Lectures on Quantum Statistics: With Applications to Dilute Gases and Plasmas

279 Pages·2019·4.122 MB·English

by  Werner Ebeling, Thorsten Pöschel

Lecture Notes in Physics 953
Werner Ebeling
Thorsten Pöschel
Lectures on
Quantum
Statistics
With Applications to Dilute Gases and
Plasmas
Lecture Notes in Physics
Volume 953
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Werner Ebeling (cid:129) Thorsten Poschel
Lectures on Quantum
Statistics
With Applications to Dilute Gases
and Plasmas
123
WernerEbeling ThorstenPöschel
FBPhysik Friedrich-Alexander-Universität
HumboldtUniversita¨tzuBerlin Erlangen-Nürnberg
Berlin,Germany Erlangen,Germany
ISSN0075-8450 ISSN1616-6361 (electronic)
LectureNotesinPhysics
ISBN978-3-030-05733-6 ISBN978-3-030-05734-3 (eBook)
https://doi.org/10.1007/978-3-030-05734-3
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Preface
In spite of the dominatingimportance of solids and liquids in our life, it is a fact
that most of the matter in our universeoccursin the state of gas and plasma. The
overwhelming part of this matter exists under extreme conditions. Therefore, the
physicsofgasesandplasmasisafieldofgrowinginterest.We believethatweare
just at the beginning of the exploration of the world beyond the narrow window
of experience given by the state of our planet. Thus, the deeper reason for the
growinginterestinthephysicsofgasesandplasmasisthedominatingroleofthis
stateofmatterintheuniverse.Thesubjectofresearchisfundamentalphysicsand
in particular the quantum-statistical thermodynamics of states of matter, starting
from rare gases and ending with plasma states. We are deeply convinced that the
thermodynamics and quantum statistics are still the solid foundation on which
even the most advanced research is operating. It remains our conviction that in
spite of a flood of modern concepts, the traditional concepts of thermodynamics
and statistics with terms such as temperature, pressure, energy, and entropy and
the theory developed by Planck, Einstein, Nernst, Saha, and other pioneers are
fundamentaltothefield.Thebodyofthesenotesisbasedonlecturesatuniversities
andpresentationsatseminarsandconferences.Inparticular,theselecturenotesare
based on the lectures on quantum statistics and plasma physics given by Werner
EbelingattheUniversityofRostockbetween1970and1979andattheHumboldt
UniversityofBerlinbetween1980and2001,includingshorterguestlecturesatthe
Universities Paris VI 1977, Minneapolis 1986, Moscow 2005, and Krakow 2008.
The body of the text is based on the last full two-semester course on quantum
statistics held jointly by Werner Ebeling and Thorsten Pöschel at the Humboldt
UniversityofBerlin,assistedbyseveralformercoworkerssuchasAndreasFörster,
BurkhardMilitzer,LutzMolgedey,andWaldemar Richertandbyseveralstudents
such as Jörn Dunkel, Hendrik Hache, Stefan Hilbert, Dirk Holste, Thomas Pohl,
MichaelSpahn,andothers.Thistextisbasedonthenotesincludingofthecomplete
lecturewrittendownbyThorstenPöschelaswellasseverallaterrevisions.
Following the personal interests of the authors and the traditions of quantum
statisticalthermodynamicsinBerlinandRostock,weconcentrateonthefundamen-
tals and applications to gases and plasmas. This is in contrast to most textbooks
v
vi Preface
andmonographsonquantumstatisticswhichrevealsomebiastocondensedmatter
and in particular to solid state systems. We believe that the focus on gases and
plasmas corresponds to trends in research evidenced, e.g., by several recent large
internationalresearchprojectsandconferences.
Thisbookiswrittenonanintermediatelevel,thatis,mostofitscontentsshould
be accessible for undergraduate and graduate students, while a few results might
be on a more advanced level and, thus, addressed to young researchers in the
field. Readersinterested in additionalinformation,in particular also on numerical
methods and extreme states of matter, are referred to a monographby one of the
current authors (W.E.) together with V.E. Fortov and V. Filinov which appeared
recently in Springer’s “Series in Plasma Science and Technology” (Ebeling et al.
2017).
On many occasions, the text contains historical remarks on the roots of quan-
tum statistics. There are two reasons why we decided to include this historical
material:
– Thefirstreasonisrelatedtothegeniusloci:Infact,alargepartofthebookgoes
backtolecturesgiveninBerlinwheremanyfundamentalsofquantumstatistics
weredevelopedbetween1900and1932.
– We are convinced—and this is based on a long experience in lecturing—that
studentsarestronglymotivatedbylookingbacktothehistoryofthedevelopment
ofourscience.
The spirit of this textbookis influenced by meetings and discussions with several
pioneersofquantumstatisticalthermodynamics,such asAlexanderA. Abrikosov,
Berni Alder, Nikolay N. Bogolyubov, Alexander S. Davydov, Hans Falkenhagen,
MichaelFisher, VitaliL. Ginzburg,GünterKelbg,HughDeWitt, and inparticular
Yuri L. Klimontovich. Further, we have to thank our colleagues in Berlin, in
particular Dietmar Ebert, Wolfgang Muschik, and Lutz Schimansky-Geier,and in
Rostock,inparticularKlausKilimann,Wolf-DietrichKraeft,DietrichKremp,and
GerdRöpke,aswellasPeterHänggifromAugsburgforcollaboration,advice,and
discussions.
Berlin,Germany WernerEbeling
Erlangen,Germany ThorstenPöschel
Summer2018
Reference
Ebeling,W.,V.E.Fortov,andV.S.Filinov.2017.QuantumStatisticsofDenseGases
and Nonideal Plasmas. Springer Series in Plasma Science and Technology.
Springer.
Contents
1 BasicPhysicsofGasesandPlasmas ....................................... 1
1.1 MatterinFormofGasesandPlasmas ................................. 1
1.2 BasicPhysicsofGases.................................................. 2
1.2.1 ClassicalGasLaws............................................. 2
1.2.2 Third Law of Thermodynamicsand the Roots
ofQuantumStatistics........................................... 5
1.3 BasicPhysicsofPlasmas............................................... 8
1.3.1 CoulombInteractionsandIonizationEquilibrium............ 8
1.3.2 DifferentPlasmaStatesonEarth............................... 12
References..................................................................... 16
2 ElementsofQuantumStatisticalTheory.................................. 19
2.1 Many-BodyQuantumTheory.......................................... 19
2.1.1 QuantumStates ................................................. 19
2.1.2 IdentityandSymmetry ......................................... 24
2.2 QuantumDynamicsofManyParticles................................. 27
2.2.1 SchrödingerEquation........................................... 27
2.2.2 PureandMixedEnsembles .................................... 32
2.3 StandardApproximationsforMany-ParticleStates................... 35
2.3.1 Hartree-FockApproximation .................................. 35
2.3.2 Born-OppenheimerApproximation,VirialTheorem
andCoulombStability.......................................... 36
2.3.3 Thomas-FermiTheoryofMulti-ElectronAtoms ............. 38
2.4 QuantumStatisticalEnsembleTheory................................. 41
2.4.1 MicrocanonicalandCanonicalEnsembles.................... 41
2.4.2 GrandCanonicalEnsembles ................................... 42
2.5 TheoryofFluctuationsandRelaxationProcesses..................... 44
2.5.1 Einstein-OnsagerRelaxationTheory.......................... 44
2.5.2 Correlations,Spectra,andSymmetryRelations............... 49
References..................................................................... 52
vii
viii Contents
3 IdealQuantumGases........................................................ 55
3.1 QuantumStatisticsofOscillatorandPhononGases .................. 55
3.1.1 EinsteinModelofOscillationsinCrystals .................... 55
3.1.2 DebyeTheoryofPhononExcitationsinLattices ............. 57
3.2 StatisticsofBose-EinsteinandFermi-DiracGases.................... 58
3.2.1 DevelopmentoftheQuantumStatisticsofGases............. 58
3.2.2 GasesasParticleSystemswithAdditiveHamiltonian........ 61
3.3 Fermi-andBoseDistributions.......................................... 62
3.3.1 Bose-EinsteinGases............................................ 62
3.3.2 Fermi-DiracGases.............................................. 63
3.4 ThermodynamicsPropertiesofBose-EinsteinGases ................. 64
3.5 BlackBodyRadiationandRelativisticGases ......................... 70
3.5.1 Planck’sLawofRadiation ..................................... 70
3.5.2 RadiationasaRelativisticGas................................. 74
3.6 ThermodynamicFunctionsofFermiGases............................ 75
3.6.1 TheIdealFermiGas............................................ 75
3.6.2 FermiGasesintheHigh-andLow-TemperatureLimits...... 76
3.7 Density-DependentFermiGasFunctions.............................. 79
3.7.1 ExpansionsforWeaklyDegeneratedFermiGases............ 79
3.7.2 ThermodynamicsinFullDensityRange ...................... 81
3.8 Hartree-FockTheoryofWeaklyInteractingElectronGases.......... 83
References..................................................................... 88
4 DensityOperatorsandOtherToolsofQuantumStatistics ............. 91
4.1 DensityMatricesandOperators........................................ 92
4.1.1 DensityMatrices................................................ 92
4.1.2 vonNeumann’sDensityOperatorsandTimeEvolution...... 94
4.1.3 MaximumEntropyPrincipleandThermodynamic
Functions........................................................ 100
4.2 RepresentationsinCoordinateSpaceandTwo-TimeFunctions...... 103
4.2.1 CoordinateRepresentationsandBlochEquations ............ 103
4.2.2 Two-TimeDensityOperators .................................. 108
4.3 Bogolyubov’sReducedDensityOperators ............................ 109
4.4 Slater-,Wigner-andKlimontovichRepresentations.................. 111
4.4.1 SlaterRepresentations.......................................... 112
4.4.2 Wigner-Representation......................................... 113
4.4.3 Klimontovich’sMicroscopicDensity.......................... 114
4.5 DensityFunctionals,VirialTheoremsandStability................... 116
4.5.1 Kohn-ShamandThomas-FermiFunctionals.................. 116
4.5.2 CoulombStabilityandVirialTheorem........................ 118
4.6 SecondQuantization.................................................... 119
4.6.1 OccupationNumberRepresentations.......................... 119
4.6.2 SecondQuantization............................................ 123
4.6.3 KlimontovichOperatorinSecondQuantization .............. 125
Contents ix
4.7 Green’sFunctions....................................................... 126
4.7.1 DefinitionandRelationsforGreen’sFunctions............... 126
4.7.2 ThermodynamicsandGreen’sFunctions...................... 129
4.8 PairBoundStatesandBethe-SalpeterEquation....................... 134
References..................................................................... 138
5 RealGasQuantumStatistics ............................................... 141
5.1 ClusterExpansionsforRealGases..................................... 141
5.2 SlaterFunctionsandVirialCoefficients ............................... 146
5.2.1 GeneralDensityExpansions ................................... 146
5.2.2 SlaterSumsforPairCorrelations.............................. 148
5.3 TheSecondVirialCoefficient.......................................... 149
5.3.1 VirialCoefficientIncludingExchangeEffects................ 149
5.3.2 Beth–UhlenbeckMethodforNon-associatingGases......... 153
5.4 EquationofStateforGaseswithDeepBoundStates................. 154
5.4.1 FugacityExpansions............................................ 154
5.4.2 ChemicalPicture................................................ 159
5.5 HeliumandOtherQuantumGasesatLowTemperature.............. 161
5.5.1 VirialExpansionforHelium ................................... 161
5.5.2 PhaseTransitionsinLow-TemperatureGases................. 162
5.6 WeaklyInteractingQuantumGases.................................... 164
5.6.1 BlochEquation ................................................. 164
5.6.2 SlaterFunctionandFreeEnergy............................... 167
References..................................................................... 169
6 QuantumStatisticsofDilutePlasmas ..................................... 171
6.1 BasicPhysicsofPlasmas............................................... 171
6.1.1 ScreeningandLatticeFormationinCoulombSystems....... 171
6.1.2 TheDivergenceofthePartitionFunction ..................... 175
6.2 PairCorrelationsonNon-degeneratePlasmas......................... 180
6.2.1 DensityMatrixofPairs......................................... 180
6.2.2 MethodofEffectivePotentials................................. 185
6.3 ThermodynamicsoftheClassicalElectronGasandQuantum
Corrections.............................................................. 190
6.3.1 ClassicalBogolyubovExpansions............................. 190
6.3.2 QuantumCorrections........................................... 192
6.4 ScreeningandThermodynamicFunctionsofNon-degenerate
Plasmas.................................................................. 193
6.4.1 Debye-HückelScreening....................................... 193
6.4.2 RingFunctions.................................................. 195
6.5 ThePairBoundStateContributions.................................... 196
6.5.1 Mean-MassApproximation—SymmetricalPlasmas ......... 196
6.5.2 TheSecondVirialCoefficientforGeneralPlasmas........... 200