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April 18, 2008 16:58 WSPC - Proceedings Trim Size: 9in x 6in qmath10 1 Repeated Interaction Quantum Systems: Deterministic and Random Alain Joye Institut Fourier Universite de Grenoble BP 74 38402 Saint Martin d'Heres, France This paper gives an overview of recent results concerning the long time dy- namics of repeated interaction quantum systems in a deterministic and ran- dom framework. We describe the non equilibrium steady states (NESS) such systems display and we present, as a macroscopic consequence, a second law of thermodynamics these NESS give rise to. We also explain in some details the analysis of products of certain random matrices underlying this dynamical problem. Keywords: Non equilibrium quantum statistical mechanics, Repeated interac- tion quantum systems, Products of random matrices 1. Introduction and Model A repeated interaction quantum system consists of a reference quantum subsystem S which interacts successively with the elements Em of a chain C = E1+E2+· · · of independent quantum systems. At each moment in time, S interacts precisely with one Em (m increases as time does), while the other elements in the chain evolve freely according to their intrinsic dynamics. The complete evolution is described by the intrinsic dynamics of S and of all the Em, plus an interaction between S and Em, for each m. The latter is characterized by an interaction time ?m > 0, and an interaction operator Vm (acting on S and Em); during the time interval [?1+· · ·+?m?1, ?1+· · ·

  • constant interaction

  • large-time asymptotics

  • interaction quantum

  • dynamics

  • ideal repeated

  • any master

  • interaction operators


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April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath10RepeatedInteractionQuantumSystems:DeterministicandRandomAlainJoyeInstitutFourierUniversite´deGrenoble47PB38402SaintMartind’He`res,FranceThispapergivesanoverviewofrecentresultsconcerningthelongtimedy-namicsofrepeatedinteractionquantumsystemsinadeterministicandran-domframework.Wedescribethenonequilibriumsteadystates(NESS)suchsystemsdisplayandwepresent,asamacroscopicconsequence,asecondlawofthermodynamicstheseNESSgiveriseto.Wealsoexplaininsomedetailstheanalysisofproductsofcertainrandommatricesunderlyingthisdynamicalproblem.Keywords:Nonequilibriumquantumstatisticalmechanics,Repeatedinterac-tionquantumsystems,Productsofrandommatrices11.IntroductionandModelArepeatedinteractionquantumsystemconsistsofareferencequantumsubsystemSwhichinteractssuccessivelywiththeelementsEmofachainC=E1+E2+∙∙∙ofindependentquantumsystems.Ateachmomentintime,SinteractspreciselywithoneEm(mincreasesastimedoes),whiletheotherelementsinthechainevolvefreelyaccordingtotheirintrinsicdynamics.ThecompleteevolutionisdescribedbytheintrinsicdynamicsofSandofalltheEm,plusaninteractionbetweenSandEm,foreachm.Thelatterischaracterizedbyaninteractiontimeτm>0,andaninteractionoperatorVm(actingonSandEm);duringthetimeinterval[τ1+∙∙∙+τm11+∙∙∙+τm),SiscoupledtoEmonlyviaVm.Systemswiththisstructureareimportantfromaphysicalpointofview,sincetheyarisenaturallyasmodelsforfunda-mentalexperimentsontheinteractionofmatterwithquantizedradiation.Asanexample,the“Oneatommaser”providesanexperimentalsetupinwhichthesystemSrepresentsamodeoftheelectromagneticfield,whereastheelementsEkdescribeatomsinjectedinthecavity,onebyone,which
April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath102interactwiththefieldduringtheirflightinthecavity.Aftertheyleavethecavity,theatomsencodesomepropertiesofthefieldwhichcanbemea-suredontheseatoms14,16Forrepeatedinteractionsystemsconsideredasideal,i.e.suchthatallatomsareidenticalwithidenticalinteractionsandtimesofflightthroughthecavity,correspondingmathematicalanalysesareprovidedin17,7Totakeintoaccounttheunavoidablefluctuationsintheexperimentsetupusedtostudytheserepeatedinteractionsystems,mod-elizationsincorporatingrandomnesshavebeenproposedandstudiedin8and.9Withadifferentperspective,repeatedquantuminteractionmodelsalsoappearnaturallyinthemathematicalstudyofmodelizationofopenquantumsystemsbymeansofquantumnoises,see4andreferencestherein.Any(continuous)masterequationgoverningthedynamicsofstatesonasystemScanbeviewedastheprojectionofaunitaryevolutiondrivingthesystemSandafieldofquantumnoisesininteraction.Itisshownin4howtorecoversuchcontinuousmodelsassomedelicatelimitofadiscretizationgivenbyarepeatedquantuminteractionmodel.Letusfinallymention15forresultsofasimilarflavourinasomewhatdifferentframework.Ourgoalistopresenttheresultsofthepapers7,8and9on(random)repeatedinteractionquantumsystems,whichfocusonthelongtimebe-haviourofthesesystems.Letusdescribethemathematicalframeworkusedtodescribethesequantumdynamicalsystems.Accordingtothefundamentalprinciplesofquantummechanics,statesofthesystemsSandEmaregivenbynormal-izedvectors(ordensitymatrices)onHilbertspacesHSandHEm,respec-tively,3,6a.WeassumethatdimHS<,whiledimHEmmaybeinfinite.ObservablesASandAEmofthesystemsSandEmareboundedopera-torsformingvonNeumannalgebrasMS⊂B(HS)andMEm⊂B(HEm).TheyevolveaccordingtotheHeisenbergdynamicsR3t7→αtS(AS)andtttR3t7→αEm(AEm),whereαSandαEmare-automorphismgroupsofMSandMEm,respectively,seee.g.6Wenowintroducedistinguishedreferencestates,givenbyvectorsψS∈HSandψEm∈HEm.TypicalchoicesforψS,ψEmareequilibrium(KMS)statesforthedynamicsαtS,αtEm,atinversetemperaturesβS,βEm.TheHilbertspaceofstatesofthetotalsystemisthetensorproductH=HS⊗HC,aAnormalizedvectorψdefinesa“pure”stateA7→hψ,Aψi=TrP(%ψA),where%ψ=|ψihψ|.Ageneral“mixed”stateisgivenbyadensitymatrix%=n1pn%ψn,wheretheprobabilitiespn0sumuptoone,andwheretheψnarenormalizedvectors.
April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath103NwhereHC=mN1HEm,andwheretheinfiniteproductistakenwithrespecttoψC=m1ψEm.Thenon-interactingdynamicsistheprod-Nuctoftheindividualdynamics,definedonthealgebraMSm1MEmNbyαtSm1αtEm.ItwillproveusefultoconsiderthedynamicsintheSchro¨dingerpicture,i.e.asactingonvectorsinH.Todothis,wefirstimplementthedynamicsviaunitaries,satisfyingαt#(A#)=eitL#A#eitL#,tR,andL#ψ#=0,(1)foranyA#M#,where#standsforeitherSorEm.Theself-adjointoperatorsLSandLEm,calledLiouvilleoperatorsor“positivetemperatureHamiltonians”,actonHSandHEm,respectively.Theexistenceandunique-nessofL#satisfying(1)iswellknown,undergeneralassumptionsonthereferencestatesψ#.6Werequirethesestatestobecyclicandseparating.Inparticular,(1)holdsifthereferencestatesareequilibriumstates.Letτm>0andVmMSMEmbetheinteractiontimeandinteractionoper-atorassociatedtoSandEm.Wedefinethe(discrete)repeatedinteractionSchro¨dingerdynamicsofastatevectorφ∈H,form0,byU(m)φ=eiLem∙∙∙eiLe2eiLe1φ,(2)erehwXLek=τkLk+τkLEn(3)k=6ndescribesthedynamicsofthesystemduringthetimeinterval[τ1+∙∙∙+τk11+∙∙∙+τk),whichcorrespondstothetime-stepkofourdiscreteprocess.HenceLkisLk=LS+LEk+Vk,(4)actingonHS⊗HEk.WeunderstandthattheoperatorLEnin(3)actsnontriviallyonlyonthen-thfactoroftheHilbertspaceHCofthechain.Asageneralrule,wewillignoretensorproductswiththeidentityoperatorinthenotation.Astate%()=Tr(ρ)givenbydensitymatrixρonHiscalledanormalstate.Ourgoalistounderstandthelarge-timeasymptotics(m→∞)ofexpectations%(U(m)OU(m))=%(αm(O)),(5)fornormalstates%andcertainclassesofobservablesOthatwespecifybelow.Wedenotethe(random)repeatedinteractiondynamicsbyαm(O)=U(m)OU(m).(6)
April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath1041.1.VanHoveLimitTypeResultsAfirststeptowardsunderstandingthedynamicsofrepeatedinteractionquantumsystemsreducedtothereferencesystemSwasperformedinthework.2ThispaperconsidersIdealRepeatedQuantumInteractionSys-temswhicharecharacterizedbyidenticalelementsEk≡EinthechainC,constantinteractiontimesτkτandidenticalinteractionoperatorsVkVMSMEbetweenSandtheelementsEofthechain.Inthissetup,aVanHovetypeanalysisofthesystemispresented,insev-eralregimes,todescribethedynamicsofobservablesonSintermsofaMarkovianevolutionequationofLindbladtype.Informally,thesimplestresultof2readsasfollows.AssumetheinteractionoperatorVisreplacedbyλV,whereλ>0isacouplingconstant,andletm,thenumberofinter-actionsduringthetimeT=,scalelikem't/λ2,where0t<andτarefixed.Assumeallelementsofthechainareinasamethermalstateattemperatureβ.Then,theweakcouplinglimitλ0oftheevolutionofanyobservableOactingonSobtainedbytracingoutthechaindegreesoffreedomfromtheevolution(6)satisfies,afterremovingatrivialfreeevolu-tion,acontinuousLindbladtypeevolutionequationint.ThetemperaturedependentgeneratorisexplicitelyobtainedfromtheinteractionoperatorVandthefreedynamics.Theasymptoticregimesoftheparameters(λ,τ)characterizedbyτ0andτλ21arealsocoveredin,2givingrisetodif-ferentLindbladgeneratorswhichallcommutewiththefreeHamiltonianonS.Thecriticalsituation,whereτ0withτλ2=1yieldsaquitegeneralLindbladgenerator,withoutanyspecificsymmetry.Inparticular,itshowsthatanymasterequationdrivenbyLindbladoperator,underreasonableassumptions,canbeviewedasaVanHovetypelimitofacertainexplicitrepeatedinteractionquantumsystem.Bycontrast,thelongtimelimitresultsobtainedin7,8and9thatwepresenthereareobtainedwithoutrescalinganycouplingconstantorpa-rameter,asisusuallythecasewithmasterequationtechniques.Itispos-sibletodowithouttheseapproximations,makinguseofthestructureofrepeatedinteractionsystemsonly,aswenowshow.2.ReductiontoProductsofMatricesWefirstlinkthestudyofthedynamicstothatofaproductofreduceddy-namicsoperators.Inordertomaketheargumentclearer,weonlyconsidertheexpectationofanobservableASMS,andwetaketheinitialstateof
April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath105theentiresystemtobegivenbythevectorψ0=ψSψC,(7)wheretheψSandψCarethereferencestatesintroducedabove.We’llcom-mentonthegeneralcasebelow.TheexpectationofASatthetime-stepmsiEDhψ0m(AS)ψ0i=ψ0,PeiLe1∙∙∙eiLemASeiLem∙∙∙eiLe10,(8)whereweintroducedOP=1lHSPψEm,(9)1mtheorthogonalprojectionontoHSCψC.AfirstimportantingredientinouranalysisistheuseofC-Liouvilleansintroducedin11,whichareoperatorsKkdefinedbythepropertieseiLekAeiLek=eiKkAeiKk,(10)KkψSψC=0,(11)whereAin(10)isanyobservableofthetotalsystem.Theidentity(10)meansthattheoperatorsKkimplementthesamedynamicsastheLekwhereasrelation(11)selectsauniquegeneratorofthedynamicsamongalloperatorswhichsatisfy(10).TheexistenceofoperatorsKksatisfying(10)and(11)isrootedtotheTomita-TakesakitheoryofvonNeumannalgebras,c.f.11andreferencestherein.ItturnsoutthattheKkarenon-normaloperatorsonH,whiletheLekareself-adjoint.Combining(10)with(8)wecanwrite hψ0m(AS)ψ0i=ψ0,PeiK1∙∙∙eiKmPASψ0.(12)Asecondimportantingredientofourapproachistorealizethattheinde-pendenceofthesub-systemsEmimpliestherelationPeiK1∙∙∙eiKmP=PeiK1P∙∙∙PeiKmP.(13)IdentifyingPeiKkPwithanoperatorMkonHS,wethusobtainfrom(12)and(13),hψ0m(AS)ψ0i=hψS,M1∙∙∙MmASψSi.(14)Itfollowsfrom(11)thatMkψS=ψS,forallk,and,becausetheoperatorsMk=PeiKkPimplementaunitarydynamics,weshow(Lemma4.1)thattheMkarealwayscontractionsforsomesuitablenorm||||||onCd.Thismotivatesthefollowing
April18,200816:586WSPC-ProceedingsTrimSize:9inx6inqmath10Definition:GivenavectorψSCdandanormon||||||onCd,wecallReducedDynamicsOperatoranymatrixwhichisacontractionfor||||||andleavesψSinvariant.Remark:IncaseallcouplingsbetweenSandEkareabsent,Vk0,Mk=ekLSisunitaryandadmits1asadegenerateeigenvalue.Wewillcomebackonthepropertiesofreduceddynamicsoperators(RDO’s,forshort)below.LetusemphasizeherethatthereductionprocesstoproductofRDO’sisfreefromanyapproximation,sothatthesetofmatrices{Mk=PeiKkP}kNencodesthecompletedynamics.Inparticu-lar,weshow,usingthecyclicityandseparabilityofthereferencevectorsψSEk,thattheevolutionofanynormalstate,notonlyhψ0,ψ0i,canbeunderstoodcompletelyintermsoftheproductoftheseRDO’s.Wearenowinapositiontostateourmainresultsconcerningtheasymp-toticdynamicsofnormalstates%actingoncertainobservablesO.TheseresultinvolveaspectralhypothesiswhichweintroduceinthenextDefinition:LetM(E)denotethesetofreduceddynamicsoperatorswhosespectrumσ(M)satisfiesσ(M)∩{zC||z|=1}={1}and1issimpleeigenvalue.WeshalldenotebyP1,MthespectralprojectorofamatrixMcorre-spondingtotheeigenvalue1.Asusual,iftheeigenvalue1issimple,withcorrespondingnormalizedeigenvectorψS,weshallwriteP1,M=|ψSihψ|forsomeψs.t.hψ|ψSi=1.3.Results3.1.IdealRepeatedInteractionQuantumSystemWeconsiderfirstthecaseofIdealRepeatedInteractionQuantumSystems,characterizedbyEk=E,LEk=LE,Vk=V,τk=τforallk1,Mk=M,k1.(15)Theorem3.1.LetαnbetherepeatedinteractiondynamicsdeterminedbyoneRDOM.SupposethatM∈M(E)sothatP1,M=|ψSihψ|.Then,forany0<γ<infzσ(M)\{1}(1−|z|),anynormalstate%,andanyASMS,%(αn(AS))=hψ,ASψSi+O(eγn).(16)
April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath107Remarks:1.Theasymptoticstatehψ|∙ψSiandtheexponentialdecayrateγarebothdeterminedbythespectralpropertiesoftheRDOM.2.Onconcreteexamples,theverificationofthespectralassumptiononMcanbedonebyrigorousperturbationtheory,see.7ItisreminiscentofaFermiGoldenRuletypeconditionontheefficiencyofthecouplingV,seetheremarkfollowingthedefinitonofRDO’s.3.Otherpropertiesofidealrepeatedinteractionquantumsystemsarediscussedin7,e.g.continuoustimeevolutionandcorrelations.Fordeterministicsystemswhicharenotideal,thequantity%(αn(AS))keepsfluctuatingasnincreases,which,ingeneral,forbidsconvergence,seeProposition5.3.That’swhyweresorttoergodiclimitsinarandomsetup,aswenowexplain.3.2.RandomRepeatedInteractionQuantumSystemToallowadescriptionoftheeffectsoffluctuationsonthedynamicsofrepeatedinteractionquantumsystems,weconsiderthefollowingsetup.Letω7→M(ω)bearandommatrixvaluedvariableonCddefinedonaprobabilityspace(Ω,F,p).WesaythatM(ω)isarandomreduceddynamicsoperator(RRDO)if(i)Thereexistsanorm||||||onCdsuchthat,forallω,M(ω)isacontractiononCdforthenorm||||||.(ii)ThereexistsavectorψS,constantinω,suchthatM(ω)ψS=ψS,forallω.ToanRRDOω7→M(ω)onΩisnaturallyassociatedaiidrandomreduceddynamicsprocess(RRDP)ω7→M(ω1)∙∙∙M(ωn)ΩN,(17)wherewedefineinastandardfashionaprobabilitymeasuredPonΩNbydPj1dpj,wheredpjdp,jN.WeshallwritetheexpectationofanyrandomvariablefasE[f].Letusdenotebyαωn,ωΩN,theprocessobtainedfrom(6),(14),wheretheMj=M(ωj)in(14)areiidrandommatrices.WecallαωntherandomrepeatedinteractiondynamicsdeterminedbytheRRDOM(ω)=PeiK(ω)P.ItistheindependenceofthesuccessiveelementsEkofthechainCwhichmotivatestheassumptionthattheprocess(17)beiid.
April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath108Theorem3.2.Letαωnbetherandomrepeatedinteractiondynamicsde-terminedbyanRRDOM(ω).Supposethatp(M(ω)∈M(E))>0.ThenthereexistsasetΩΩN,s.t.P(Ω)=1,ands.t.foranyωΩ,anynormalstate%andanyASMS,N1Xlim%(αωn(AS))=hθ,ASψSi,(18)N→∞Nn=1whereθ=P1,E[M]ψS.Remarks:1.Oursetupallowsustotreatsystemshavingvarioussourcesofrandomness.Forexample,randominteractionsortimesofinteractions,aswellasrandomcharacteristicsofthesystemsEmandSsuchasrandomtemperaturesanddimensionsoftheEmandofS.2.Theasymptoticstatehθ,ψSiisagaindeterminedbythespectraldataofamatrix,theexpectationE[M]oftheRRDOM(ω).Actually,ourhypothesesimplythatE[M]belongstoM(E),seebelow.3.Theexplicitcomputationoftheasymptoticstate,inthisTheoremandinthepreviousone,isingeneraldifficult.Nevertheless,theycanbereachedbyrigorousperturbationtheory,seetheexamplesin7,8and.93.3.InstantaneousObservablesThereareimportantphysicalobservablesthatdescribeexchangeprocessesbetweenSandthechainCand,which,therefore,arenotrepresentedbyoperatorsthatactjustonS.Totakeintoaccountsuchphenomena,weconsiderthesetofobservablesdefinedasfollows.Definition:TheinstantaneousobservablesofS+CareoftheformO=ASjr=lB(mj),(19))j(whereASMSandBmMEm+j.Instantaneousobservablescanbeviewedasatrainofl+r+1observables,roughlycenteredatEm,whichtravelalongthechainCwithtime.FollowingthesamestepsasinSection2,wearriveatthefollowingexpressionfortheevolutionofthestateψ0actingonaninstantaneousobservableOattimem:mhψ0(O)ψ0i=hψ0,PM1∙∙∙Mml1Nm(O)0i.(20)
April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath109Hereagain,PistheorthogonalprojectionontoHS,alongψC.TheoperatorNm(O)actsonHSandhastheexpression(Proposition2.4in9)Nm(O)ψ0=(21)PeiτmlLeml∙∙∙eiτmLem(ASrB(j))eiτmLem∙∙∙eiτmlLemlψ0.ml=jWewanttoanalyzetheasymptoticsm→∞of(20),allowingforran-domnessinthesystem.Wemakethefollowingassumptionsontherandominstantaneousobservable:(R1)TheoperatorsMkareRRDO’s,andwewritethecorrespondingiidrandommatricesMk=M(ωk),k=1,2,∙∙∙,.(R2)TherandomoperatorNm(O)isindependentoftheMkwith1kml1,andithastheformN(ωml,...,ωm+r),whereNr+l+1→B(Cd)isanoperatorvaluedrandomvariable.TheoperatorMkdescribestheeffectoftherandomk-thinteractiononS,asbefore.TherandomvariableNin(R2)doesnotdependonthetimestepm,whichisaconditionontheobservables.Itmeansthatthenatureofthequantitiesmeasuredattimemarethesame.Forinstance,theB(mj)in(19)canrepresenttheenergyofEm+j,orthepartoftheinteractionenergyVm+jbelongingtoEm+j,etc.Bothassumptionsareverifiedinawidevarietyofphysicalsystems:wemaytakerandominteractiontimesτk=τ(ωk),randomcouplingoperatorsVk=V(ωk),randomenergylevelsoftheEkencodedinLEk=LE(ωk),randomtemperaturesβEk=βE(ωk)oftheinitialstatesofEk,andsoon.Theexpectationvalueinanynormalstateofsuchinstantaneousob-servablesreachesanasymptoticvalueintheergodiclimitgiveninthenextTheorem3.3.Supposethatp(M(ω)∈M(E))6=0.ThereexistsasetΩeΩNofprobabilityones.t.foranyωΩe,foranyinstantaneousobservableO,(19),andforanynormalinitialstate%,wehaveµ1Xmµlimµ%αω(O)=hθ,E[N]ψSi,E[N]MS.(22)1=mRemarks1.Theasymptoticstateinwhichonecomputestheexpectation(w.r.ttherandomness)ofNisthesameasinTheorem3.2,withθ=P1,E[M]ψS.2.Incasethesystemisdeterministicandideal,thesameresultholds,droppingtheexpectationontherandomnessandtakingθ=ψ,asinThe-orem3.1,see.7
April18,200816:58WSPC-ProceedingsTrimSize:9inx6inqmath10013.4.EnergyandEntropyFluxesLetusconsidersomemacroscopicpropertiesoftheasymptoticstate.Thesystemsweconsidermaycontainrandomness,butwedropthevariableωfromthenotation.Sincewedealwithopensystems,wecannotspeakaboutitstotalen-ergy;however,variationsintotalenergyareoftenwelldefined.Usinganargumentof7onegetsaformalexpressionforthetotalenergywhichisconstantduringalltime-intervals[τm1m),andwhichundergoesajumpj(m):=αm(Vm+1Vm)(23)attimestepm.Hence,thevPariationofthetotalenergybetweentheinstants0andmisthenΔE(m)=km=1j(k).Therelativeentropyof%withrespectto%0,twonormalstatesonM,isdenotedbyEnt(%|%0).Ourdefinitionofrelativeentropydiffersfromthatgivenin6byasign,sothatinourcase,Ent(%|%0)0.Forathermodynamicinterpretationofentropyanditsrelationtoenergy,weassumeforthenextresultthatψSisa(βStS)–KMSstateonMS,andthattheψEmare(βEmtEm)–KMSstateonMEm,whereβSistheinversetemperatureofS,andβEmarerandominversetemperaturesoftheEmN.Let%0bethestateonMdeterminedbythevectorψ0=ψSψC=ψSmψEm.ThechangeofrelativeentropyisdenotedΔS(m):=Ent(%αm|%0)Ent(%|%0).ThisquantitycanbeexpressedintermsoftheLiouvilleanandinteractionoperatorsbymeansofaformulaprovenin.12Onechecksthatboththeenergyvariationandtheentropyvariationscanbeexpressedasinstantaneousobservables,towhichwecanapplytheresultsofthepreviousSection.Definingtheasymptoticenergyandentropyproductionsbythelimits,iftheyexist,ΔE(mΔS(m)mlim%m=:dE+andmlimm=:dS+,(24)weobtainTheorem3.4(2ndlawofthermodynamics).Let%beanormalstateonM.Then  dE+=θ,EP(LS+VeiτL(LS+V)eiτL)Sa.s.  dS+=θ,EβEP(LS+VeiτL(LS+V)eiτL)Sa.s.Theenergy-andentropyproductionsdE+anddS+areindependentoftheinitialstate%.IfβEisdeterministic,i.e.,ω-independent,thenthesystemsatisfiesthesecondlawofthermodynamics:dS+=βEdE+.
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