Lecture notes in statistical mechanics

148 Pages·2002·1.841 MB·English

by  Vilfan I.

Lecture Notes
in
Statistical Mechanics
Igor Vilfan
The Abdus Salam ICTP, Trieste, Italy
and
The J. Stefan Institute, Ljubljana, Slovenia
Trieste, Autumn 2002
I.Vilfan StatisticalMechanics
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Preface
The present Lecture Notes in Statistical Mechanics were written for the students
of the ICTP Diploma Course in Condensed Matter Physics at the Abdus Salam
ICTPinTrieste,Italy.
Thelecturescoverclassicalandquantumstatisticalmechanicswithsomeem-
phasisonclassicalspinsystems. IgivealsoanintroductiontoBosecondensation
andsuperfluiditybutIdonotdiscussphenomenaspecifictoFermiparticles,being
coveredbyotherlecturers.
I.V.
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I.Vilfan StatisticalMechanics
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Contents
Preface iii
Tables ix
1 Foundations 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 EnsemblesinStatisticalMechanics . . . . . . . . . . . . . . . . . 4
1.2.1 MicrocanonicalEnsembleandtheEntropy
(IsolatedSystems) . . . . . . . . . . . . . . . . . . . . . 4
1.2.2 CanonicalEnsembleandtheFreeEnergy
(SystemsatFixedTemperature) . . . . . . . . . . . . . . 6
1.2.3 GrandCanonicalEnsemble
(Opensystems) . . . . . . . . . . . . . . . . . . . . . . . 10
1.3 AnalogyBetweenFluidsandMagneticSystems . . . . . . . . . . 12
1.4 Fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.5 PhaseTransitions . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.5.1 OrderParameter . . . . . . . . . . . . . . . . . . . . . . 18
1.5.2 CriticalPoint . . . . . . . . . . . . . . . . . . . . . . . . 19
1.5.3 CriticalExponents . . . . . . . . . . . . . . . . . . . . . 21
2 ClassicalModels 25
2.1 RealGases . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
2.2 IsingModel . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.2.1 LatticeGas . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2.2 BinaryAlloys . . . . . . . . . . . . . . . . . . . . . . . . 32
2.2.3 Isingmodelind = 1 . . . . . . . . . . . . . . . . . . . . 33
2.2.4 IsingModelind = 2 . . . . . . . . . . . . . . . . . . . . 38
2.3 PottsandRelatedModels . . . . . . . . . . . . . . . . . . . . . . 40
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2.4 Two-dimensionalxy Model . . . . . . . . . . . . . . . . . . . . . 42
2.5 SpinGlasses . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.5.1 SomePhysicalPropertiesofSpinGlasses . . . . . . . . . 49
3 QuantumModels 53
3.1 QuantumStatistics . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.1.1 TheDensityMatrix . . . . . . . . . . . . . . . . . . . . . 54
3.1.2 EnsemblesinQuantumStatisticalMechanics . . . . . . . 55
3.2 BoseSystems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.1 IdealBoseGas . . . . . . . . . . . . . . . . . . . . . . . 58
3.2.2 BoseEinsteinCondensation . . . . . . . . . . . . . . . . 61
3.3 Liquid4He . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
3.3.1 Superfluidity . . . . . . . . . . . . . . . . . . . . . . . . 68
3.3.2 VortexExcitations . . . . . . . . . . . . . . . . . . . . . 69
3.3.3 OrderParameter . . . . . . . . . . . . . . . . . . . . . . 70
4 MethodsofStatisticalMechanics 73
4.1 Mean-FieldTheories . . . . . . . . . . . . . . . . . . . . . . . . 73
4.1.1 WeissMolecularFieldofanIsingSystem . . . . . . . . . 74
4.1.2 Bragg-WilliamsApproximation . . . . . . . . . . . . . . 75
4.1.3 LandauTheoryofContinuousPhaseTransitions . . . . . 79
4.1.4 CriticalExponentsintheLandauTheory . . . . . . . . . 85
4.1.5 Short-RangeOrderandFluctuations . . . . . . . . . . . . 86
4.1.6 ValidityofTheMean-fieldTheory . . . . . . . . . . . . . 88
4.2 MonteCarloSimulations . . . . . . . . . . . . . . . . . . . . . . 90
4.2.1 ImportanceSampling . . . . . . . . . . . . . . . . . . . . 90
4.2.2 MetropolisAlgorithm . . . . . . . . . . . . . . . . . . . 91
4.2.3 PracticalDetailsandDataAnalysis . . . . . . . . . . . . 91
4.3 TheRenormalizationGroupMethods . . . . . . . . . . . . . . . 96
4.3.1 ATrivialExample: Thed = 1IsingModel . . . . . . . . 101
4.3.2 RGTransformationsonthed = 2IsingModel . . . . . . 104
4.3.3 GeneralPropertiesoftheRGTransformations . . . . . . . 107
4.4 Momentum-SpaceRenormalization
Group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
4.4.1 TheGaussianModel . . . . . . . . . . . . . . . . . . . . 118
4.4.2 TheLandau-WilsonModel . . . . . . . . . . . . . . . . . 121
4.4.3 CriticalExponents . . . . . . . . . . . . . . . . . . . . . 123
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FurtherReading 129
Appendices 131
A DistributionFunctions 131
B MaxwellRelations 133
C BasicThermodynamicRelationsofMagneticSystemsRevisited 135
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Tables
Table1: Ensemblesinstatisticalmechanics
Ensemble Partitionfunction Thermodynamicpotential
Microcanonical ∆Ω(E,V,N) S(E,V,N) = k ln∆Ω(E,V,N)
B
Canonical Z(T,V,N) F(T,V,N) = −k T lnZ(T,V,N)
B
= P e−Ei(V,N)/kBT
i
= R ∆Ω(E,V,N)e−E(V,N)/kBTdE
Y(T,p,N) G(T,p,N) = −k T lnY(T,p,µ)
B
Grandcanonical Ξ(T,V,µ) J(T,V,µ) = F −µN = −pV
= P eµN/kBTZ(T,V,N) = −k T lnΞ(T,V,µ)
N B
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Table2: Thermodynamicfunctions
Thermodynamicfunctions Natural Totaldifferential
(Definition) variables
Entropy
S hEi,V,N dS = dhEi/T +pdV/T −µdN/T
Internalenergy
hEi S,V,N dhEi = TdS −pdV +µdN
Enthalpy
H = hEi+pV S,p,N dH = TdS +Vdp+µdN
Helmholtzfreeenergy
F = hEi−TS T,V,N dF = −SdT −pdV +µdN
Gibbsfreeenergy
G = F +pV = Nµ T,p,N dG = −SdT +Vdp+µdN
Grandpotential
J = F −G = −pV T,V,µ dJ = −SdT −pdV −Ndµ
x