Theoretical Physics 8: Statistical Physics PDF
Preview Theoretical Physics 8: Statistical Physics
Wolfgang Nolting Theoretical Physics 8 Statistical Physics Theoretical Physics 8 Wolfgang Nolting Theoretical Physics 8 Statistical Physics 123 WolfgangNolting InstituteofPhysics Humboldt-UniversityatBerlin Germany ISBN978-3-319-73826-0 ISBN978-3-319-73827-7 (eBook) https://doi.org/10.1007/978-3-319-73827-7 LibraryofCongressControlNumber:2016943655 ©SpringerInternationalPublishingAG2018 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. 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Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAG Theregisteredcompanyaddressis:Gewerbestrasse11,6330Cham,Switzerland General Preface TheninevolumesoftheseriesBasicCourse:TheoreticalPhysicsarethoughttobe textbookmaterialforthestudyofuniversity-levelphysics.Theyareaimedtoimpart, in a compact form, the most important skills of theoretical physics which can be usedasbasisforhandlingmoresophisticatedtopicsandproblemsintheadvanced study of physics as well as in the subsequent physics research. The conceptual designofthepresentationisorganizedinsuchawaythat ClassicalMechanics(volume1) AnalyticalMechanics(volume2) Electrodynamics(volume3) SpecialTheoryofRelativity(volume4) Thermodynamics(volume5) are considered as the theory part of an integrated course of experimental and theoretical physics as is being offered at many universities starting from the first semester.Therefore,thepresentationisconsciouslychosentobeveryelaborateand self-contained,sometimessurelyatthe costofcertainelegance,sothatthecourse is suitableevenforself-study,at firstwithoutanyneedof secondaryliterature.At anystage,nomaterialisusedwhichhasnotbeendealtwithearlierinthetext.This holds in particular for the mathematical tools, which have been comprehensively developed starting from the school level, of course more or less in the form of recipes,suchthatrightfromthebeginningofthestudy,onecansolveproblemsin theoreticalphysics. The mathematicalinsertionsare always then pluggedin when theybecomeindispensabletoproceedfurtherintheprogramoftheoreticalphysics. It goes without saying that in such a context, not all the mathematical statements canbeprovedandderivedwithabsoluterigor.Instead,sometimesareferencemust be made to an appropriate course in mathematics or to an advanced textbook in mathematics. Nevertheless, I have tried for a reasonably balanced representation so that the mathematical tools are not only applicable but also appear at least ‘plausible’. v vi GeneralPreface The mathematical interludes are of course necessary only in the first volumes of this series, which incorporatemore or less the material of a bachelor program. InthesecondpartoftheserieswhichcomprisesthemodernaspectsofTheoretical Physics, QuantumMechanics:Basics(volume6) QuantumMechanics:MethodsandApplications(volume7) StatisticalPhysics(volume8) Many-BodyTheory(volume9), mathematical insertionsare no longer necessary. This is partly because, by the time one comes to this stage, the obligatory mathematics courses one has to take in order to study physics would have provided the required tools. The fact that training in theory has already started in the first semester itself permits inclusion ofpartsofquantummechanicsandstatisticalphysicsinthebachelorprogramitself. Itisclearthatthecontentofthelastthreevolumescannotbepartofanintegrated coursebutratherthesubjectmatterofpuretheorylectures.Thisholdsinparticular forMany-BodyTheorywhichisoffered,sometimesunderdifferentnamesas,e.g., AdvancedQuantumMechanics,intheeighthorsosemesterofstudy.Inthispartnew methodsandconceptsbeyondbasicstudiesareintroducedanddiscussed,whichare developedinparticularforcorrelatedmanyparticlesystemswhichinthemeantime havebecomeindispensablefora studentpursuingmaster’sora higherdegreeand forbeingabletoreadcurrentresearchliterature. In all the volumes of the series Basic Course: Theoretical Physics numerous exercisesareincludedtodeepentheunderstandingandtohelpcorrectlyapplythe abstractlyacquiredknowledge.Itisobligatoryforastudenttoattemptonhisown to adapt and apply the abstract concepts of theoretical physics to solve realistic problems.Detailed solutionsto the exercisesare givenat the endof eachvolume. The idea is to help a student to overcomeany difficulty at a particular step of the solutionortocheckone’sowneffort.Importantlythesesolutionsshouldnotseduce thestudenttofollowtheeasywayoutasasubstituteforhisowneffort.Attheend ofeachbiggerchapterIhaveaddedself-examinationquestionswhichshallserveas aself-testandmaybeusefulwhilepreparingforexaminations. Ishouldnotforgettothankallthepeoplewhohavecontributedonewayorother to the success of the book series. The single volumes arose mainly from lectures that I gave at the universities of Muenster, Wuerzburg, Osnabrueck, and Berlin (Germany),Valladolid(Spain),andWarangal(India).Theinterestandconstructive criticismofthestudentsprovidedmethedecisivemotivationforpreparingtherather extensivemanuscripts.AfterthepublicationoftheGermanversionIreceivedalot ofsuggestionsfromnumerouscolleaguesforimprovementandthishelpedtofurther develop and enhance the concept and the performance of the series. In particular I appreciate very much the support by Prof. Dr. A. Ramakanth, a long-standing scientificpartnerandfriend,whohelpedmeinmanyrespects,e.g.,whatconcerns thecheckingofthetranslationoftheGermantextintothepresentEnglishversion. GeneralPreface vii SpecialthanksareduetotheSpringercompany,inparticulartoDr.Th.Schneider and his team. I remember many useful motivations and stimulations. I have the feelingthatmybooksarewelltakencareof. Berlin,Germany WolfgangNolting December2017 Preface to Volume 8 IntheprefacesoftheprecedingvolumesIhavealreadysetoutthegoalofthebasic course in Theoretical Physics. This goal, explained and justified in the General Preface, remains of course unchangedfor the present eighth volume of the series onStatisticalPhysicsalso. The Statistical Physics represents in almost all courses of study on physics the closure of the basic education in Theoretical Physics and is offered, as a rule, in the sixth semester, at least when the training in Theoretical Physics starts already in the first semester. It belongs, besides Quantum Mechanics (Vols. 6 and 7), to the modern disciplines of Theoretical Physics, whose understanding is mandatory either in elementary form for the bachelor program or in an advanced version for the master program. In contrast, Classical and Analytical Mechanics (Vols.1and2),Electrodynamics(Vol.3),SpecialTheoryofRelativity(Vol.4),and Thermodynamics (Vol. 5) are ascribed to the classical disciplines. Normally they arepartsofthebachelorprograminthecourseofstudyonphysics. The underlying volume on Statistical Physics is subdivided into four larger chapters.Inthefirstchapter,themostimportantconceptsandmethodsforclassical systems are explained and exercised. It is demonstrated how the large number of degrees of freedom of macroscopic systems can lead to completely novel phenomena.Asanexampleitmaybementionedheretheirreversibletransitionof athermodynamicsystemintoequilibrium,which,althoughactuallyallmicroscopic equations of motion are time-reversal invariant, has to, as everyday observation, beacceptedandunderstood.TheMethodofStatisticalEnsembles(microcanonical, canonical,grandcanonical)turnsouttobeasuccessfulapproachforthedescription of macroscopic physical systems. The proof of the equivalence of these three ensemblesisanimportantsubjectofthefirstchapter. The second chapter deals with Quantum Statistics. A double indeterminacy is characteristicofit,whichrequirestwoaveragingprocessesofcompletelydifferent nature.Besides the indeterminacydue to the large numberof degreesof freedom, whichis ofcoursepresentalso forclassical systems, thereappearsthe principally unavoidable quantum-mechanical uncertainty (measurement process!). This fact necessitatesthedevelopmentofgenuinequantum-statisticalconcepts. ix x PrefacetoVolume8 A firstimportantapplicationof the generaltheoryconcernstheIdealQuantum GasesinChap.3,forwhichthequantum-mechanicalPrincipleofIndistinguishabil- ityofIdenticalParticlesplaysanextraordinaryrole.SystemsofidenticalFermions and systems of identical Bosonsunderliedifferentphysicalprinciples, which lead to physicalbehaviorsstrongly deviating from one another. As a further important applicationoftheStatisticalPhysicsIhavechosenthehighlytopicalbranchofthe PhaseTransitionsandCriticalPhenomenainChap.4. This volume on Statistical Physics arose from lectures I gave at the German universitiesinWürzburg,Münster,andBerlin.Theanimatinginterestofthestudents inmylecturenoteshasinducedmetopreparethetextwithspecialcare.Thepresent oneaswellastheothervolumesarethoughttobethetextbookmaterialforthestudy ofbasicphysics,primarilyintendedforthestudentsratherthanfortheteachers. I am thankful to the Springer company, especially to Dr. Th. Schneider, for acceptingandsupportingtheconceptofmyproposal.Thecollaborationwasalways delightfulandveryprofessional.Adecisivecontributiontothebookwasprovided by Prof. Dr. A. Ramakanth from the Kakatiya University of Warangal (India), a long-standingscientificpartnerandfriend,whohelpedmeinmanyrespects.Many thanksforit! Berlin,Germany WolfgangNolting December2017 Contents 1 ClassicalStatisticalPhysics................................................. 1 1.1 Preparations ............................................................. 1 1.1.1 FormulationoftheProblem................................... 1 1.1.2 SimpleModelSystem......................................... 3 1.1.3 Exercises....................................................... 10 1.2 Micro-CanonicalEnsemble............................................. 11 1.2.1 State,PhaseSpace,TimeAverage............................ 11 1.2.2 StatisticalEnsemble,EnsembleAverage..................... 16 1.2.3 LiouvilleEquation............................................. 18 1.2.4 Micro-CanonicalEnsemble................................... 22 1.2.5 Exercises....................................................... 26 1.3 ConnectiontoThermodynamics ....................................... 28 1.3.1 ConsiderationsonThermalEquilibrium..................... 28 1.3.2 EntropyandTemperature..................................... 35 1.3.3 SecondLawofThermodynamics............................. 41 1.3.4 ChemicalPotential ............................................ 43 1.3.5 BasicRelationofThermodynamics.......................... 45 1.3.6 EquipartitionTheorem........................................ 49 1.3.7 IdealGas....................................................... 51 1.3.8 Exercises....................................................... 57 1.4 CanonicalEnsemble .................................................... 60 1.4.1 PartitionFunction ............................................. 61 1.4.2 FreeEnergy.................................................... 65 1.4.3 Fluctuations.................................................... 68 1.4.4 EquivalenceofMicro-CanonicalandCanonical Ensemble....................................................... 69 1.4.5 Exercises....................................................... 72 1.5 Grand-CanonicalEnsemble ............................................ 77 1.5.1 Grand-CanonicalPartitionFunction.......................... 78 1.5.2 ConnectiontoThermodynamics.............................. 81 xi
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