Density operators that extremize Tsallis entropy a

arXiv:cond-mat/0505158v1[cond-mat.stat-mech]6May2005DensityoperatorsthatextremizeTsallisentropyandthermalstabilityeffectsC.VignatE.E.C.S.,UniversityofMichigan,U.S.A.andL.I.S.,Grenoble,FranceA.PlastinoNationalUniversityLaPlataandArgentina’sCONICETC.C.727,1900LaPlata,ArgentinaFebruary2,2008AbstractQuitegeneral,analytical(bothexactandapproximate)formsfordiscreteprobabilitydistributions(PD’s)thatmaximizeTsallisentropyforafixedvariancearehereinvestigated.Theyapply,forinstance,inawidevarietyofscenariosinwhichthesystemischaracterizedbyaseriesofdiscreteeigenstatesoftheHamiltonian.UsingthesediscretePD’sas“weights”leadstodensityoperatorsofarathergeneralcharacter.Thepresentstudyallowsonetovividlyexhibittheeffectsofnon-extensivity.VaryingTsallis’non-extensivityindexqoneisseentopassfromunstabletostablesystemsandeventounphysicalsituationsofinfiniteenergy.1IntroductionTsallis’thermostatisticsistodayanewparadigmforstatistics,withapplica-tionstoseveralscientificdisciplines[1,2,3,4,5].Notwithstandingitsmanifoldapplications,somedetailsofthebasicthermostatisticalformalismremainun-explored.Thisiswhyanalyticalresultsaretobewelcome,speciallyiftheyare,astheonestobehereinvestigated,ofaverygeneralnature.WewillprovidethistypeofresultsfordiscreteprobabilitydistributionsoffixedvariancethatmaximizeTsallis’entropy.Givenadiscreteprobabilitydistribution(DPD)p={pk},itsTsallis’informa-tionmeasure(orentropicform)isdefinedasHq(p)=1q−1(1−+∞Xk=−∞pqk).(1)1Itisaclassicalresultthatasq→1,TsallisentropyreducestoShannonentropyH1(p)=−+∞Xk=−∞pklogpk.Withoutlossofgenerality,wewillhereconsideronlycenteredrandomvariablesoffixedvariance.Theaimofthispaperistoprovideaccurateestimatesofi)theparametersoftheDPD’sandii)theirbehavior.ThemaximizersofTsallis’informationmeasureundervarianceconstraintinthecontinuous,multivariatecasehavebeendiscussedin[6].Consideraquantumsystemwhoseeigenstatesarecharacterizedbyasetofquantumnumbersthatwecollectivelydenotewithaninteger,runningindexk(Cf.Eq.(1)),thatwillofcoursealsolabeltheeigensolutions(|ψki,ǫk)ofthepertinenttime-independentSchrœdingerequationH|ψki=ǫk|ψki,(2)withHtheHamiltonian.Letpk≡|ψk|2betheprobabilityoffindingoursysteminthestate|ψki.Themixedstateρ=Xkpk|ψkihψk|,(3)commuteswiththeHamiltonianbyconstructionandrepresentsthusabonafidepossiblestationarystateofthesystem.IfwenowfindaphysicalquantityZwhosemeanvalueisproportionaltothevariance,wecaninterpretρasthemixedstatethatmaximizesTsallismeasuresubjecttothea-prioriknownexpectationvalueofsuchaphysicalquantity.Wewillshowbelowthat,inthesecircum-stances,universalexpressionscanbegivenforthepk’s,andthusforρ.Wediscusspossibleapplicationsintheforthcomingsection.Afterwards,afterhavingintroducedsomedefinitionsandnotations,wecharacterizethediscreteTsallismaximizersforfixedvarianceinboththeq1andtheq1cases,anddiscussthermalstabilityquestions.Wepassthentoanalyzeextensionstothemultivariatecases.Forthesakeofcompleteness,someoftheproofsofassertionsreferencedtoinwhatfollowsaregiveninAnnex.2PossiblephysicalapplicationsSeveralphysicalmodelscanbeadaptedtothescenariodescribedabove(Cf.Eqs.(2)and(3))Theweightspkin(3))willbeoftheTsallis-powerlawform.Withtheseweights,ρmaximizesTsallis’entropysubjecttotheconstraintofaconstantvariance,whichintroducesaLagrangemultiplierthatwewillcallβ.GivenaphysicalquantityZwhosemeanvalueisproportionaltothevariance,ρisthatmixedstatewhichmaximizesTsallismeasuresubjecttothea-priori2knownhZi−value.IntheexamplesbelowZisthesystem’senergyE,butmanyotherpossibilitiescanbeimagined.Thus,ρwillbethestatemaximizingTsallis’HqforafixedvalueoftheexpectationvalueU=hEi=Tr[Hρ]oftheHamiltonian.Asaconsequence,themultiplierβcanbethoughtofasan“inversetemperature”T,thatis,setβ=1/(kBT),withkBtheBoltzmannconstant.ThisissobecauseweareatlibertyofimaginingthatUiskeptconstantbecauseitisincontactwithaheatreservoir[7].Ofcourse,thisisnotnecessarilythecase.ρexistsbyitselfandisalegitimatestationarymixedstateofoursystem.Butwecanthinkofβaseithera“real”oran“equivalent”inversetemperature.Consider,forexample,asystemforwhichtheenergyspectrumconsistsofadenumerablesetofN(Npossiblyinfinite)energylevelslabeledbyaquantumnumberkwithp−k=pk,sothatalllevelsexhibitadegeneracygk=2fork6=0;k0;andg0=1,(4)i.e.,thesumsin(1)runfrom0upto∞andeachsummandismultipliedbygk.Withinthepresentframework,U1U=hEi=+∞Xk=0gkpkEk,(5)becomesnumericallyequaltotheDPD’svariance,whichbydefinitionisfixedandassumedlyknownapriori.Asjuststated,wemaythink,ifwewish,thatoursystemisincontactwithaheatreservoir,whichfixesthemeanenergy,andthattheassociatedLagrangemultiplierβcanbeassimilatedtoaninversetemperatureT.Amongmanyexamplesofsuchascenariowementionhere:•theplanarrotor[8],wherekisthemagneticquantumnumbercorrespond-ingtotheazimuthalangleusuallydenotedbyφ.Thelevel-energiesEkareproportionaltok2andEk=CEk2;CEhasdimensionofenergy.(6)WehaveCE=~2/2MI,withMIthesystem’smomentofinertia[9].Forsimplicity’ssakewetakehereCE=1,butretain,ofcourse,itsenergy-units.•thethree-dimensionalrigidrotator[8],althoughknowmeansthequan-tumnumberLassociatedtoorbitalangularmomentum(forlargek,thespect