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Statistical Physics PDF

217 Pages·2012·2.24 MB·English
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StatisticalPhysics Statistical Physics Second Revised and Enlarged Edition by Tony Guénault EmeritusProfessorofLowTemperaturePhysics LancasterUniversity,UK AC.I.P.CataloguerecordforthisbookisavailablefromtheLibraryofCongress. ISBN978-1-4020-5974-2(PB) ISBN978-1-4020-5975-9(e-book) PublishedbySpringer, P.O.Box17,3300AADordrecht,TheNetherlands. www.springer.com Printedonacid-freepaper Firstedition1988 Secondedition1995 Reprinted1996,2000,2001,2003 Reprintedrevisedandenlargedsecondedition2007 AllRightsReserved ©1988,1995A.M.Guénault ©2007Springer Nopartofthisworkmaybereproduced,storedinaretrievalsystem,ortransmittedinanyformorby anymeans,electronic,mechanical,photocopying,microfilming,recordingorotherwise,withoutwritten permissionfromthePublisher,withtheexceptionofanymaterialsuppliedspecificallyforthepurposeof beingenteredandexecutedonacomputersystem,forexclusiveusebythepurchaserofthework. Table of contents Preface ix 1 Basicideas 1 1.1 Themacrostate 1 1.2 Microstates 2 1.3 Theaveragingpostulate 3 1.4 Distributions 4 1.5 Thestatisticalmethodinoutline 6 1.6 Amodelexample 7 1.7 Statisticalentropyandmicrostates 10 1.8 Summary 11 2 Distinguishableparticles 13 2.1 TheThermalEquilibriumDistribution 14 2.2 Whatareαandβ? 17 2.3 Astatisticaldefinitionoftemperature 18 2.4 Theboltzmanndistributionandthepartitionfunction 21 2.5 Calculationofthermodynamicfunctions 22 2.6 Summary 23 3 Twoexamples 25 3.1 ASpin-1 solid 25 2 3.2 Localizedharmonicoscillators 36 3.3 Summary 40 4 Gases:thedensityofstates 43 4.1 Fittingwavesintoboxes 43 4.2 Otherinformationforstatisticalphysics 47 4.3 Anexample–heliumgas 48 4.4 Summary 49 5 Gases:thedistributions 51 5.1 Distributioningroups 51 5.2 Identicalparticles–fermionsandbosons 53 5.3 Countingmicrostatesforgases 55 5.4 Thethreedistributions 58 5.5 Summary 61 v vi Tableofcontents 6 Maxwell–Boltzmanngases 63 6.1 ThevalidityoftheMaxwell–Boltzmannlimit 63 6.2 TheMaxwell–Boltzmanndistributionofspeeds 65 6.3 Theconnectiontothermodynamics 68 6.4 Summary 71 7 Diatomicgases 73 7.1 Energycontributionsindiatomicgases 73 7.2 Heatcapacityofadiatomicgas 75 7.3 Theheatcapacityofhydrogen 78 7.4 Summary 81 8 Fermi–Diracgases 83 8.1 PropertiesofanidealFermi–Diracgas 84 8.2 Applicationtometals 91 8.3 Applicationtohelium-3 92 8.4 Summary 95 9 Bose–Einsteingases 97 9.1 PropertiesofanidealBose–Einsteingas 97 9.2 Applicationtohelium-4 101 9.3 Phoneybosons 104 9.4 Anoteaboutcoldatoms 109 9.5 Summary 109 10 Entropyinothersituations 111 10.1 Entropyanddisorder 111 10.2 Anassemblyatfixedtemperature 114 10.3 Vacanciesinsolids 116 11 Phasetransitions 119 11.1 Typesofphasetransition 119 11.2 Ferromagnetismofaspin-1 solid 120 2 11.3 Realferromagneticmaterials 126 11.4 Order–disordertransformationsinalloys 127 12 Twonewideas 129 12.1 Staticsordynamics? 129 12.2 Ensembles–alargerview 132 13 Chemicalthermodynamics 137 13.1 Chemicalpotentialrevisited 137 13.2 Thegrandcanonicalensemble 139 13.3 Idealgasesinthegrandensemble 141 13.4 Mixedsystemsandchemicalreactions 146 Tableofcontents vii 14 Dealingwithinteractions 153 14.1 Electronsinmetals 154 14.2 Liquidhelium-3:AFermiliquid 158 14.3 Liquidhelium-4:ABoseliquid? 163 14.4 Realimperfectgases 164 15 Statisticsunderextremeconditions 169 15.1 SuperfluidstatesinFermi–Diracsystems 169 15.2 Statisticsinastrophysicalsystems 174 AppendixASomeelementarycountingproblems 181 AppendixBSomeproblemswithlargenumbers 183 AppendixCSomeusefulintegrals 187 AppendixDSomeusefulconstants 191 AppendixEExercises 193 AppendixFAnswerstoexercises 199 Index 201 Preface Preface to the first edition Statistical physics is not a difficult subject, and I trust that this will not be found a difficultbook. ItcontainsmuchthatanumberofgenerationsofLancasterstudents havestudiedwithme,aspartoftheirphysicshonoursdegreework.Thelecturecourse wasof20hours’duration,andIhaveaddedcomparativelylittletothelecturesyllabus. Aprerequisiteisthatthereadershouldhaveaworkingknowledgeofbasicthermal physics(i.e.thelawsofthermodynamicsandtheirapplicationtosimplesubstances). ThebookThermalPhysicsbyColinFinninthisseriesformsanidealintroduction. Statisticalphysicshasathousandandonedifferentwaysofapproachingthesame basic results. I have chosen a rather down-to-earth and unsophisticated approach, withoutIhopetotallyobscuringtheconsiderableinterestofthefundamentals.This enablesapplicationstobeintroducedatanearlystageinthebook. As a low-temperature physicist, I have always found a particular interest in sta- tisticalphysics,andespeciallyinhowtheabsolutezeroisapproached.Ishouldnot, therefore,apologizeforthelow-temperaturebiasinthetopicswhichIhaveselected fromthemanypossibilities. Withoutburdeningthemwithanyresponsibilityformycompetence,Iwouldlike toacknowledgehowmuchIhavelearnedinverydifferentwaysfrommyfirstthree ‘bosses’asatraineephysicist:BrianPippard,KeithMacDonaldandSydneyDugdale. More recently my colleagues at Lancaster, George Pickett, David Meredith, Peter McClintock,ArthurCleggandmanyothershavedonemuchtokeepmeontherails. Finally,butmostofall,IthankmywifeJoanforherencouragement. A.M.Guénault 1988 Preface to the second edition Some new material has been added to this second edition, whilst leaving the organization of the rest of the book (Chapters 1–12) unchanged. The new chapters aim to illustrate the basic ideas in three rather distinct and (almost) independent ways. Chapter 13 gives a discussion of chemical thermodynamics, including something about chemical equilibrium. Chapter 14 explores how some interactingsystemscanstillbetreatedbyasimplestatisticalapproach,andChapter15 looks at two interesting applications of statistical physics, namely superfluids and astrophysics. ix

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