizability.Finallywearriveattheextendedmodel

σ0∂tϕ=∂2ϕ−τ0ϕ−λ0ϕ3/6+f,

󰀃󰀄

󰀈f(x)f(x′)󰀉=2σ0δ(t−t′)w0k4−d−y+1

(9)(10)

whichhasbecomemultiplicativelyrenormalizableford<4;thespecialcasew0=0givesthemodelA(whichismultiplicativelyrenormalizableinitself).Interpretationoftheadditionallocaltermin(10)canbetwofold.Ontheonehand,thefactthat

Effectsofmixingandstirringonthecriticalbehaviour8

itisgeneratedbytherenormalizationproceduremeansthatitisnotforbiddenbydimensionalityorsymmetryconsiderationsand,therefore,itisnaturaltoincludeitinthemodelfromtheverybeginning.InthelanguageoftheWilsonianRG,thismeansthatsuchtermnecessarilyarisesintheeffectivemodelfortheproperlysmoothed(coarse-grained)field;itbecomesIRrelevantford<4,whereitaffectsthecriticalbehaviourandcannotbeneglected.Ontheotherhand,onecaninsistonstudyingtheoriginalmodelwithapurelypower-lawcorrelationfunction.ThentheextensionofthemodelisonlyneededtoensurethemultiplicativerenormalizabilityandtoderivetheRGequations;thelattershouldbesolvedwiththespecialinitialdatathatcorrespondtothepower-lawcorrelator.SincetheIRattractivefixedpointoftheRGequationsisuniqueforanygivenchoiceoftheparametersεandy(seesection4),theresultingIRbehaviouristhesameasforthecaseofthegeneralcorrelationfunction(10)withtheinclusionofthelocalterm.

Bydimensionthecouplingsλ0∼Λεandw0∼Λy−εwithε=4−d,sothatweexpectthedoubleRGexpansioninyinεinsteadofasingleexpansioniny(asford>4)orinε(asforthemodelA).However,aswewillsee,therealsituationappearsslightlymorecomplicated.

Therenormalizedactionis

󰀃󰀄

SR(ϕ,ϕ′)=σϕ′wµy−εkε−y+Z1ϕ′

󰀃󰀄′2ε3

+ϕ−Z2σ∂tϕ+Z3∂ϕ−τZ5ϕ−Z4λµϕ/6,(11)whichisequivalenttothemultiplicativerenormalizationofthefieldsϕ→Zϕϕ,′′

ϕ′→Zϕϕandparameters

σ0=Zσσ,

τ0=Zττ,

u0=µεZuu,

w0=µy−εZww,

(12)

whereweintroducedthenewcouplingconstantu=λ/16π2(thecoefficientsinRGfunctionsbecomeslightlysimpler)andµisthereferencemass–additionalparameteroftherenormalizedtheory.

TherenormalizationconstantsZi=Zi(ε,y,g,w)capturethedivergencesatε,y→0,sothatthecorrelationfunctionsoftherenormalizedmodel(11)havefinitelimitsforε,y=0(whenexpressedinrenormalizedparametersu,w,τandµ).

TherelationsbetweentheZ’sin(11)and(12)havetheforms

2ZσZϕ′=Z1,

2

ZσZϕ′Zw=1,

ZσZϕ′Zϕ=Z2,

3

Zϕ′=Z4.ZuZϕ

ZϕZϕ′=Z3,ZτZϕZϕ′=Z5,

(13)

TherenormalizationconstantsZ1–Z5arecalculatedaccordingtostandardrules

fromtheperturbationtheory;thentheconstantsin(12)areeasilyfoundusingtherelations(13).TheexpansionparameterintheZ’sisu,whilethedependenceonthesecondcouplingconstantwshouldbecalculatedexactlyineachorderoftheexpansioninu.LikeforthemodelA,thefirstnontrivialcontributionstotheconstantsZ4,5aredeterminedbyone-loopFeynmangraphs,sothatZ4,5=1+O(u).TheleadingcontributionstotheconstantsZ1–Z3aredeterminedbytwo-loop(“watermelon”)graphsdepictedinfigure1,sothatZ1,2,3=1+O(u2).Thesedetailswillbeimportantin

Effectsofmixingandstirringonthecriticalbehaviour9

Figure1.Two-loopgraphsgivingrisetorenormalizationconstantsZ1–Z3.Thesimplelinescorrespondtothebarepropagator󰀈ϕϕ󰀉0=2σ{w(k/µ)y−ε+1}/(ω2σ2+k4)infrequency-momentumrepresentation,whereasthelineswithaslashtothepropagator󰀈ϕϕ′󰀉0=(−iσω+k2)−1,theslashindicatingthevariablesofthethefieldϕ′.Thevertexfactorisλµε.

theanalysisofthefixedpointsoftheRGequations.Thetwo-loopcalculationoftherenormalizationconstantsinourmodelisillustratedbytheexampleofZ1inAppendixA.

3.RGequationsandRGfunctions

LetusrecallanelementaryderivationoftheRGequations;see[21].TheRGequationsarewrittenfortherenormalizedcorrelationfunctionsGR=󰀈Φ···Φ󰀉R,whichdifferfromtheoriginal(unrenormalized)onesG=󰀈Φ···Φ󰀉onlybynormalizationandchoiceofparameters,andthereforecanbeequallyusedforanalyzingthecriticalbehaviour.TherelationSR(Φ,e,µ)=S(Φ,e0)betweenthefunctionals(8)and(11)resultsin

N′N

therelationsG(e0,...)=ZϕϕZϕ′ϕGR(e,µ,...)betweenthecorrelationfunctions.Here,asusual,NϕandNϕ′arethenumbersofcorrespondingfieldsenteringintoΓ;e0={σ0,τ0,w0,λ0}isthefullsetofbareparametersande={σ,τ,w,λ∝u}aretheirrenormalizedanalogs;thedotsstandfortheotherarguments(times,coordinates,

󰀌µtodenotethedifferentialoperationµ∂µforfixede0andmomentaetc).WeuseD

operateonbothsidesofthisequationwithit.ThisgivesthebasicRGdifferentialequation:

󰀌µexpressedintherenormalizedvariables:whereDRGistheoperationD

DRG≡Dµ+βu∂u+βw∂w−γσDσ−γτDτ.{DRG+Nϕγϕ+Nϕ′γϕ′}GR(e,µ,...)=0,

(14)

(15)

Inequation(15),wehavewrittenDx≡x∂xforanyvariablex,theRGanomalous

dimensionsγaredefinedas

󰀌µlnZFγF≡D

foranyquantityF,

(16)

andtheβfunctionsforthetwodimensionlesscouplingsuandware

˜µu=u[−ε−γu],βu≡D

˜µw=w[−y+ε−γw],βw≡D

wherethelastequalitiescomefromthedefinitionsandrelations(12).

(17a)(17b)

Effectsofmixingandstirringonthecriticalbehaviour10

Theanomalousdimensionsγ1–γ5arefoundfromtheknownconstantsZ1–Z5(seeAppendixAforγ1).Thentherelations(13)give

γσ+2γϕ′=γ1,γϕ+γϕ′=γ3,

Resolvingtheserelationsgives

γw=−γ1,γσ=γ2−γ3,

2γϕ=γ3+γ2−γ1,γτ=γ5−γ3,

γu=γ1−γ2−2γ3+γ4,2γϕ′=γ1+γ3−γ2,

(19)(20)

γσ+2γϕ′+γw=0,γτ+γϕ+γϕ′=γ5,

γσ+γϕ′+γϕ=γ2,γu+3γϕ+γϕ′=γ4.

(18)

andfortheβfunctions(17)oneobtains:

βu=u[−ε−γ1+γ2+2γ3−γ4],

4.Fixedpointsandscalingregimes

βw=w[−y+ε+γ1].

ItiswellknownthatpossiblescalingregimesofarenormalizablemodelareassociatedwiththeIRattractivefixedpointsofthecorrespondingRGequations.Inourmodel,thecoordinatesu∗,w∗ofthefixedpointsarefoundfromtheequations

βu(u∗,w∗)=0,

βw(u∗,w∗)=0

(21)

withthebetafunctionsgivenin(20).ThetypeofafixedpointisdeterminedbythematrixΩ={Ωij=∂βi/∂gj},whereβidenotesthefullsetofthebetafunctionsandgj={u,w}isthefullsetofcouplings.ForIRstablefixedpointsthematrixΩispositive,i.e.,therealpartsofallitseigenvaluesarepositive.

Fromtheformsoftherenormalizationconstants(seetheremarkintheendofsection2)itfollowsthatγ4,5=O(u)whileγ1,2,3,4=O(u2).Thereforeonlyγ4givestheleadingcontributiontothefunctionβuin(20).Theactualcalculationgives(seeAppendixA)

γ1=bu2(1+w)3+O(u3),

γ4=−au(1+w)+O(u2)

(22)

witha=3andb=ln(4/3).Thustheleading-orderexpressionsfortheβfunctionsare

βu=u[−ε+au(1+w)+O(u2)],βw=w[−y+ε+bu2(1+w)3+O(u3)].

(23)

From(23)oneimmediatelyfindsthelocalGaussianfixedpointu∗=w∗=0,whichisIRattractiveforε<0,ε>y.Thecaseu∗=0,w∗=0appearsmoresubtle.Substitutingu∗=0intoβwgivesβw=w(−y+ε),whichsuggeststhatforε<0,ε0,ε<0requiresabitmoreelaborateanalysis,which

Effectsofmixingandstirringonthecriticalbehaviour11

showsthatthelarge-scalebehaviouralsointhisregionisthatoftheGaussianmodelwiththenonlocalcorrelator.Thus,wefinallymayconcludethattheregionε<0,εInmodelswithtworegulatorslikeεandy,itisusuallyimpliedthattheyareofthesameorder,y=O(ε),andthecoordinatesofthefixedpointsandthevaluesofanomalousdimensionsatthosepointsaresoughtintheformofdoubleseriesinεandy;seee.g.[32].Nosuchsolution,however,canbeconstructedinthecaseathand,duetotheabsenceofanO(u)terminthesquarebracketsforβwin(23).Aregularsolutioncanbeconstructedifweassumethaty=O(ε),buttheirdifferencey−󰀋εisoforderε2.Inotherwords,theactualexpansionparametersappeartobeεand

εexpansionfortheIsingferromagnetwithquenched

disorder[33].Thereitwasaconsequenceofanaccidentaldegeneracyoftheβfunctionsofthetwoinvolvedcouplings,whichfortheirratio(theanalogofourw)alsoimpliesvanishingofthefirst-ordercontribution.Asimilarsituationwasalsoencounteredin[17]wherethemixingofaconservedorderparameterbyaGaussianvelocityensemblewasstudied.

Tobedefinite,letuswritey=ε+Bε2withsomeB.Nowwehave

βu=u[−ε+au(1+w)+O(u2)],βw=w[−Bε2+bu2(1+w)3+O(u3)]

(24)

andthesolutionsforu∗andw∗canbefoundasregularseriesinε,whilethedependenceonBshouldbetakenintoaccountexactlyineachorderoftheεexpansion.Nowwecanidentifytwonontrivialfixedpoints,whichwedenoteasIandII.

ForthepointI,wefindw∗=0,u∗=ε/a.ThispointclearlycorrespondstothemodelAofcriticaldynamics;w∗vanishesidenticallytoallordersoftheεexpansion(themodelwithw0=w=0islocalandthereforeclosedwithrespecttotherenormalization),whileu∗hasnontrivialcorrectionsoforderε2andhigher.TheΩmatrixatthispointis:

βuu=au∗(1+w∗)=ε,

2

βuw=au2∗=ε/a,

βwu=2u∗bw∗(1+w∗)=0,

βww=−Bε2+bu2=−Bε2+b(ε/a)2.

(25)

Itistriangular,sothefixedpointisIR-attractiveifthediagonalelementsβuuandβww

arepositive.Thisgivesε>0(ofcourse)andB0,ε>y,butisslightlywider:italsoinvolvesanarrow“beak”adjacenttotherayε=yintheregionεFurthermore,wecanseethatthefixedpointwithbothu∗=0andw∗=0alsoexistsforthebetafunctions(24);wedenoteitaspointII.Indeed,βu∗=0andu∗=0

Effectsofmixingandstirringonthecriticalbehaviour12

Figure2.Regionsofstabilityofthefixedpointsoftheextendedmodel(9),(10).TheregionlabelledGNL(Gaussiannonlocal)correspondstothefixedpointwithvanishinginteraction(d>4)andpurelynonlocalcorrelationfunctionofthenoise;thelabelNL,u=0referstothenonlocalregimeIIoftheextendedmodel;thefirst-orderboundarybetweenthelatterandtheregionofstabilityoftheuniversalityclassofmodelA(thelocalregimeIoftheextendedmodel)isy=εandtheimprovedboundaryisy=ε+Bcε2.TheboundarybetweentheuniversalityclassofmodelAandthatofthelinearmodeldrivenbytherandomnoisewithlocalcorrelationsonly–labelledGL(Gaussianlocal)–isthenegativeyaxis.

givesu∗(1+w∗)=ε/a,substitutinginβw∗=0andassumingw∗=0gives

sothatatthefixedpoint

󰀃󰀄

0=−Bε2+b(1+w∗)[u∗(1+w∗)]2=ε2−B+b(1+w∗)/a2,w∗=−1+Ba2/b,

u∗=εb/Ba3

(26)

withcorrectionsoforderO(ε)andO(ε2)respectively.

TheΩmatrixatthispointis:

βuu=au∗(1+w∗)=ε,

2225

βuw=au2∗=εb/Ba,

βwu=2u∗bw∗(1+w∗)3=2bw∗(ε/a)(a2B/b)2,

22βww=3w∗bu2∗(1+w∗)=3bw∗(ε/a).

(27)

ThustheΩmatrixhastheform

󰀁󰀂αAε

Ω=ε.

βBε

(28)

Effectsofmixingandstirringonthecriticalbehaviour13

Althoughtheelementsintherightcolumnaresmallinεincomparisontotheleftcolumn,theyareneededtofindtheeigenvaluestoleadingorder.ItisimportantthattheO(ε2)correctionstotheleftelementsarenotneeded(theygiveonlycorrections),despitethefactthattheywouldbeofthesameorderastherightelements.

Theeigenvaluesare:

Ω1=αε+O(ε2),

Ω2=(B−βA/α)ε2+O(ε3).

(29)

ThepointisIR-attractiveifΩ1,2>0.Substituting(27),(26)gives:ε>0,B>b/a2.

Thusthecurvey=ε+Bcε2withBc=b/a2istheboundarybetweentheregionsofIR-stabilityforthepointI(ε>0,B0,B>b/a2).Thereisneithergapnoroverlap(atleastinthisapproximation).

Thephysicsrequiresthatthecoordinatesofanyphysicalfixedpoint,u∗andw∗,benon-negative(uanduwareamplitudesinpaircorrelators).OnecancheckthatthisconditionisautomaticallysatisfiedintheregionoftheirIRstability.Forexample,w∗>0in(26)givesB>b/a2,whichisalsoΩ2>0,andsoon.

Theresultingpatternoftheregionsofstabilityofthenontrivialfixedpointsofthemodel(9),(10),showninfigure2,isasfollows:thequadrantε>0,y>0isdividedintotwopartsbytheparabolay=ε+Bcε2withBc=b/a2.ThepartbelowitistheregionofIRstabilityofthepointI(universalityclassoftheequilibriummodelA);itincludesthewholesectorε>0,ε>y.ThepartaboveitistheregionofIRstabilityofthepointII.Itcorrespondstoanewnonequilibriumuniversalityclass,wherethenonlocaltermintherandomforceisimportant.ItisworthnotingthatBcappearsrathersmall:b=ln(4/3)≈0.287683andBc=ln(4/3)/9≈0.032.5.ScalingbehaviourintheIRrange

ExistenceofIR-attractivefixedpointsimpliesscalingbehaviourwithdefinitecriticaldimensions∆FofallquantitiesF(fieldsandparameters):

ω∗

∆F=dkF+∆ωdF+γF,

∗

∆ω=2+γσ,

(30)

ThisistheleadingtermoftheasymptoticbehaviourintheIRrange,determinedbythe

inequalitiesk∼1/r≪Λ,whereΛistheUVmomentumscaledefinedbytherelationu0∼Λy.Fisauniversalscalingfunctionoftwoarguments,whicharesupposedtobeoforderunity;thiscompletesthedefinitionoftheIRrange:τ0r∆τ∼1,t/r∆ω∼1.(Inthefreetheorywewouldhaveτ0r2∼1,t/r2∼1).Thecorrectcanonicaldimensionsin(31)areguaranteedbytheamplitudesbuiltfromIRirrelevantparametersσandΛ,notshownexplicitly.OneusuallyassumesthatFhasfinitelimitsforτ0=0(thatis,

∗

wheredk,ωarethecanonicaldimensionsofF,givenintable1,andγFisthevalueofF

∗

γFatthefixedpointinquestion;seee.g.[21,26].InthecaseathandγF=γF(u∗,w∗).

Inparticular,forthepaircorrelatorofthefieldϕthisgives:

󰀅∆τ󰀆′′−2∆ϕ∆ω

󰀈ϕ(x+r,t+t)ϕ(x,t)󰀉=rFτ0r,t/r.(31)

Effectsofmixingandstirringonthecriticalbehaviour

Table1.Canonicaldimensionsofthefieldsandparameters;d=4−ε.

14

Fϕ

2

ϕ′

−1−1

d

vi

′vi

σ0,σν0,νm,µ,Λτλ0w0g0λ,w,g

2

d

Effectsofmixingandstirringonthecriticalbehaviour15

Toavoidpossiblemisunderstandings,itisworthnotingthatthescalingbehaviourofthemodelAandtheextendedmodel(11)nearitsfixedpointIcoincideonlytotheleadingordersgivenbytheexpressions(31),(32).Thecorrectionstothoseexpressionsaredifferent.Inparticular,theleadingcorrectionfromtheUVrangehastheform󰀇󰀉Ω1Ω2

1+c1(k/Λ)+c2(k/Λ),whereΩ1,2aretheeigenvaluesofmatrixΩforthefixedpointIgivenin(25).OneoftheminvolvestheparameterywhichisabsentinthepuremodelA.

Letusturntothenonlocalregimeoftheextendedmodel(11),describedbythefixedpointII.Substituting(34)and(26)into(33)gives

∆ϕ=1−ε/2+(ε2/2)(1/+b/9−B)+O(ε3),∆ϕ′=3−ε/2+(ε2/2)(−1/+b/9−B)+O(ε3),∆ω=2+(ε/3)2(b−1/6)+O(ε3),∆τ=2−ε/3+O(ε2).

(36)

WerecallthatBcomesfromtherelationy=ε+Bε2andthenonlocalfixedpointisIR-attractivewhenB>b/9≈0.032,see(29).Notethatthedimension∆ω=2+0.0013ε2+O(ε3)appearssurprisinglyclosetoitscanonicalvalue∆ω=2.

Considertherealcased=3,theny=1+Bandthenoisecorrelatoris1/r3−B.Theexponentη,definedbythesame“equilibrium”relation∆ϕ=d/2−1+η/2,takesontheformη=(1/+b/9−B)≈0.05−B.SoηcanbemadenegativeforreasonableB:inparticular,forB=1andnoisecorrelator1/r2wehaveη≈−1.Inthisrespect,thenonequilibriumsteady-statescalingdiffersfromtheequilibriumcase,describedbyalocalLandau–Ginzburgaction:forthelatter,theexactinequalityη>0canbederivedfromtheunitarityofthecorrespondingpseudo-Euclideanquantumfieldtheory[35].Bearinginmindpossiblecosmologicalapplicationofthemodel[30],itistemptingtonotethatη=−1correspondstotheZeldovichspectrum[36].6.Inclusionofthevelocityfield

Letusturntothefullstochasticproblem(2),(9),(10).Thefieldtheoreticactionfunctionalthenbecomes

󰀃󰀄′′′′23

S(Φ)=Sv(v,v)+ϕDϕϕ/2+ϕ−σ0∇tϕ+∂ϕ−τ0ϕ−λ0ϕ/6,(37)

Sv(v,v)=vDvv/2+v

′

′

′

′

where

istheactionfunctionalforthestochasticproblem(2),DϕandDvarethecorrelationfunctionsoftherandomforcesfandfi,respectively,∇t=∂t+vi∂i,andalltherequiredintegrationsoverx={t,x}andsummationsoverthevectorindicesareunderstood.The

′′

newfullsetoffieldsΦ={ϕ,ϕ′,vi,vi}involvestheauxiliaryvectorfieldvi.Itisalso

′

transverse,∂ivi=0,whichallowsonetoomitthepressuretermontheright-handside

󰀃

−∇t+ν0∂

2

󰀄

v(38)

Effectsofmixingandstirringonthecriticalbehaviour16

ofrelation(38).CorrelationfunctionDϕisgivenby(10),whileDvwillbetakenintheform

wherePij(k)=δij−kikj/k2isthetransverseprojector,y′anewarbitraryparameter

′3

analogoustoyfrom(10),g0∼Λyisanewpositivecouplingconstant,andthefactorν0isexplicitlyisolatedforconvenience.TheIRcutoffatk=misalsoimplied.Canonicaldimensionsofallthenewparametersandtheirfuturerenormalizedcounterpartsaregivenintable1.

ThestochasticNavier–Stokesequation(2)withapower-lawnoisespectrumwasintroducedalongago[23,24,25]andisbynowverywellstudied,atleastforsmallvaluesofy′.Thetwo-loopresultshavebeenderivedrecentlyin[37].DetailedexpositionoftheRGapproachcanbefoundin[21,26];belowweconfineourselvestoonlythenecessaryinformation.

Themodel(38)islogarithmic(thecouplingconstantg0isdimensionless)aty′=0,andtheUVdivergenceshavetheformofthepolesiny′inthecorrelationfunctionsofthefieldsv,v′.Dimensionalanalysis,augmentedbysomeadditionalconsiderations(Galileansymmetryandstructureofthevertex),showsthatforalld>2,thesuperficialUVdivergences,whoseremovalrequirescounterterms,arepresentonlyinthe1-irreduciblefunction󰀈v′v󰀉,andthecorrespondingcountertermreducestotheformv′∂2v.Owingtotheformofthevertex(thederivativecanalwaysbemovedontov′usingintegrationbyparts),thedivergenceinthefunction󰀈v′v′󰀉(allowedbydimensionford<4)isinfactabsentforalld>2.Sothelocalcountertermv′v′,analogoustoϕ′ϕ′in(11),isnotneededhere.Forthisreason,wedidnotincludetheconstantcontributionto(39),incontrasttoitsscalarcounterpart(10).ThenforthecompleteeliminationoftheUVdivergencesitissufficienttoperformthemultiplicativerenormalizationoftheparametersν0andg0withtheonlyindependentrenormalizationconstantZν:

ν0=νZν,

g0=gµyZg,

′

3

󰀈fi(x)fj(x′)󰀉=g0ν0δ(t−t′)k4−d−yPij(k),

′

(39)

−3

Zg=Zν.

(40)

HereµisthereferencemassintheMSscheme,gandνarerenormalizedanalogsofthe

bareparametersg0andν0,andZ=Z(g,y′,d)aretherenormalizationconstants.Incontrasttothemodel(11),norenormalizationofthefieldsisneeded,Zv.v′=1.TherelationbetweentheZ’sin(40)resultsfromtheabsenceofrenormalizationofthenoisetermin(38).NowthestandardRGequationsarereadilyderived,thecorresponding

󰀌µgintheone-loopapproximationisfunctionβg=D

󰀊

3(d−1)Sdg

βg(g)=g(−y′+3γν)=g−y′+

Effectsofmixingandstirringonthecriticalbehaviour17

∗

γνatthenontrivialfixedpointisfoundexactly:γν=y′/3(nocorrectionsoforder(y′)2andhigher).Asaresult,criticaldimensionsofthefrequencyandthefieldsarealsofoundexactly:

∆ω=2−y′/3,∆v=1−y′/3,∆v′=d−1+y′/3.

(42)

Theseresultsremainintactinthefullmodel(37),becausetheinclusionofthescalarfieldsdoesnotaffectthevelocity(thefieldϕis“passive”).Wethusconcludethatfory′<0,wheretheIRbehaviourofthevelocitybecomesGaussian,thescalingregimesofthefullproblem(37)aredescribedbythefixedpointsIandIIfromsection4dependingontherelationbetweenyandε=4−d.

Fory′>0thecontributionsofthevelocityfieldbecomeIRrelevant,andtheRGanalysisofthefullproblem(37),(38)isneeded.Ford>4,wherenolocalcountertermϕ′ϕ′isrequired,themodelinquestionisformallyequivalenttothestochasticmodeloftheturbulentadvectionofachemicallyactivescalarfield,studiedearlierin[38].Wewillbeinterestedinthecased<4,wherethelocalcountertermshouldbeincludedfromtheverybeginningtoensuremultiplicativerenormalizability.ThentheUVdivergencesmanifestthemselvesaspolesinthefullsetofregulatorsy,y′andε=4−d.Dimensionalanalysisandsymmetryconsiderationsshowthat,foralld>2,thefullmodelismultiplicativelyrenormalizable,andthecorrespondingrenormalizedactionhastheform

󰀃󰀄

SR(Φ)=SvR(v′,v)+ϕ′wµy−εkε−y+Z1ϕ′/2

󰀃󰀄+ϕ′−Z2σ∇tϕ+Z3∂2ϕ−τZ5ϕ−Z4λµεϕ3/6,(43)whereSvR(v′,v)istherenormalizedanalogoftheaction(38)expressedinrenormalized

variablesusingrelation(40).TherenormalizationconstantsZ1–Z5nowcontainthepolesinthefullsetofregulatorsy,y′,ε=4−dand,incomparisonto(11),dependontwoadditionaldimensionlesscouplingsgands=σν.Onecaneasilyseethattherelations(12),(13)forthenewconstantsZFand(18),(19)forthecorrespondinganomalousdimensionsγFremainvalidintheextendedrenormalizedmodel(43).TheRGequationtakesontheformwhereDRG

{DRG+Nϕγϕ+Nϕ′γϕ′+Nvγv+Nv′γv′}GR(e,µ,...)=0,

󰀌µexpressedintherenormalizedvariables:istheoperationD

DRG≡Dµ+βg∂g+βu∂u+βw∂w+βs∂s−γνDν−γτDτ,βs=−sγs=−s[γν+γσ]=−s[γν+γ2−γ3].

(44)(45)(46)

withβgfrom(41),βu,wfrom(17),andthenewfunctionβsis

Furthermore,itiseasilycheckedthatintheleadingorder,theanomalousdimensions

γ1,2,4,5remainthesameasinthemodelwithoutvelocityandcanbetakenfrom(34).Onthecontrary,γ3acquiresanadditionalO(g)term,incomparisontowhichtheO(u2)termin(34)isonlyacorrectionandshouldbeneglected.Sotheleading-orderexpressionforγ3becomes

3gs2

γ3=

Effectsofmixingandstirringonthecriticalbehaviour18

Asusual,scalingregimesofthefullmodelareassociatedwiththeIRattractivefixedpoints,whosecoordinatesarefoundfromtheequationsβx=0withx={g,s,u,w}.Duetopassivityofthefieldϕ,thefunctionβgisindependentofx={s,u,w}andthecorrespondingelements∂βg/∂xofthematrixΩvanish.ThusΩistriangular,itselements∂βx/∂gwithx={s,u,w}donotaffecttheeigenvalues,andwecansetg=g∗inthefunctionsβxwithx=gfromtheverybeginning.Nowwearetreatingε=4−dasoneofthesmallexpansionparameters,andintheleading-orderapproximationweshouldsetd=4in(41)andg∗.Intheleadingapproximation,thefunctionβsisindependentofuandw,andtheelements∂βs/∂xwithx={u,w}alsovanish.Theelement∂βs/∂scoincideswithoneoftheeigenvaluesofΩ.Itiseasytoseethat,fory′>0,thiseigenvalue∂βs/∂scannotbepositivefors∗=0,sothattheconditionβs=0impliesγν+γ2−γ3=0,whichinourapproximationgives3s2−s∗−1=0.Physical

√∗

considerationsrequires∗>0,whichfinallygivess∗=(1+

Effectsofmixingandstirringonthecriticalbehaviourinagreementwith(42).CombiningEqs.(19)and(50)gives

∗∗∗∗∗∗

2γϕ′=γ3+γ1−γ2=γν+γ1,∗∗∗∗∗∗∗2γϕ=γ3+γ2−γ1=γν+2γ2−γ1,

19

(52)

∗

which,alongwith(51),allowsonetodeterminethedimensions∆ϕ=(d/2−1)+γϕ

∗

and∆ϕ′=(d/2−1)+∆ω+γϕ′inthesecondorderoftheεexpansionwithoutpracticalcalculationofthetwo-loopcorrectionstoγ3andv∗:

∆ϕ=1−ε/2+y′/6+bε¯2/18,

forthefixedpointIIIand

¯2/18∆ϕ′=3−ε/2−y′/6+bε

(53)

¯ε∆ϕ=1−ε/2+y′/6+bε¯2/9−B¯2/2,¯ε¯2/2()∆ϕ′=3−ε/2−y′/6+B

forthefixedpointIV,withcubic-in-εcorrections.(Werecallthatincountingtheorders

weimplyε∼ε¯∼y′,ε−y∼ε2).Infact,theexpressionfor∆ϕ′in()holdstoall

∗

orders(nocorrectionsoforderε3andhigher),becauseγ1=y−ε=Bε2exactly,asaconsequenceoftherelationsβw=0andw∗=0.Finally,forthebothpointsIIIandIVoneobtains∆τ=2+ε/3+y′/9,withquadraticcorrections.

Forε¯<0,theself-interactionofthescalarfieldbecomesirrelevant(u∗=0),andweobtaintwomorefixedpoints,whichcorrespondtothescalarfieldsubjecttoalineardiffusion-advectionequation,withthevelocityensemblegivenbytheactionSv(v′,v)from(38).Theregionsε¯>y¯andε¯ThelinearpassivescalarcaseisverywellunderstoodwithintheRGframework(seee.g.chapter2inthebook[26]andreferencestherein).Theonlysuperficiallydivergent1-irreduciblecorrelationfunctionis󰀈ϕ′ϕ󰀉withthecountertermϕ′∂2ϕ,and

−1

thecorrespondingrenormalizationconstant(inournotationidentifiedwithZϕ=Zσ)isindependentoftheformofthenoisecorrelator(thelatteronlydeterminesthecanonicaldimensions).TheabsenceofrenormalizationofthenoisetermresultsintheexactrelationZϕ′=1(itisimpliedthattheamplitudeisscaledoutfromtheaction).Alongwiththerelations(50)and(51),whichremainvalidforthepassivelinearcase,thisgivesexactresultsforthedimensions:

∆ϕ=d/2−1+y′/3,

∆ϕ′=d/2+1−y′/3,

∆τ=2,

(55)

whichinthestandard“equilibrium”notationcorrespondstoη=y′/3andν=1/2,differentfromtheircounterpartsforthestandardmodelA.7.Conclusion

Wehavestudiedstochasticmodelthatdescribesdynamicsofanonconservedscalarfield(orderparameter)nearitscriticalpoint,subjecttorandomexternalstirringandmixing,indspatialdimensions.ThestirringwasmodelledbyanadditiverandomGaussiannoise

Effectsofmixingandstirringonthecriticalbehaviour20

withthepaircorrelationfunction∝δ(t−t′)k4−d−y.Themixingwasmodelledbytheconvectiontermwithadivergence-free(duetotheincompressibilitycondition)velocityfield,governedbythestochasticNavier–StokesequationwitharandomGaussianforce

′

withpaircorrelationfunction∝δ(t−t′)k4−d−y.PossiblescalingregimesofthemodelareassociatedwithnontrivialIRattractivefixedpointsofthecorrespondingRGequations.Theircoordinates,regionsofstability,andthecorrespondingcriticaldimensionscanbecalculatedwithinasystematicexpansioniny,y′andε=4−d(oronlyyandε=4−dforthemodelwithoutvelocity)withtheadditionalassumptionthaty−ε=O(ε2).Dependingontherelationsbetweenthoseparameters,themodelrevealsseveraltypesofscalingregimes.Someofthemarewellknown:modelAofequilibriumcriticaldynamicsandlinearpassivescalarfieldadvectedbyarandomturbulentflow,buttherearethreenewnonequilibriumuniversalityclasses,associatedwithnewnontrivialfixedpoints.Inthissense,thecriticalbehaviourofthemodelappearsricherandlessuniversalthanthatoftheequilibriumcriticaldynamics.

Thecriticalexponents(dimensions)forthenewuniversalityclassesarederivedinthesecondorderoftheexpansioniny,y′andε(two-loopapproximation).

Itremainstonotethatthelarge-scalemixing(y=y′=4)inthreedimensions(ε=1)belongstotheuniversalityclassofthelinearpassivescalarwiththenonlocalnoisecorrelatorandthereforecorrespondstothedimensions(55).Ofcourse,theresultsofourperturbativeRGanalysisareabsolutelyreliableandinternallyconsistentonlyforsmallvaluesoftheexpansionparametersε,yandy′,whilethepossibilityoftheirnaiveextrapolationtofinite(andnotsmall)realvaluesisfarfromobvious.Ontheotherhand,theobservationthattheϕ4-interactionbecomesirrelevantforthelarge-scaleforcingisreminiscentoftheresultsderivedin[13,14].There,itwasarguedthatanon-randomshearflowstronglysuppressescriticalfluctuations,andthebehaviourofthesystembecomesclosetomeanfieldinthestrongshearlimit;seealsodiscussionin[15].

OuranalysiscanbedirectlygeneralizedtothecasesofaN-componentorderparameter,presenceofanisotropy,compressibilityetc.Thegeneralizationsarestraightforwardbutrathercumbersome(forthestochasticNavier–Stokesequation,seee.g.chapter3inthebook[26]andreferencestherein).Onthecontrary,thecaseofaconservedorderparameterappearsratherdifferentfromboththeconceptualandtechnicalviewpoints(namely,itinvolvestwodifferentdispersionlaws:ω∼k2forthevelocityandω∼k4forthescalar).Theseissueswillbeaddressedelsewhere.Acknowledgments

TheauthorsthankLTsAdzhemyan,MassimoCencini,PaoloMuratoreGinanneschi,FilippoVernizzi,AngeloVulpianiandANVasil’evfordiscussions.NVAwassupportedinpartbytheRFFIgrantno05-02-17524andtheRNPgrantno2.1.1.1112.MHwassupportedinpartbytheVEGAgrant6193ofSlovakAcademyofSciences,byScienceandTechnologyAssistanceAgencyundercontractNoAPVT-51-027904.NVAand

Effectsofmixingandstirringonthecriticalbehaviour21

MHthanktheDepartmentofPhysicalSciencesintheUniversityofHelsinkiandtheNNBogoliubovLaboratoryofTheoreticalPhysicsintheJointInstituteforNuclearResearch(Dubna)fortheirkindhospitality.NVAthankstheDepartmentofMathematicsintheUniversityofHelsinkifortheirkindhospitalityduringhisvisits,financedbytheproject“ExtendedDynamicalSystems.”

AppendixA.Calculationoftherenormalizationconstants

ConsiderasanexamplethecalculationoftheconstantZ1andtheanomalousdimensionγ1forthemodel(10)withoutthevelocityfieldindetail.Theleadingcontributionhereisgivenbyatwo-loopFeynmangraph,sothisexampleisrepresentative:calculationoftheotherrenormalizationconstants(includingthevelocityfield)canbeperformedinasimilarway(fortwo-loopcontributions)orismucheasier(forone-loopgraphs).

The1-irreduciblefunctionΓ≡󰀈ϕ′ϕ′󰀉intherenormalizedcritical(τ=0)theorytoorderO(λ2)hastheform

󰀃󰀄λ2µ2ε

y−ε

Γ=2σw(k/µ)+Z1+

ω2σ2+k4

infrequency-momentumrepresentationand

󰀃󰀄1

󰀈ϕϕ󰀉0=w(k/µ)y−ε+1

(A.2)

Effectsofmixingandstirringonthecriticalbehaviour22

impossibleforthecaseτ≤0,whichwearemostlyinterestedinhere.TheadequatelanguageisthenprovidedbytheLegendretransform(effectiveaction)anduseoftheloopexpansionorthe1/Nexpansioninsteadoftheprimitiveperturbationtheory.Thisisnotconvenient,however,forthepracticalcalculationoftherenormalizationconstants.Fortunately,intheMSschemethecountertermsarepolynomialinIRregulators,andtheresultsobtainedforthemintheregionτ>0canbedirectlyusedforτ≤0;seee.g.thediscussionsection3.36in[21].Furthermore,theconstantsZareindependentonthespecificchoiceoftheIRregularization.Fromthecalculationalviewpoints,itismoreconvenienttosetτ=0intheaction(andinthepropagator(A.3))andcutoffthemomentumintegralsatk=m(bydimension,τ∼m2).Integralsoverfrequencies(ortimes)areelementary,andoneobtains:

󰀍

D(k)D(q)D(|k+q|)dk

Diagram=2σ

(2π)d

k>m

(2π)d

q>m

󰀍

dq

(2π)2d

m

󰀍∞

dk

q1+ε

󰀈󰀈

k2q2kα1qα2|k+q|α3

(2ε−α123)

DmI(m)|m=1.

(A.10)

Herethepoleisisolatedexplicitly.TheexpressionDmI(m)|m=1isfiniteatε=y=0,andwecansetε=y=0init.Thenalltheseintegralsbecomeequal(allαialsobecome0).

Effectsofmixingandstirringonthecriticalbehaviour23

Thefactorsm−2ε+α123willformdimensionlessratioslike(m/µ)O(ε)or(m/µ)O(y)withtheµ-dependentfactorsinexpressions(A.1)and(A.6).Sinceweareinterestedonlyinthepolepartswewillreplacesuchratiosbyunities.ThuswehavetocalculateSincemappearsinI(m)onlyinthelowerlimitsofintegration,thedifferentiationgives

󰀍∞

q

R=dq󰀈󰀈

1

R=−DmI(m)|m=1,ε=y=0.

(A.11)

π

󰀍

π

sin2ϑ....

0

CalculatingtheresultingdoubleintegralgivesR=(3/2)ln(4/3).

Nowconsiderthetotalcofactorwhichcontainsthepolesinεandy.Itcomesfromthedenominatorsin(A.12)andhastheform:

󰀊󰀎13w

P=.(A.13)+

2ε−2(ε−y)2ε−3(ε−y)ThenconstantZ1whichcancelsthepolesin(A.1)willhavetheform

Z1=1−

λ2

6

Thisfinallygives

γ1=

λ2

RP.

Effectsofmixingandstirringonthecriticalbehaviour24

Inthesamemannerwecanderivetheotherresultsgivenin(34).Forγ2,3theintegralisquadraticanditshouldbeexpandedintheexternalfrequencyandmomentumtoΩandp2;thecoefficientswillbelogarithmicintegralsofthetype(A.5)andwecanproceedasbeforefor(A.7).Forγ4,5thisismuchsimplerbecausethediagramsareone-loopones,theyarelogarithmic,thereisonlyonemomentumkandthetrickinvolvingthedifferentiationDmisnotneeded.References

[1][2][3][4][5]

BrayAJ1994Adv.Phys.43357

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