Equilibrium statistical physics: phases of matter and phase transitions

374 Pages·2008·2.14 MB·English

by  M. Baus, Carlos F. Tejero

Equilibrium Statistical Physics
Marc Baus Carlos F. Tejero
Equilibrium
Statistical Physics
Phases of Matter and Phase Transitions
With 82 Figures and 8 Tables
MarcBaus CarlosF.Tejero
Universite´LibredeBruxelles UniversidadComplutensedeMadrid
Faculte´desSciences FacultaddeCienciasF´ısicas
BoulevardduTriomphe CiudadUniversitaria
1050Bruxelles,Belgium 28040Madrid,Spain
[email protected] cftejero@fis.ucm.es
ISBN978-3-540-74631-7 e-ISBN978-3-540-74632-4
LibraryofCongressControlNumber:2007936163
(cid:2)c MarcBaus,CarlosF.Tejero2008
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Preface
The purpose of this textbookis to introducethe student to a basic area of macro-
scopic physics, namely the statistical mechanical study of the different phases of
matter, as well as the phase transitions between them. Although many books on
statistical physics, for both equilibrium and non-equilibriumsystems, are already
available,theylargelydifferincontents.Thisgenerallyreflectsnotonlythediffer-
ent interests of their authors, but also the epoch in which they were written. For
instance, the early books did usually devote much space to problems of the solid
state,whereaslateronesdoinclude,moreover,severalaspectsoftheliquidstate.At
present,however,the main emphasisin physicsis on softmatter (e.g.liquid crys-
tals, colloids, polymers), and therefore these particular states of matter have also
beenincludedinthisvolume.Themainpurposeofthistextbookwillconsist,hence,
inprovidingitsstudentswitha firstintroduction,withinthegeneralframeworkof
equilibriumstatisticalphysics,toamuchlargervarietyofphasesandphasetransi-
tionsthanwaspreviouslythe case fortextbooksof statistical mechanics.Manyof
thesenoveltopicsdo,ofcourse,deserveamoredetailedstudythantheonewhich
canbe providedhere.Indeed,in the spiritof a firstintroduction,onlyverysimple
modelsofthesephaseswillbegiven,butmoredetailedinformationcanbefoundin
thesuggestionsforfurtherreadinggivenintheReferences.
For pedagogicalreasons, the subject matter of this book has been divided into
four parts (Parts I–IV), and a series of appendices(AppendicesA–D), devoted to
somemathematicaltoolsusedinthemaintext.
InPartI (Chaps.1–3),a summaryisprovidedofthe mechanical(Chap.1) and
thermodynamical(Chap.2)basisofthepostulatesofequilibriumstatisticalphysics
(Chap.3).Althoughmoststudentsofequilibriumstatisticalmechanicswillingen-
eralhaveapriorknowledgeof,classicalandquantum,mechanicsandofequilibrium
thermodynamics,byincludingasummaryofthesetopicsthepresentbookbecomes
moreself-contained.Afactgenerallywellappreciatedbythestudents.
In PartII (Chaps. 4–6),the generalprinciplesof equilibriumstatistical physics
areillustratedforthesimplecaseofthenon-interactingoridealsystems.Thethree
main Gibbs ensembles, microcanonical (Chap. 4), canonical (Chap. 5) and grand
canonical(Chap.6),arestudied,togetherwithsomeoftheirstandardapplications.
vii
viii Preface
Thesetopics,althoughsimpleinprinciple,doneverthelessdrawthestudents’atten-
tiontosomeofthesubtletiesofstatisticalphysics.
In Part III (Chaps. 7–9), these general principlesare applied to the less simple
interactingor non-idealsystems. This, of course, can onlybe doneapproximately
and the most currentapproximationmethodsfor the study of classical interacting
systems are summarized(Chap.7). Based on these methods,some simple models
formostofthecurrentphasesofmatterareintroduced(Chap.8),andthephasetran-
sitionsbetweensomeofthesephasesaresubsequentlystudied(Chap.9).Although
the simple modelsof these phasesconsideredhereare onlycaricaturesof the real
systems, their study will preparethe studentsforthe more realistic, butalso more
difficult,studiesfoundinthecurrentliterature.
Finally,PartIV(Chaps.10–13)isdevotedto anintroductiontosomemoread-
vancedmaterial.Thisincludesanintroductiontocriticalphenomenaandtherenor-
malizationgroupcalculationofcriticalexponents(Chap.10),thestudyofinterfaces
andthecalculationofthesurfacetension(Chap.11),thestudyoftopologicaldefects
(innematicliquidcrystals)andtheresultingtexture(Chap.12),andtheclassicalthe-
oriesofthephasetransformationkinetics(Chap.13).Itishopedthatexposingthe
studentsto anelementarytreatmentofthese moreadvancedtopicswillencourage
themtoalsostudythesetopicsinmoredetail.
Thepresenttextbookisaimedatstudentsofcondensedmatterphysics,physical
chemistryormaterialsscience, butthroughoutthe levelof rigoris onewhichwill
bemostfamiliartostudentsoftheoreticalphysics.Likewise,thereferencesquoted
attheendofeachchaptermainlyfocusonbookswheresupplementarymaterialcan
befoundwithasimilardegreeofrigor,i.e.thosetextsmostusefultothereadersof
thepresentone.
Thisvolumeisanenlargedversionofaprevioustext,originallywritteninSpan-
ish(C. F. TejeroyM.Baus, F´ısicaestad´ısticadelequilibrio.Fasesdelamateria,
A.D.I.,Madrid(2000),ISBN84-931805-0-5).WeareindebtedtoM.Lo´pezdeHaro
foritstranslationintoEnglishandalsoforhelpfulandinterestingsuggestions.We
alsothankT.deVosforhishelpwiththefigures.Thisfinalversionownsmuchto
themanydiscussionswithourstudentsattheFaculte´desSciences(Universite´Libre
deBruxelles)andattheFacultaddeCienciasF´ısicas(UniversidadComplutensede
Madrid)andwithourcolleagues,M.Fisher,R.Lovett,J.P.RyckaertandJ.M.Ortiz
deZa´rate,allofwhicharegratefullyacknowledged.Finally,wewouldalsoliketo
thankourwifes,MyriamandIsabel,fortheirsupport.
BrusselsandMadrid, M.Baus
June2007 C.F.Tejero
Contents
PartI Basics
1 Mechanics..................................................... 3
1.1 ClassicalMechanics ........................................ 3
1.2 Hamilton’sEquations ....................................... 4
1.3 ExternalParameters ........................................ 5
1.4 DynamicalFunctions ....................................... 7
1.5 QuantumMechanics........................................ 9
1.6 Self-adjointOperators....................................... 10
1.7 EigenvalueEquation........................................ 11
1.8 Schro¨dinger’sEquation...................................... 13
1.8.1 FreeParticle ........................................ 14
1.8.2 HarmonicOscillator.................................. 16
1.8.3 ParticleinaMagneticField............................ 16
1.9 SystemofIdenticalParticles ................................. 18
References..................................................... 20
2 Thermodynamics .............................................. 21
2.1 FundamentalEquation ...................................... 21
2.2 IntensiveVariables ......................................... 23
2.3 LawofEntropyIncrease..................................... 25
2.4 ThermodynamicPotentials................................... 26
2.5 EquilibriumConditions ..................................... 28
2.6 StabilityConditions ........................................ 29
2.7 CoexistenceConditions ..................................... 32
2.8 PhaseDiagrams............................................ 33
2.8.1 GibbsFreeEnergy ................................... 33
2.8.2 HelmholtzFreeEnergy ............................... 37
2.8.3 vanderWaalsLoop .................................. 39
2.8.4 AnExampleofaPhaseDiagram ....................... 41
References..................................................... 43
ix
x Contents
3 StatisticalPhysics .............................................. 45
3.1 DynamicalFunctionsandFields .............................. 46
3.2 Liouville’sEquation ........................................ 48
3.3 SystemsinEquilibrium ..................................... 49
3.4 DensityOperator........................................... 51
3.5 Ergodicity................................................. 53
3.6 ThermodynamicLimit ...................................... 55
3.7 SymmetryBreaking ........................................ 61
References..................................................... 63
PartII IdealSystems
4 MicrocanonicalEnsemble....................................... 67
4.1 ClassicalMicrocanonicalEnsemble ........................... 67
4.2 ClassicalIdealGas ......................................... 69
4.3 EntropyandtheGibbsParadox ............................... 71
4.4 TemperatureandThermalEquilibrium......................... 75
4.5 IdealSystems.............................................. 77
4.6 EquipartitionTheorem ...................................... 78
4.7 EquationofState........................................... 79
4.8 EntropyandIrreversibility ................................... 81
4.9 QuantumMicrocanonicalEnsemble........................... 83
4.10 AbsoluteNegativeTemperatures.............................. 86
References..................................................... 88
5 CanonicalEnsemble............................................ 89
5.1 ClassicalCanonicalEnsemble................................ 89
5.2 MeanValuesandFluctuations................................ 91
5.3 HelmholtzFreeEnergy...................................... 93
5.4 ClassicalIdealGas ......................................... 94
5.5 IdealGasinanExternalPotential ............................. 95
5.6 EquipartitionTheorem ...................................... 97
5.7 ClassicalTheoryofRadiation ................................ 99
5.8 ClassicalTheoryofSolids ...................................101
5.9 QuantumCanonicalEnsemble................................103
5.10 IdealQuantumSystems .....................................105
5.11 Maxwell–BoltzmannStatistics ...............................106
5.12 Maxwell–Boltzmann’sIdealGas..............................108
5.13 Brillouin’sParamagnetism...................................109
5.14 PhotonGas................................................111
5.15 PhononGas ...............................................114
References.....................................................119
Contents xi
6 GrandCanonicalEnsemble .....................................121
6.1 ClassicalGrandCanonicalEnsemble ..........................121
6.2 MeanValuesandFluctuations................................125
6.3 GrandPotential ............................................127
6.4 ClassicalIdealGas .........................................128
6.5 ClassicalIdealGasinanExternalPotential.....................129
6.6 Two-ParticleDistributionFunction............................131
6.7 DensityFluctuations........................................133
6.8 CorrelationsattheCriticalPoint ..............................135
6.9 QuantumGrandCanonicalEnsemble..........................138
6.10 Bose–EinsteinandFermi–DiracStatistics ......................139
6.11 VirialExpansionsintheClassicalLimit........................142
6.12 BosonGas:Bose–EinsteinCondensation.......................144
6.12.1 SpecificHeat........................................147
6.12.2 EquationofState ....................................150
6.13 FermionGas...............................................152
References.....................................................158
PartIII Non-idealSystems
7 ClassicalSystemswithInteractions ..............................163
7.1 ThermodynamicIntegration..................................163
7.2 ThermodynamicPerturbationTheory..........................165
7.3 VirialExpansions ..........................................167
7.4 DirectCorrelationFunction ..................................171
7.5 DensityFunctionalTheory...................................173
7.6 MeanFieldTheory .........................................175
7.7 NumericalSimulations......................................177
7.7.1 MolecularDynamics .................................178
7.7.2 MonteCarloMethod .................................179
References.....................................................183
8 PhasesofMatter ..............................................185
8.1 Crystals...................................................186
8.1.1 CrystalStructure.....................................186
8.1.2 CellTheory.........................................188
8.1.3 vanderWaalsTheory.................................191
8.1.4 VariationalTheory ...................................192
8.2 Fluids ....................................................196
8.2.1 DenseFluids........................................196
8.2.2 FluidStructure ......................................199
8.2.3 FluidsandGlasses ...................................203
8.2.4 VaporandLiquid ....................................206
xii Contents
8.3 Mixtures..................................................209
8.3.1 BinaryMixtures .....................................209
8.3.2 ColloidalSuspensions ................................211
8.3.3 Asakura–OosawaPotential ............................213
8.3.4 DLVOPotential .....................................215
8.4 LiquidCrystals ............................................216
8.4.1 Maier–SaupeTheory .................................217
8.4.2 OnsagerTheory .....................................220
8.5 Polymers .................................................223
8.5.1 RadiusofGyration...................................224
8.5.2 Flory–HugginsTheory................................226
References.....................................................231
9 PhaseTransitions ..............................................233
9.1 StructuralTransitions .......................................233
9.1.1 Fluid–SolidTransition................................234
9.1.2 Isotropic–NematicTransition ..........................237
9.2 IsostructuralTransitions.....................................239
9.2.1 Liquid–VaporTransition ..............................239
9.2.2 Solid–SolidTransition................................241
9.3 SymmetryBreakingandOrderParameters .....................242
9.4 LandauTheory.............................................245
9.4.1 ContinuousTransitions ...............................247
9.4.2 DiscontinuousTransitions.............................247
9.5 BifurcationTheory .........................................248
9.6 CriticalPoints .............................................251
9.6.1 IsolatedCriticalPoints................................251
9.6.2 Liquid–VaporCriticalPoint ...........................253
9.6.3 Solid–SolidCriticalPoint .............................254
9.6.4 ConsoluteCriticalPoint ..............................256
9.6.5 CriticalLines .......................................259
9.7 Summary .................................................261
9.8 TriplePoints...............................................262
9.8.1 OrdinaryTriplePoint.................................262
9.8.2 CriticalEndpoint ....................................263
9.8.3 BicriticalPoint ......................................264
9.8.4 TricriticalPoint......................................265
References.....................................................265