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1、Problems&SolutionsforStatisticalPhysicsofFieldsUpdatedJuly2008byMehranKardarDepartmentofPhysicsMassachusettsInstituteofTechnologyCambridge,Massachusetts02139,USATableofContents1.CollectiveBehavior,FromParticlestoFields................12.StatisticalFields............................183.Fluctu
2、ations..............................314.TheScalingHypothesis.........................555.PerturbativeRenormalizationGroup...................636.LatticeSystems.............................907.SeriesExpansions..........................1068.BeyondSpinWaves..........................132Solutionst
3、oproblemsfromchapter1-CollectiveBehavior,FromParticlestoFields1.Thebinaryalloy:Abinaryalloy(asinβbrass)consistsofNAatomsoftypeA,andNBatomsoftypeB.Theatomsformasimplecubiclattice,eachinteractingonlywithitssixnearestneighbors.Assumeanattractiveenergyof−J(J>0)betweenlikeneighborsA−AandB−B,butar
4、epulsiveenergyof+JforanA−Bpair.(a)Whatistheminimumenergyconfiguration,orthestateofthesystematzerotemper-ature?•TheminimumenergyconfigurationhasaslittleA-Bbondsaspossible.Thus,atzerotemperatureatomsAandBphaseseparate,e.g.asindicatedbelow.AB(b)Estimatethetotalinteractionenergyassumingthattheatom
5、sarerandomlydistributedamongtheNsites;i.e.eachsiteisoccupiedindependentlywithprobabilitiespA=NA/NandpB=NB/N.•Inamixedstate,theaverageenergyisobtainedfromE=(numberofbonds)×(averagebondenergy) =3N·−Jp2−Jp2+2JppABAB2NA−NB=−3JN.N(c)Estimatethemixingentropyofthealloywiththesameapproximation.As
6、sumeNA,NB≫1.•FromthenumberofwaysofrandomlymixingNAandNBparticles,weobtainthemixingentropyofN!S=kBln.NA!NB!UsingStirling’sapproximationforlargeN(lnN!≈NlnN−N),theaboveexpressioncanbewrittenasS≈kB(NlnN−NAlnNA−NBlnNB)=−NkB(pAlnpA+pBlnpB).1(d)Usingtheabove,obtainafreeenergyfunctionF(x),wherex=(
7、NA−NB)/N.ExpandF(x)tothefourthorderinx,andshowthattherequirementofconvexityofFbreaksdownbelowacriticaltemperatureTc.Fortheremainderofthisproblemusetheexpansionobtainedin(d)inplaceofthefullfunctionF(x).•Intermsofx=pA−pB,thefreeenergycanbewrittenasF=
