Phase transitions and configuration space topology

arXiv:cond-mat/0703401v2[cond-mat.stat-mech]8Oct2007PhasetransitionsandconfigurationspacetopologyMichaelKastner∗PhysikalischesInstitut,Universit¨atBayreuth,95440Bayreuth,Germany(Dated:October08,2007)Equilibriumphasetransitionsmaybedefinedasnonanalyticpointsofthermodynamicfunctions,e.g.,ofthecanonicalfreeenergy.Givenacertainphysicalsystem,itisofinteresttounderstandwhichpropertiesofthesystemaccountforthepresence,ortheabsence,ofaphasetransition,andaninvestigationofthesepropertiesmayleadtoadeeperunderstandingofthephysicalphenomenon.Onepossiblewaytoapproachthisproblem,reviewedanddiscussedinthepresentpaper,isthestudyoftopologychangesinconfigurationspacewhich,remarkably,arefoundtoberelatedtoequilibriumphasetransitionsinclassicalstatisticalmechanicalsystems.Forthestudyofconfigurationspacetopology,oneconsidersthesubsetsMv,consistingofallpointsfromconfigurationspacewithapotentialenergyperparticleequaltoorlessthanagivenv.Forfinitesystems,topologychangesofMvareintimatelyrelatedtononanalyticpointsofthemicrocanonicalentropy(which,asasurprisetomany,doexist).Inthethermodynamiclimit,amorecomplexrelationbetweennonanalyticpointsofthermodynamicfunctions(i.e.,phasetransitions)andtopologychangesisobserved.Forsomeclassofshort-rangesystems,atopologychangeoftheMvatv=vtwasproventobenecessary,butnotsufficient,foraphasetransitiontotakeplaceatapotentialenergyvt.Incontrast,phasetransitionsinsystemswithlong-rangeinteractionsorinsystemswithnonconfiningpotentialsneednotbeaccompaniedbysuchatopologychange.Instead,forsuchsystemsthenonanalyticpointinathermodynamicfunctionisfoundtohavesomemaximizationprocedureatitsorigin.Theseresultsmayfosterinsightintothemechanismswhichleadtotheoccurrenceofaphasetransition,andthusmayhelptoexploretheoriginofthisphysicalphenomenon.ContentsPreface1I.Introduction2II.Definitionsandpreliminaries3A.StandardHamiltoniansystems3B.Configurationspacesubsets4C.Thermodynamicfunctions41.Microcanonicalthermodynamicfunctions42.Canonicalthermodynamicfunctions43.Relationofmicrocanonicalandcanonicalthermodynamicfunctions5D.Nonanalyticpointsandphasetransitions5III.Computationoftopologicalquantities6A.Morsetheory6B.Modelcalculation:Mean-fieldk-trigonometricmodel7C.Numericalcomputationoftopologicalquantities81.EulercharacteristicviaGauss-Bonnettheorem82.Criticalpointsofthepotential9IV.Nonanalyticitiesinfinitesystems9V.Phasetransitionsandconfigurationspacetopology10A.Conjectures10B.Franzosi-Pettinitheorem11C.ModelsnotcoveredbyTheoremV.712VI.Limitationsoftherelationbetweenphasetransitionsandconfigurationspacetopology12A.Long-rangeinteractions12B.Nonconfiningpotentials14∗Electronicaddress:[e-mail]-fieldϕ4modelrevisited152.Solid-on-solidmodelsrevisited16D.Twononanalyticitygeneratingmechanisms16VII.Searchforasufficiencycriterion17A.SimultaneousattachmentofO(N)differenthandles17B.NonanalyticitiesoftheEulercharacteristic17C.Nonpurelytopologicalsufficiencyconditions18VIII.Summary19IX.Epilog19Acknowledgments19A.Configurationspacetopologyofsolid-on-solidmodels20References20PrefaceItwasattheendofthe1990swhentheapplicationofconceptsfromdifferentialgeometrytoHamiltoniandy-namicalsystemsledtoaconjecturedconnectionbetweentheoccurrenceofequilibriumphasetransitionsinclassi-calHamiltoniansystemsandsometopologicalquantitiesofconfigurationspacesubsetsofthesesystems.Sincethen,theinterestinthisapproachandthenumberofpeopleworkingonthetopichasincreased,andsohasthenumberofresults.Atthetimeofthiswriting,anoverviewofthesub-jectisdifficulttoattain.First,theresultsarescattered2amongaconsiderablenumberofpublicationsand,sec-ond,severaloftheresults,althoughcorrect,demandareinterpretationasaconsequenceofrecentfindingsanddevelopments.Thepurposeofthepresentpaperistoassemblefromtheknownresults,asfaraspossible,aco-herentpictureoftherelationbetweenphasetransitionsandconfigurationspacetopology,andtoindicatenewlinesofresearchwhichmightopenupfromthesecon-cepts.I.INTRODUCTIONPhasetransitions,liketheboilingandevaporatingofwateratacertaintemperatureandpressure,arecom-monphenomenabothineverydaylifeandinalmostanybranchofphysics.Looselyspeaking,aphasetransitionbringsaboutasuddenchangeofthemacroscopicproper-tiesofamany-particlesystemwhilesmoothlyvaryingaparameter(thetemperatureorthepressureintheaboveexample).Probablythemainreasonfortheunabatedinterestthatphasetransitionshavereceivedalreadyformorethanacenturyistheiromnipresenceinallbranchesofphysics(andalsoinrelatedfieldslikebiologyorengi-neering):beittheformationofstarsinastrophysics,thetransitiontosuperconductivityinsolidstatephysics,ortheopeningoftheDNAhelixinbiophysics,examplesofmany-particlesystemswhichundergoaphasetransitionarewidespreadandofindisputablerelevanceinscience.Phasetransitionscanoccurinbothequilibriumandnonequilibriumsystems,butthefocuswillbeexclusivelyontheequilibriumcaseinthisexposition.Thereisaplethoraofbooksonthesubject,rangingfromexperi-mentaltotheoreticalandmathematicaltreatises.Espe-ciallythetheorybooksaretoalargeextentconcernedwiththeclassificationofdifferenttypesofphasetr