Statistical Physics

205 Pages·1995·2.778 MB·English

by  Tony Guénault (auth.)

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
x Preface
Thebookwill,Ihope,beusefulforuniversitycoursesofvariouslengthsandtypes.
Severalexamplesfollow:
1. Basic general course for physics undergraduates (20–25 lectures): most of
Chapters1–12,omittinganyofChapters7,10,11and12iftimeisshort;
2. Short introductory course on statistical ideas (about 10 lectures): Chapters 1, 2
and3possiblywithmaterialaddedfromChapters10and11;
3. Following (2), a further short course on statistics of gases (15 lectures):
Chapters4–6 and 8–9, with additional material available from Chapter 14 and
15.2;
4. Forchemicalphysics(20lectures):Chapters1–7and10–13;
5. Asanintroductiontocondensedmatterphysics(20lectures):Chapters1–6,8–12,
14,15.1.
In addition to those already acknowledged earlier, I would like to thank Keith
Wigmore for his thorough reading of the first edition and Terry Sloan for his
considerableinputtomyunderstandingofthematerialinsection15.2.1.
A.M.Guénault
1994
Preface to the revised and enlarged
second edition
ThisthirdeditionofStatisticalPhysicsfollowstheorganizationandpurposeofthe
secondedition,withcomparativelyminorupdatingandchangestothetext.Ihopeit
continuestoprovideanaccessibleintroductiontothesubject,particularlysuitablefor
physicsundergraduates.Chaptersummarieshavebeenaddedtothefirstnine(basic)
chapters,inordertoencouragestudentstorevisetheimportantideasofeachchapter–
essentialbackgroundforaninformedunderstandingoflaterchapters.
A.M.Guénault
2007
Preface xi
A SURVIVAL GUIDE TO STATISTICAL PHYSICS
Chapter 1Assembly of NN identical particles
volumeVVV, in thermal equilibrium at
temperature T
are the particles weakly interacting?
YES NO
gaseous particles? Chapter 12 or 14 could help
orr read a thicker book
orrgive up!
NO YES
Chapters 2,3 Chapters 4,5
use Boltzmann statistics is occupation number of each statef<<1?
nnjj= NZexp (–εεj/kkkBTTT) i.e. isA=NV(2(cid:2)Mh2kkkTTT(3/2<<1?
partition function: B
Z=Σ(–εεj/kkkBTTT)
all YES NO
states
Chapters 6,7 is ΨAor S?
use MB statistics
F=––NkkkBTlnZ f(ε) =NZexp (–ε/kkkBTTT) A S
(2(cid:2)MkkkT(3/2
Z = V B
h2 Chapter 8 Chapter 9
for a monatomic gas use FD statistics use BE statistics
f(ε) = 1 f(ε)= 1
exp {(ε –(cid:2))/kkkT} + 1 Bexp {ε/kkkT} – 1
B B
F=–kkkBTln(NZN( asppipnl ipeas rttoic oledsd (-ehlaelcft irnotnesg,ral asapptoipnml ipes,as r4ttHoic ezl)ee sSr oe( sto oBrm i=ne1t ec igofrldal
3He, nucleons, some cold no particle conservation
atoms) (photons, phonons)
1
Basic ideas
Thereisanobviousproblemaboutgettingtogripswithanunderstandingofmatterin
thermalequilibrium.Letussupposeyouareinterested(asadesignerofsaucepans?)
in the thermal capacity of copper at 450K. On the one hand you can turn to ther-
modynamics, but this approach is of such generality that it is often difficult to see
thepoint.Relationshipsbetweentheprincipalheatcapacities,thethermalexpansion
coefficient and the compressibility are all very well, but they do not help you to
understandtheparticularmagnitudeandtemperaturedependenceoftheactualheat
capacity of copper. On the other hand, you can see that what is needed is a micro-
scopicmechanicalpictureofwhatisgoingoninsidethecopper.However,thispicture
becomesimpossiblydetailedwhenonestartstodiscussthelawsofmotionof1024
orsocopperatoms.
Theaimofstatisticalphysicsistomakeabridgebetweentheover-elaboratedetail
ofmechanicsandtheobscuregeneralitiesofthermodynamics.Inthischapterweshall
lookatonewayofmakingsuchabridge.Mostreaderswillalreadybefamiliarwith
thekinetictheoryofidealgases.Thetreatmentgivenherewillenableustodiscuss
a much wider variety of matter than this, although there will nevertheless be some
limitationstothetrafficthatcantravelacrossthebridge.
1.1 THEMACROSTATE
The basic task of statistical physics is to take a system which is in a well-defined
thermodynamic state and to compute the various thermodynamic properties of that
systemfroman(assumed)microscopicmodel.
The‘macrostate’isanotherwordforthethermodynamicstateofthesystem.Itis
aspecificationofasystemwhichcontainsjustenoughinformationforitsthermody-
namicstatetobewelldefined,butnomoreinformationthanthat.Asoutlinedinmost
booksonthermalphysics(e.g. Finn’sbookThermalPhysics inthisseries), forthe
simplecaseofapuresubstancethiswillinvolve:
• thenatureofthesubstance–e.g.naturalcopper;
• theamountofthesubstance–e.g.1.5moles;
1