Universal behavior in the static and dynamic prope

arXiv:cond-mat/0007422v2[cond-mat.stat-mech]8Nov2000Universalbehaviorinthestaticanddynamicpropertiesoftheα-XYmodelAndreaGiansantia,§,DanieleMoronia,†andAlessandroCampab,‡aDipartimentodiFisica,Universit`adiRoma“LaSapienza”andINFMUnit`adiRoma1,PiazzaleAldoMoro2,00185Roma,ItalybLaboratoriodiFisica,IstitutoSuperiorediSanit`aandINFNSezionediRoma1,GruppoCollegatoSanit`aVialeReginaElena299,00161Roma,Italy(8thNovember2000)AcceptedforpublicationinChaosSolitonsandFractals:Specialissue.ProceedingsoftheInternationalWorkshop:ClassicalandQuantumComplexityandNonextensiveThermodynamics,Denton(Texas),April3-62000.AbstractTheα-XYmodelgeneralizes,throughtheintroductionofapower-lawdecayingpoten-tial,awellstudiedmean-fieldhamiltonianmodelwithattractivelong-rangeinterac-tions.Intheα-model,theinteractionbetweenclassicalrotatorsonalatticeisgaugedbytheexponentαinthecouplingsdecayingasrα,whereraredistancesbetweensites.Wereviewandcommenthereafewrecentresultsonthestaticanddynamicpropertiesoftheα-model.Wediscusstheappropriateαdependentrescalingsthatmapthecanonicalthermodynamicsoftheα-modelintothatofthemeanfieldmodel.Wealsoshowthatthechaoticpropertiesofthemodel,studiedasafunctionofαdisplayauniversalbehaviour.§Correspondingauthor;[e-mail]‡[e-mail]†[e-mail]1IntroductionEquilibriumClassicalStatisticalMechanicsisoneofthegroundsofmodernphysics;itoriginatedfromtheworksofBoltzmannandGibbsattheendof19thcentury.ThecomputationaltechniquesofStatisticalMechanics,generalizedtoencompasstherightenumerationofquantumstates,havebeensuccessfullyappliedtothestudyofallstatesofmatter,alongallthe20thcentury.Ononeside,therigorousanalysisofthisbranchoftheoreticalphysicshasbeenimpressiveandfarreaching[1][2]andthebeliefthateverythinginthisfieldhasbeenrobustlyfoundedseemstobequitewidespread.OntheothersidetherehasbeenaserendipitouspragmaticuseoftheBoltzmann-Gibbsapproachbeyondtheboundssetbytheorems.TheuseofEquilibriumStatisticalMechanicsoutsidetheallowedboundariesledtothediscoveryofanomaliesorparadoxes,particularlyinthefieldofself-gravitatingsystems[3][4].Infactalltheedificeofstatisticalmechanicsrestsonfewstringentassumptionsontheinteractionsthatarenotfullfilledbylong-rangeforces.Inspiteoftheirfundamentalrelevance,gravitationalandCoulombicforcesdonotfitinwithEquilibriumClassicalStatisticalMechanics[5].Short-rangeinteractionsguaranteeextensivity:thatisenergyandentropy,asfunctionsofintensiveinternalparameters,growlinearlywiththenumberNofmicroscopiccomponentsofthesystems.Ifanextensivesystemisdividedintomacroscopicpartsthetotalenergyandentropyisthesumoftheenergiesandentropiesoftheparts.Onthecontrarythatisnottrueinsystemswherelong-rangeinteractionscanreflectthemselvesinthermodynamicpotentialsthatdonotscalewiththesizeofthesystemandcannotbedefinedinthethermodynamiclimit(i.e.forNgoingtoinfinity).Moreover,long-rangeforcesmaywellinducestrongspatialandtemporaldynamiccorrelationsthatcontrastmixingandresulteitherinverylongrelaxationtimesorinequilibriumdistributionsdifferentfromtheexpectedBoltzmann-Gibbs.AimingatamicroscopicfoundationofnonextensivethermodynamicsagoodstartingpointisthestudyoftheergodicpropertiesinthethermodynamiclimitofsimpleandmeaningfulHamiltonianmodelswithlong-rangeinteractions[6].IntherecentpastStefanoRuffoandco-workershaveintroducedaclassofmean-fieldmodelswithinfiniterangeoftheinteractions[7].Thesemodelsarerelatedwiththestatisticalphysicsofself-gravitatingsystemsandthatofphasetransi-tionsandbecameparadigmaticinthestudyofnon-extensivity.Inparticular,thedynamicsandthermodynamicsofthesocalledHamiltonianMean-Fieldmodel(HMF)havebeenextensivelystudiedinthelastfewyearsandacompre-hensivereviewcanbefoundinthecontributiontotheseproceedingsbyAndreaRapisardaandVitoLatora[8].Theα-XYmodelwerefertointhispaperhasbeenintroducedbyCeliaAnteneodoandConstantinoTsallis[9];thismodelcleverlycombinesthephysicsofthemean-fieldmodelsquotedabovewiththestatisticalphysicsoflatticemodelsoftheIsingtypewithlong-rangecouplingsdecayingastheinversepowerαofthedistancesbetweensites.SinceclassicalworksonthefoundationsofStatisticalMechanics[10]ithasbeenknownthatthesesystemsarenonextensivefor0≤α/d1,wheredisthedimensionalityoftheambientspace;α/disthenthenaturalcontrolparameterofnonextensivityinthisclassofmodels.Ourworkoriginatedfrompreviousworksbyotherresearchersactiveinthefields:inthesectiondevotedtothethermodynamicsoftheα-XYmodela1canonicalsolutionispresentedthathasbeenlargelyinspiredbytheanalyticalworkofAntoniandRuffoontheHMFmodel[11]andbyanumericalworkbyTamaritandAnteneodo[12][13];inthesectionaboutdynamicstheresultsontheLyapunovexponentsextendthepreviousworkbyAnteneodoandTsallis[9].IntheconclusionswetrytoexpressourpointofviewabouttheconnectionofourresultswiththeparticularformofnonextensivethermodynamicsproposedbyTsallis[14].2ThermodynamicsInthissectionwedefinethemodelandwecomputeitscanonicalpartitionfunc-tion.WeusethetypicalmethodsofGaussiantransformationandsaddlepointintegrationtogetherwithaFourierdiagonalizationoftheinteractionpotential.2.1Definitionofthemo