Dissipative Time Evolution of Observables in Non-e

arXiv:hep-th/9807145v120Jul1998February1,2008HD-THEP-98-32hep-th/9807145DissipativeTimeEvolutionofObservablesinNon-equilibriumStatisticalQuantumSystemsHerbertNachbagauer1Institutf¨urTheoretischePhysik,Universit¨atHeidelberg,Philosophenweg16,D-69120Heidelberg,GermanyAbstractWediscussdifferential–versusintegral–equationbasedmethodsdescribingout–ofthermalequilibriumsystemsandemphasizetheimportanceofawelldefinedreductiontostatisticalobservables.Applyingtheprojectionoperatorapproach,weinvestigateonthetimeevolutionofexpectationvaluesoflinearandquadraticpolynomialsinpositionandmomentumforastatisticalanhar-monicoscillatorwithquarticpotential.Basedontheexactintegro-differentialequationsofmotion,westudythefirstandnaivesecondorderapproximationwhichbreaksdownatseculartime-scales.Amethodisproposedtoimprovetheexpansionbyanon–perturbativeresummationofallquadraticoperatorcorrelatorsconsistentwithenergyconservationforalltimes.MotioncannotbedescribedbyaneffectiveHamiltonianlocalintimereflectingnon-unitarityofthedissipativeentropygeneratingevolution.Wenumericallyintegratetheconsistentlyimprovedequationsofmotionforlargetimes.Werelateentropytotheuncertaintyproduct,bothbeingexpressibleintermsoftheobservablesunderconsideration.PACSnumbers:05.70.Ln,11.10.Wx,11.15.TkKeywords:Non-equilibriumtimeevolution,projectionoperatormethod,[e-mail]1IntroductionNon-equilibriumaspectsofquantumfieldsystemsareofactualinterest.Amongotherprob-lems,onechallengeisthedescriptionoftimeevolutionofmacroscopicquantitiessuchastheexpectationvalueofthefieldstrengthinsystemsoutofthermalequilibrium.Theproblem,essentiallythebasicquestionformacroscopicdynamicsofstatisticalsystems,andthusinter-estinginitself,alsoplaysaprominentroleincontextwithcosmologicalinflationaryphases.Althoughaplethoraofequivalentdescriptionsofthecollisionlesslimitareavailable,itisdifficulttoelaboratemethodswhichallowtogobeyondthisthermodynamicallytriviallimit.Onepurposeofthispaperistopromoteastrategybasedonwhichtimeevolutioncanbedescribedinaconsistentandsystematicway.Althoughthebasicconceptsofthermodynamicstatisticaltheoryareknownforalongtime,therestillappearstobesomeconfusionaboutterminologyandunderlyingconceptsofnon-equilibriumsystemsandtheirdynamicaldescription.Anothertaskofthisworkistointroduce,reviewanddiscusssomeofthebasicideasinabriefbutsystematicway.Tech-nically,weadopttheprojectionoperatormethodasoneconsistentmethodtodescribetimeevolutionofphysicalquantitiesinstatisticalsystems.Tokeepcomplexityminimal,weconsiderthetoymodelofthequantumDuffingoscillatorforpracticalcalculations,whichneverthelessisasystemofsufficientcomplexitytoexhibitimportantfeaturesofnon-equilibriumthermodynamics.Forsystematicreasons,wediscusssomepropertiesofthatsysteminthestatisticallytrivial,entropyconserving,approximationfirst.Wepresentcriteriaforpermissibleinitialdata,studystaticsolutionsandtheirstability,constructthefirstorderintegralsofmotionandthefirstordereffectiveHamiltonianandLagrangian.Thesystemexhibitsphenomenaofparametricresonancewhichbestmaybedisplayedinthesmallcouplinglimit.Finally,weattackourmaintaskandinvestigateonpropernon-equilibriumfeatures.Inafirststep,thatrequiresthestudyofsecondordereffectswhich,however,turnouttoviolateenergyconservationatlongtimescalesinthestrictpower–seriesexpansion.Themainresultofthisinvestigationistoimprovethesecondorderresultbytakingintoaccountthenon–unitaryeffectiveevolutionofobservableswhichrenderstheapproximationschemeself–consistentandsolvestheproblemofnon–conservationofconstantsofmotion.Intheappendix,weexplicitlyderivethebasicuncertainty–entropyrelationforthesetofquadraticobservables.2Definitionsandgeneralproperties(i)Amixedstateofaquantumsystem(configuration),bothinzerodimensionalquantummechanicsaswellasinquantumfieldtheory,isdescribedbyadensityoperator,which,inordertoallowaprobabilityinterpretation,mustbeahermitiantrace-classoperatorwithpositiveeigenvalues.Itisanintrinsicfeatureofquantumsystemsthatthedensitymatrixisfictiveinthesensethatonlyitsdiagonalelementscorrespondtophysicalprobabilities,theotherdependenciesbeingphaseswhichenterinexpectationvaluesviainterferenceeffects2.Alternatively,onemaycharacterizeaconfigurationbytheexpectationvaluesofhermitianoperators.Agenericsetofthoserepresentingacompletesetofobservablesisgivenbythemutuallyorthogonalhermitianprojectorsconstructedfromtheeigenvectorsofthedensitymatrix.Acompletesetof(notnecessarilycommuting)observablescontainsallinformation2Thatisthefundamentaldifferencetoclassicalphasespaceaverages.2aboutthedensitymatrixsuchthatanyobservablecanbeexpressedintermsofthecompleteset.(ii)Thesystemmaybeassignedadynamicalstructure.MotionisdefinedasasequenceofpossiblestateshavingaconstantexpectationvalueoftheHamiltonianoperator,theenergyoftheconfiguration.Quantummechanicaltimeparameterizesthoseconfigurationswhichareassumedtohavetime–independentprobabilitiesandphasesforautonomoussystems.ThetimeevolutiongeneratedbytheHamiltoniancanbeexpressedbyfirstorderdifferentialoperator–equationsintimeforthedensitymatrix(vonNeumann