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1、Scaling,universality,andrenormalization:ThreepillarsofmoderncriticalphenomenaH.EugeneStanleyCenterforPolymerStudiesandDepartmentofPhysics,BostonUniversity,Boston,Massachusetts02215Thisbriefoverviewisdesignedtointroducesomeoftheadvancesthathaveoccurredinourunderstandingofphase
2、transitionsandcriticalphenomena.Thepresentationisorganizedaroundthreesimplequestions:(i)Whatarethebasicphenomenaunderconsideration?(ii)Whydowecare?(iii)Whatdoweactuallydo?Toanswerthethirdquestion,theauthorshallbrie¯yreviewscaling,universality,andrenormalization,threeofthemany
3、importantthemeswhichhaveservedtoprovidetheframeworkofmuchofourcurrentunderstandingofcriticalphenomena.Thestyleisthatofacolloquium,notthatofamini-reviewarticle.[S0034-6861(99)02902-5]I.THEFIRSTQUESTION:``WHATARECRITICALAthirdreasonisawe.WewonderhowitisthatspinsPHENOMENA?''``kn
4、ow''toalignsosuddenlyasT!T1.HowcanthecspinspropagatetheircorrelationssoextensivelySupposewehaveasimplebarmagnet.WeknowitisthroughouttheentiresystemthatMÞ0andxT!`?aferromagnetbecauseitiscapableofpickingupthumb-tacks,thenumberofwhichiscalledtheorderparameterM.Asweheatthissystem
5、,Mdecreasesandeventually,atacertaincriticaltemperatureTc,itreacheszero:nomorethumbtacksremain!Infact,thetransitionisre-III.THETHIRDQUESTION:``WHATDOWEDO?''markablysharp,sinceMapproacheszeroatTcwithin-®niteslope.SuchsingularbehaviorisanexampleofaTheanswertothisquestionwilloccu
6、pytheremainder``criticalphenomenon.''ofthisbriefoverview.Therecentpastofthe®eldofCriticalphenomenaarebynomeanslimitedtothecriticalphenomenahasbeencharacterizedbyseveralorderparameter.Forexample,theresponse-functionsimportantconceptualadvances,threeofwhicharescal-constant-®eld
7、speci®cheatCHandisothermalsuscepti-ing,universality,andrenormalization.bilityxTbothbecomein®niteatthecriticalpoint.A.ScalingII.THESECONDQUESTION:``WHYDOWECARE?''ThescalinghypothesiswasindependentlydevelopedOnereasonforinterestinany®eldisthat,simplyput,byseveralworkers,includi
8、ngWidom,Dombandwedonotfullyunderstandthebasicphenomena.ForHunter,Kad
