Quantum chaos, random matrix theory, and statistic

arXiv:chao-dyn/9609013v119Sep1996QuantumChaos,RandomMatrixTheory,andStatisticalMechanicsinTwoDimensions-AUnifiedApproachSudhirR.Jain1andDanielAlonso1,21Facult´edesSciencesandCenterforNonlinearPhenomenaandComplexSystems,Universit´eLibredeBruxelles,CampusPlaineC.P.231,BoulevardduTriomphe,1050Bruxelles,Belgium2DepartamentodeF´ısicaFundamentalyExperimentalFalcultaddeF´ısicas,UniversidaddeLaLagunaLaLaguna38204,CanaryIslands,Tenerife,SpainAbstractWepresentatheorywherethestatisticalmechanicsfordiluteidealgasescanbederivedfromrandommatrixapproach.WeshowtheconnectionofthisapproachwithSrednickiapproachwhichconnectsBerryconjecturewithstatisticalmechanics.WefurtherestablishalinkbetweenBerryconjectureandrandommatrixtheory,thusprovidingaunifiededificeforquantumchaos,randommatrixtheoryandstatisticalmechanics.Inthecourseofarguingfortheseconnections,wealsoobservesumrulesassociatedwiththeoutstandingcountingprobleminthetheoryofBraidgroups.PACSnumbers:05.30.-d05.45.+bTypesetusingREVTEX11.IntroductionTounderstandthethemeofthepaper,wepresentanoverviewofvariousdifferentlinksthathavebeendiscoveredinlastfewdecadesbetweenclassicallychaoticsystemsandtheirquantalcounterparts.Anystudymotivatedtobringaboutthisconnectioniswhatweunder-standhereby”quantumchaos”[1,2].Anoverwhelmingnumberofnumericalexperimentsonspectralstatistics[3,4]andtheircorrespondingsemiclassicalanalysis[5,6]suggestthattheuniversalfeaturesobservedinchaoticquantumsystemscanbemodelledintermsofrandommatrixtheory.Apartfromenergyspectra,ithasbeenfoundthattheconjecture[7]whereaneigenstateofachaoticquantumsystemisrepresentedasaGaussianrandomsuperpositionofplanewavesentailsresultswhicharefoundinagreementwithnumericalstudies[8,4].Webelievethatanimportantstephasbeeninestablishingtheresultthatthisconjectureleadstomomentumdistributionofidealgases,thusbringingoutstatisticalmechanics[9].However,inordertobringoutthepuzzlingresultsintwo-dimensionalstatis-ticalmechanics,itisnecessarythatthechoiceofthecorrelationsbetweenamplitudesoftheeigenstatesbespecified.Thus,inthispursuit,weareledtorandommatrixtheorywhereonecansystematicallychoosetheensemble.Recently,ithasbeenshownhowonecangofromrandommatrixtheorytostatisticalmechanics[10]-aworkthathasbroughttogethertwoimportantstatisticaltheorieswhichhavebeen,hitherto,consideredquiteapart.However,thereremainmanyquestionsandwedescribewhatwebelieveisanimpor-tantone.Althoughonecanargueforthermalization[9]andassociateatemperaturebyasuitably-definedcoarsegraining,isthereawaywecanmakethisassociationmoreprecise?Bytheendofthispaper,wehopetoconvincethereaderthat,inacertainsense,thewayofexplainingthermalizationisconsistentwiththetraditionalapproaches,andmoreimportantly,withthesecondlawofthermodynamics.Weknowthattheconceptoftemper-atureassociatedwithheatliesinacuriousandsubtlecombinationofentropyandenergy.IfwecouldarguethatthepremisethatleadsustothederivationofquantumthermalizationfromtheBerryconjecturealsoallowsustoshowtheacceptablebehaviourofentropy,we2wouldhaveamorecoherentlogicalschemetiedwithsecondlawofthermodynamics.Aswillbeseen,thisalsoshowsatwhichlevelofdescriptionwearewithrespecttotheprojectionmethodssowidelyused[11].Throughoutthepaper,wewillbeconcentratingontwodimensionsasthatisthemostdifficultcaseinstatisticalmechanics[12–14].Insection2,wegiveabriefdiscussionofthechoiceofrandommatrixensemblewhentime-reversalandparityarebroken.Thisisfundamentalindealingsuccessfullywiththeproblemofmomentumdistributionfunctionandvirialcoefficientsinsection3.Thefactthatquantummechanicscanbedoneonrealfieldiftheantiunitarysymmetriesarewell-specified[16],and,theclassificationtheoremofassociativedivisionalgebra[18]leadstothreebasicensemblesinrandommatrixtheory[17].Incorporatingtheviolationofparityisanimportantstep.Insection4,weunifythedifferentstreamsofthoughtfromquantumchaos,randommatrixtheory,andstatisticalmechanicsbydiscussingentropywhichisfundamentaltoallthethree.Wewouldliketomentionthatarecentwork[19]isaninterestingcompanionofthispaper.Weconcludethepaperwithasummary.2.RandomMatrixEnsembleinTwoDimensionsInusualdiscussionofrandommatrixtheory,thespacedimensionalityofthephysicalsystemplaysnoexplicitrole.Ofcourse,itismisleadingtobeinthatthought-frame.Duetothecomplicationsarisingfromthefactthatweareworkingintwodimensions,wepresenthereacomparativediscussionaboutthefundamentalsymmetriesintwoandthree(orgreater)spacedimensionswhichdecisivelyrestrictthepossibilitiesoftherandommatrixensemble.DenotingthetimereversaloperatorbyT,thepositionoperator,qandthemomentumoperatorsatisfyTqT−1=q,TpT−1=−p.(1)Inordertopreservethecommutatorbetweenqandp,weseethrough3TiT−1=−i(2)(iisthesquarerootof-1)thatTisantilinear.Moreover,sinceTT†=1,(3)wesaythatTisantiunitary.Tcanalwaysbewrittenasaproductofaunitaryoperator,Uandaconjugationoperator,K.OnastateΨ,onapplicationofT2,wegetaconstantλtimesΨ.NotethatT2=UKUK=UU∗(4)whichgivesUU∗=λ,UU†=1.(5)ItnowfollowsthatU=λU,andhence|λ|2=1.Butthen,sinceTisantilinear,λ∗=λwhichentailsT2=±1.(6)Thisthengivesus,afterproperintroductionofangularmomentumoperator,thetwopossible-symmetricandant