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ClassicalandQuantumStatisticalPhysics Statisticalphysicsexaminesthecollectivepropertiesoflargeensemblesofparticles, andisapowerfultheoreticaltoolwithimportantapplicationsacrossmanydifferent scientific disciplines. This book provides a detailed introduction to classical and quantum statistical physics, including links to topics at the frontiers of current research. The first part of the book introduces classical ensembles, provides an extensive review of quantum mechanics, and explains how their combination leads directly to the theory of Bose and Fermi gases. It contains a detailed analysis of the quantum properties of matter and introduces the exotic features of vacuum fluctuations. The second part discusses more advanced topics such as the two- dimensional Ising model and quantum spin chains. This modern text is ideal for advanced undergraduate and graduate students interested in the role of statistical physics in current research. One hundred and forty homework problems reinforce keyconceptsandfurtherdevelopreaders’understandingofthesubject. CarloHeissenberg is a postdoctoral scholar at the Nordic Institute for Theoretical Physics (Nordita) in Stockholm, and at Uppsala University. He received his PhD from Scuola Normale Superiore in Pisa, with a thesis on asymptotic symmetries and higher spin theories, and his research is now focused on the interface between scatteringamplitudesandgravitationalwaves. AugustoSagnottiisProfessorofTheoreticalPhysicsatScuolaNormaleSuperioreand has taught the Statistical Physics course there since 2017. His research is focused ongravitationalphysicsandconformalfieldtheory,andhispioneeringcontribution ledtotheintroduction oforientifoldvacua instringtheory.Heisarecipientofthe HumboldtResearchAward. Classical and Quantum Statistical Physics Fundamentals and Advanced Topics CARLO HEISSENBERG NordicInstituteforTheoreticalPhysicsandUppsalaUniversity AUGUSTO SAGNOTTI ScuolaNormaleSuperiorePisaandIstitutoNazionalediFisicaNucleare(INFN) UniversityPrintingHouse,CambridgeCB28BS,UnitedKingdom OneLibertyPlaza,20thFloor,NewYork,NY10006,USA 477WilliamstownRoad,PortMelbourne,VIC3207,Australia 314–321,3rdFloor,Plot3,SplendorForum,JasolaDistrictCentre,NewDelhi–110025,India 103PenangRoad,#05–06/07,VisioncrestCommercial,Singapore238467 CambridgeUniversityPressispartoftheUniversityofCambridge. ItfurtherstheUniversity’smissionbydisseminatingknowledgeinthepursuitof education,learning,andresearchatthehighestinternationallevelsofexcellence. www.cambridge.org Informationonthistitle:www.cambridge.org/9781108844628 DOI:10.1017/9781108952002 ©CarloHeissenbergandAugustoSagnotti2022 Thispublicationisincopyright.Subjecttostatutoryexception andtotheprovisionsofrelevantcollectivelicensingagreements, noreproductionofanypartmaytakeplacewithoutthewritten permissionofCambridgeUniversityPress. Firstpublished2022 AcataloguerecordforthispublicationisavailablefromtheBritishLibrary. LibraryofCongressCataloging-in-PublicationData Names:Heissenberg,Carlo,author.|Sagnotti,Augusto,other. Title:Classicalandquantumstatisticalphysics:Fundamentalsandadvancedtopics/Carlo HeissenbergandAugustoSagnotti. Description:NewYork:CambridgeUniversityPress,2021.| Includesbibliographicalreferencesandindex. Identifiers:LCCN2021041905(print)|LCCN2021041906(ebook)| ISBN9781108844628(hardback)|ISBN9781108952002(epub) Subjects:LCSH:Statisticalmechanics.|Quantumstatistics.|Quantum theory.|BISAC:SCIENCE/Physics/Mathematical&Computational Classification:LCCQC174.7.H452021(print)|LCCQC174.7(ebook)| DDC530.13–dc23/eng/20211005 LCrecordavailableathttps://lccn.loc.gov/2021041905 LCebookrecordavailableathttps://lccn.loc.gov/2021041906 ISBN978-1-108-84462-8Hardback CambridgeUniversityPresshasnoresponsibilityforthepersistenceoraccuracyof URLsforexternalorthird-partyinternetwebsitesreferredtointhispublication anddoesnotguaranteethatanycontentonsuchwebsitesis,orwillremain, accurateorappropriate. Contents Preface pageix Acknowledgments xi PartI 1 1 ElementsofThermodynamics 3 1.1 TheLawsofThermodynamics 3 1.2 ThermodynamicPotentials 5 1.3 ComparisonbetweenC andC 8 P V 1.4 Fluctuations 9 1.5 Stability 10 1.6 SpecificHeatandCompressibility 11 1.7 TheIdealMonatomicGas 12 BibliographicalNotes 13 Problems 13 2 TheClassicalEnsembles 17 2.1 TimeAveragesandEnsembleAverages 17 2.2 TheMicrocanonicalEnsemble 17 2.3 TheCanonicalEnsemble 26 2.4 TwoExamples 31 BibliographicalNotes 35 Problems 35 3 AspectsofQuantumMechanics 40 3.1 SomeGeneralProperties 40 3.2 TheFreeParticle 41 3.3 TheHarmonicOscillator 44 3.4 EvolutionKernelsandPathIntegrals 49 3.5 TheFreeParticleonaCircle 53 3.6 TheHydrogenAtom 56 3.7 TheWKBApproximation 64 3.8 Instantons 77 3.9 TheDensityMatrix 87 BibliographicalNotes 89 Problems 90 4 SystemsofQuantumOscillators 97 v 4.1 TheEffectofanEnergyGap 97 vi Contents 4.2 BlackbodyRadiation 99 4.3 DebyeTheoryofSpecificHeats 106 BibliographicalNotes 107 Problems 107 5 VacuumFluctuations 112 5.1 TheCasimirEffect 112 5.2 TheLambShift 114 5.3 TheCosmological-ConstantProblem 117 BibliographicalNotes 121 6 ThevanderWaalsTheory 122 6.1 TheRoleofInteractions 122 6.2 ThePartitionFunction 124 6.3 ThevanderWaalsEquationofState 125 6.4 PhaseEquilibriaandthevanderWaalsGas 126 6.5 TheGibbsPhaseRule 130 BibliographicalNotes 131 Problems 131 7 TheGrandCanonicalEnsemble 134 7.1 TheGrandCanonicalEquations 134 7.2 TwoInstructiveExamples 137 BibliographicalNotes 140 8 QuantumStatistics 141 8.1 IdenticalParticlesinQuantumMechanics 141 8.2 IdenticalOscillatorsintheCanonicalEnsemble 145 8.3 NonrelativisticFermiandBoseGases 147 8.4 High-TemperatureLimits 148 8.5 TheFreeFermiGasatLowTemperatures 150 8.6 FermiGasesinSolids 154 8.7 TheFreeBoseGasatLowTemperatures 159 8.8 BosonsinanExternalPotential 162 8.9 AtomicandMolecularSpectra 167 8.10 SomeApplications 179 BibliographicalNotes 187 Problems 187 9 MagnetisminMatter,I 193 9.1 OrbitsinaUniformMagneticField 193 9.2 LandauLevels 194 9.3 LandauDiamagnetism 199 9.4 High-TParamagnetism 202 9.5 Low-TParamagnetism 202 BibliographicalNotes 203 Problems 203 vii Contents 10 MagnetisminMatter,II 207 10.1 EffectiveSpin–SpinInteractions 207 10.2 TheOne-DimensionalIsingModel 209 10.3 TheRoleofBoundaryConditions 212 10.4 TheContinuumLimit 215 10.5 TheInfinite-RangeIsingModel 217 10.6 Mean-FieldandVariationalMethod 219 10.7 Mean-FieldAnalysisoftheIsingModel 221 10.8 CriticalExponentsandScalingRelations 226 10.9 Landau–GinzburgTheory 229 10.10AToyModelofaPhaseTransition 233 BibliographicalNotes 235 Problems 235 PartII 243 11 The2DIsingModel 245 11.1 ClosedPolygonsinTwoDimensions 245 11.2 Kramers–WannierDuality 246 11.3 TheOnsagerSolution 250 BibliographicalNotes 254 12 TheHeisenbergSpinChain 255 12.1 NoninteractingSystems 255 12.2 TheSpectrumoftheHeisenbergModel 258 12.3 ThermodynamicBetheAnsatz 270 BibliographicalNotes 273 13 ConformalInvarianceandtheRenormalizationGroup 274 13.1 ConformalInvariance 274 13.2 1DIsingModelandRenormalizationGroup 279 13.3 Percolation 282 13.4 TheXYModel 283 13.5 (cid:2)-ExpansionandtheD=3IsingModel 301 BibliographicalNotes 314 14 TheApproachofEquilibrium 315 14.1 TheLangevinEquation 315 14.2 TheFokker–PlanckEquation 318 14.3 TheBoltzmannEquation 320 14.4 TheH-Theorem 321 14.5 TransportPhenomena 323 14.6 NondissipativeHydrodynamics 325 14.7 TheEmergenceofViscosity 327 14.8 TheFluctuation–DissipationTheorem 329 BibliographicalNotes 335 viii Contents AppendixA ProbabilityDistributions 336 AppendixB EquilibriumandCombinatorics 340 AppendixC WKBattheBottom 343 AppendixD SomeAnalyticFunctions 348 AppendixE Euler–MaclaurinandAbel–PlanaFormulas 352 AppendixF SpinandthePauliEquation 356 AppendixG TheG Operator 358 n,m References 362 Index 365 Preface This book grew out of a 40-hour course on statistical physics for third-year undergraduatestudentsthattheseniorauthor(A.S.)gaveatScuolaNormaleduring the 2017–2018 and 2018–2019 academic years, and also includes a few relevant additions.SomeofthelecturesoriginatedinpreviouscoursesgivenattheUniversity of Rome “Tor Vergata,” initially with the late Prof. Roberto Petronzio, and then at Scuola Normale. The two courses offered during the 2017–2018 and 2018–2019 academicyearsincluded20hoursofadditionallectures,givenbythejuniorauthor (C.H.)anddevotedtovariouscomplementsandtotheexplicitsolutionofinstructive problems,whichalsogrewintoimportantportionsofthebook. A peculiarity of this book is perhaps that, while it is aimed at undergraduate studentsatthebeginningoftheirthirdyear,italsoexploresconnectionswithmore advanced topics without resorting, insofar as possible, to the full machinery of quantum field theory. This qualifies the end result as a hybrid, beyond the level of elementary introductions and yet below the level of many excellent textbooks devoted to statistical field theory. Optimistically, we might have blended somehow thevirtuesofboth,butmorerealisticallyweendedupperhapshalfwaybetweenthis goal and the opposite end. Our choice of topics and the level of the presentation, however, were motivated by the advanced level of the undergraduate students at ScuolaNormale,whosecoursesshouldprovidestimulatingviewsaroundandbeyond themainpathsetoutbytheUniversityofPisa. In order to stress its twofold nature, we have decided to divide the book into two parts. The first part (Chapters1–10) is devoted to aspects of classical and quantumstatisticalphysicsthataremorestandardforatextbookofthislevel.These includethedifferentensemblesandtheBoltzmann,Fermi–Dirac,andBose–Einstein statistics; the van der Waals theory; and some key magnetic properties of matter, withemphasisonphasetransitionsandsingularbehavior,butalsoanintroductionto zero-pointfluctuations.Ontheotherhand,thesecondpart(Chapters11–14)explores a selection of more advanced topics, some aspects of which lie at the forefront of currentresearch.Itismeantasarewardforthestudentswhohavemadetheeffortto getthatfar,andshouldalsoserveasastimulustogofartherinmoreadvancedtexts. InChapter11wedescribetheOnsagersolutionofthetwo-dimensionalIsingmodel, and along the way we discuss Kramers–Wannier duality, while also stressing the utmost importance of dualities in more general contexts. In Chapter12 we explore the one-dimensional quantum Heisenberg model, the XXX chain, thus providing a first glimpse at integrability in that context. In Chapter13 we discuss aspects of second-order phase transitions from the vantage point of symmetry. We thus address conformal invariance, taking note of its peculiar form in two dimensions andhintingattheroleoftheAdS/CFTcorrespondence,beforetakingafirstlookat ix the renormalization group. We first consider the XY model, whose treatment rests