Statistical physics: including applications to condensed matter

278 Pages·2005·1.413 MB·English

by  Claudine Hermann

Statistical Physics
Advanced Texts in Physics
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particular areas of physics will also benefit from the titles in this collection.
Claudine Hermann
Statistical Physics
Including Applications
to Condensed Matter
With 63 Figures
ClaudineHermann
LaboratoiredePhysiquedelaMatie`reCondense´e
EcolePolytechnique
91128Palaiseau
France
[email protected]
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Table of Contents
Introduction xi
Glossary xv
1 Statistical Description of Large Systems. Postulates 1
1.1 Classical or Quantum Evolution of a Particle; Phase Space . . 2
1.1.1 Classical Evolution . . . . . . . . . . . . . . . . . . . . . 2
1.1.2 Quantum Evolution . . . . . . . . . . . . . . . . . . . . 4
1.1.3 Uncertainty Principle and Phase Space. . . . . . . . . . 4
1.1.4 Other Degrees of Freedom . . . . . . . . . . . . . . . . . 5
1.2 Classical Probability Density; Quantum Density Operator . . . 5
1.2.1 Statistical Approach for Macroscopic Systems . . . . . . 6
1.2.2 Classical Probability Density . . . . . . . . . . . . . . . 8
1.2.3 Density Operator in Quantum Mechanics . . . . . . . . 9
1.3 Statistical Postulates; Equiprobability . . . . . . . . . . . . . . 10
1.3.1 Microstate, Macrostate . . . . . . . . . . . . . . . . . . 10
1.3.2 Time Averageand Ensemble Average . . . . . . . . . . 11
1.3.3 Equiprobability . . . . . . . . . . . . . . . . . . . . . . . 12
1.4 General Properties of the Statistical Entropy . . . . . . . . . . 15
1.4.1 The Boltzmann Definition . . . . . . . . . . . . . . . . 15
1.4.2 The Gibbs Definition . . . . . . . . . . . . . . . . . . . . 15
1.4.3 The Shannon Definition of Information . . . . . . . . . 18
Summary of Chapter 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 19
Appendix 1.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
Appendix 1.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Appendix 1.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 The Different Statistical Ensembles. General Methods 31
2.1 Energy States of an N-Particle System . . . . . . . . . . . . . . 32
2.2 Isolated System in Equilibrium : “MicrocanonicalEnsemble” . 35
2.3 Equilibrium Conditions for Two Systems in Contact . . . . . . 36
2.3.1 Equilibrium Condition : Equal β Parameters . . . . . . 36
2.3.2 Fluctuations of the Energy Around its Most Likely Value 37
v
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2.4 Contact with a Heat Reservoir, “Canonical Ensemble” . . . . . 40
2.4.1 The Boltzmann Factor . . . . . . . . . . . . . . . . . . . 41
2.4.2 Energy, with Fixed Average Value . . . . . . . . . . . . 42
2.4.3 Partition Function Z . . . . . . . . . . . . . . . . . . . . 44
2.4.4 Entropy in the Canonical Ensemble . . . . . . . . . . . 45
2.4.5 Partition Function of a Set of Two Independent Systems 46
2.5 Grand Canonical Ensemble . . . . . . . . . . . . . . . . . . . . 48
2.5.1 Equilibrium Condition : Equality of Both T’s and µ’s . 49
2.5.2 Heat Reservoirand Particles Reservoir . . . . . . . . . . 50
2.5.3 Grand Canonical Probability and Partition Function . . 51
2.5.4 Average Values . . . . . . . . . . . . . . . . . . . . . . . 52
2.5.5 Grand Canonical Entropy . . . . . . . . . . . . . . . . . 53
2.6 Other Statistical Ensembles . . . . . . . . . . . . . . . . . . . . 53
Summary of Chapter 2 . . . . . . . . . . . . . . . . . . . . . . . . . 55
Appendix 2.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3 Thermodynamics and Statistical Physics 59
3.1 Zeroth Law of Thermodynamics. . . . . . . . . . . . . . . . . . 60
3.2 First Law of Thermodynamics. . . . . . . . . . . . . . . . . . . 60
3.2.1 Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.2.2 Heat . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.2.3 Quasi-Static General Process . . . . . . . . . . . . . . . 63
3.3 Second Law of Thermodynamics . . . . . . . . . . . . . . . . . 64
3.4 Third Law of Thermodynamics . . . . . . . . . . . . . . . . . . 66
3.5 The Thermodynamical Potentials;the Legendre Transformation 67
3.5.1 Isolated System . . . . . . . . . . . . . . . . . . . . . . . 67
3.5.2 Fixed N, Contact with a Heat Reservoirat T . . . . . . 69
3.5.3 Contact with a Heat and Particle Reservoir at T . . . . 70
3.5.4 Transformationof Legendre; Other Potentials. . . . . . 72
Summary of Chapter 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 75
4 The Ideal Gas 79
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2 Kinetic Approach . . . . . . . . . . . . . . . . . . . . . . . . . . 80
4.2.1 Scattering Cross Section, Mean Free Path . . . . . . . . 80
4.2.2 Kinetic Calculation of the Pressure . . . . . . . . . . . . 81
4.3 Classical or Quantum Statistics? . . . . . . . . . . . . . . . . . 83
4.4 Classical Statistics Treatmentof the Ideal Gas . . . . . . . . . 85
4.4.1 Calculation of the Canonical Partition Function. . . . . 85
4.4.2 Average Energy; Equipartition Theorem. . . . . . . . . 87
4.4.3 Free Energy; Physical Parameters (P,S,µ) . . . . . . . 89
4.4.4 Gibbs Paradox . . . . . . . . . . . . . . . . . . . . . . . 90
4.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91
Summary of Chapter 4 . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Appendix 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Tableof Contents vii
Appendix 4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
5 Indistinguishability, the Pauli Principle 113
5.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.2 States of Two Indistinguishable Particles. . . . . . . . . . . . . 116
5.2.1 General Case . . . . . . . . . . . . . . . . . . . . . . . . 116
5.2.2 Independent Particles . . . . . . . . . . . . . . . . . . . 116
5.3 Pauli Principle; Spin-Statistics Connection . . . . . . . . . . . 119
5.3.1 Pauli Principle; Pauli Exclusion Principle . . . . . . . . 119
5.3.2 Theorem of Spin-Statistics Connection . . . . . . . . . . 120
5.4 Case of Two Particles of Spin 1/2. . . . . . . . . . . . . . . . . 121
5.4.1 Triplet and Singlet Spin States . . . . . . . . . . . . . . 121
5.4.2 WaveFunction of Two Spin 1/2 Particles . . . . . . . . 123
5.5 Special Case of N Independent Particles . . . . . . . . . . . . . 124
5.5.1 WaveFunction . . . . . . . . . . . . . . . . . . . . . . . 124
5.5.2 Occupation Numbers . . . . . . . . . . . . . . . . . . . . 125
5.6 Return to the Introduction Examples . . . . . . . . . . . . . . 126
5.6.1 Fermions Properties . . . . . . . . . . . . . . . . . . . . 126
5.6.2 Bosons Properties . . . . . . . . . . . . . . . . . . . . . 126
Summary of Chapter 5 . . . . . . . . . . . . . . . . . . . . . . . . . . 129
6 General Properties of the Quantum Statistics 131
6.1 Use of the Grand Canonical Ensemble . . . . . . . . . . . . . . 132
6.1.1 2 Indistinguishable Particles at T, Canonical Ensemble. 132
6.1.2 Description in the Grand Canonical Ensemble . . . . . . 133
6.2 Factorizationof the Grand PartitionFunction . . . . . . . . . . 134
6.2.1 Fermions and Bosons. . . . . . . . . . . . . . . . . . . . 134
6.2.2 Fermions . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.3 Bosons. . . . . . . . . . . . . . . . . . . . . . . . . . . . 135
6.2.4 Chemical Potential and Number of Particles. . . . . . . 136
6.3 Average Occupation Number;Grand Potential. . . . . . . . . . 136
6.4 Free Particle in a Box;Density of States . . . . . . . . . . . . . 138
6.4.1 Quantum States of a Free Particle in a Box . . . . . . . 138
6.4.2 Density of States . . . . . . . . . . . . . . . . . . . . . . 143
6.5 Fermi-Dirac Distribution; Bose-Einstein Distribution . . . . . . 147
6.6 Average Values of Physical Parameters at T . . . . . . . . . . . 148
6.7 Common Limit of the Quantum Statistics . . . . . . . . . . . . 149
6.7.1 Chemical Potential of the Ideal Gas . . . . . . . . . . . 149
6.7.2 Grand Canonical Partition Function of the Ideal Gas . . 151
Summary of Chapter 6 . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7 Free Fermions Properties 155
7.1 Properties of Fermions at Zero Temperature . . . . . . . . . . . 155
7.1.1 Fermi Distribution, Fermi Energy. . . . . . . . . . . . . 155
7.1.2 Internal Energy and Pressure at Zero Temperature . . . 158
viii Tableof Contents
7.1.3 Magnetic Properties. Pauli Paramagnetism . . . . . . . 159
7.2 Properties of Fermions at Non-Zero Temperature . . . . . . . . 160
7.2.1 Temperature Ranges and Chemical Potential Variation. 160
7.2.2 Specific Heat of Fermions . . . . . . . . . . . . . . . . . 163
7.2.3 Thermionic Emission . . . . . . . . . . . . . . . . . . . . 165
Summary of Chapter 7 . . . . . . . . . . . . . . . . . . . . . . . . . . 169
Appendix 7.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171
8 Elements of Bands Theory and Crystal Conductivity 175
8.1 What is a Solid, a Crystal? . . . . . . . . . . . . . . . . . . . . 176
8.2 The Eigenstates for the Chosen Model . . . . . . . . . . . . . . 177
8.2.1 Recall : the Double PotentialWell . . . . . . . . . . . . 177
8.2.2 Electron on an Infinite and Periodic Chain . . . . . . . 180
8.2.3 Energy Bands and Bloch Functions . . . . . . . . . . . . 182
8.3 The Electron States in a Crystal . . . . . . . . . . . . . . . . . 184
8.3.1 WavePacketof Bloch Waves . . . . . . . . . . . . . . . 184
8.3.2 Resistance; Mean Free Path. . . . . . . . . . . . . . . . 184
8.3.3 Finite Chain, Density of States, Effective Mass . . . . . 185
8.4 Statistical Physics of Solids . . . . . . . . . . . . . . . . . . . . 189
8.4.1 Filling of the Levels . . . . . . . . . . . . . . . . . . . . 189
8.4.2 Variation of Metal Resistance versusT . . . . . . . . . . 190
8.4.3 Insulators’ Conductivity Versus T ; Semiconductors . . . 191
8.5 Examples of Semiconductor Devices . . . . . . . . . . . . . . . 196
8.5.1 The Photocopier : Photoconductivity Properties . . . . 196
8.5.2 The Solar Cell : an Illuminated p−n Junction . . . . . 196
8.5.3 CD Readers : the Semiconductor Quantum Wells . . . . 197
Summary of Chapter 8 . . . . . . . . . . . . . . . . . . . . . . . . . . 199
9 Bosons : Helium 4, Photons,Thermal Radiation 201
9.1 Material Particles . . . . . . . . . . . . . . . . . . . . . . . . . . 201
9.1.1 Thermodynamics of the Boson Gas . . . . . . . . . . . . 202
9.1.2 Bose-Einstein Condensation . . . . . . . . . . . . . . . . 203
9.2 Bose-Einstein Distribution of Photons . . . . . . . . . . . . . . 206
9.2.1 Description of the Thermal Radiation; the Photons . . 206
9.2.2 Statistics of Photons, Bosons in Non-ConservedNumber 207
9.2.3 Black Body Definition and Spectrum . . . . . . . . . . . 210
9.2.4 Microscopic Interpretation. . . . . . . . . . . . . . . . . 212
9.2.5 Photometric Measurements : Definitions . . . . . . . . . 214
9.2.6 Radiative Balances . . . . . . . . . . . . . . . . . . . . . 217
9.2.7 Greenhouse Effect . . . . . . . . . . . . . . . . . . . . . 218
Summary of Chapter 9 . . . . . . . . . . . . . . . . . . . . . . . . . . 221
Solving Exercises and Problems of Statistical Physics 223
Units and Physical Constants 225
Tableof Contents ix
A few useful formulae 229
Exercises and Problems 231
Ex. 2000 : Electrostatic Screening. . . . . . . . . . . . . . . . . . . . 233
Ex. 2001 : Magnetic Susceptibility of a “Quasi-1D” Conductor . . . 234
Ex. 2002 : Entropies of the HC(cid:2) Molecule . . . . . . . . . . . . . . . 236
Pr. 2001 : Quantum Boxes and Optoelectronics . . . . . . . . . . . . 237
Pr. 2002 : Physical Foundations of Spintronics . . . . . . . . . . . . 245
Solution of the Exercises and Problems 251
Ex. 2000 : Electrostatic Screening. . . . . . . . . . . . . . . . . . . . 253
Ex. 2001 : Magnetic Susceptibility of a “Quasi-1D” Conductor . . . 255
Ex. 2002 : Entropies of the HC(cid:2) Molecule . . . . . . . . . . . . . . . 258
Pr. 2001 : Quantum Boxes and Optoelectronics . . . . . . . . . . . . 259
Pr. 2002 : Physical Foundations of Spintronics . . . . . . . . . . . . 266
Index 275
Introduction
A glosssary at the end of this introduction defines the terms specificto Statis-
tical Physics. In the text, these terms are marked by an asterisk in exponent.
A courseof QuantumMechanics,like the one taughtat Ecole Polytechnique,
is devotedto the descriptionofthe state ofanindividualparticle,orpossibly
ofafew ones.Conversely,the topic ofthis book willbe the study ofsystems∗
containing very many particles, of the order of the Avogadro number N, for
example the molecules in a gas, the components of a chemical reaction, the
adsorption sites for a gas on a surface, the electrons of a solid. You certainly
previouslystudied this type ofsystem,using Thermodynamicswhichis ruled
by“exact”laws,suchastheidealgasone.Itsphysicalparameters,thatcanbe
measuredinexperiments,aremacroscopicquantitieslikeitspressure,volume,
temperature, magnetization, etc.
It is now well-known that the correct microscopic description of the state of
a system, or of its evolution, requires the Quantum Mechanics approachand
thesolutionoftheSchroedingerequation,buthowcanthisequationbesolved
when such a huge number of particles comes into play? Printing on a listing
the positions and velocities of the N molecules of a gas would take a time
muchlongerthattheoneelapsedsincetheBigBang!Astatisticaldescription
is the only issue, which is the more justified as the studied system is larger,
since the relative fluctuations are then very small (Ch. 1).
This course is restricted to systems in thermal equilibrium∗ : to reach such
an equilibrium it is mandatory that interaction terms should be present in
thehamiltonianofthe totalsystem:evenifthey areweak,they allowenergy
exchangesbetween the system and its environment(for example one may be
concerned by the electrons of a solid, the environment being the ions of the
samesolid).Theseinteractionsprovidethewaytoequilibrium.Thisapproach
takesplace during a characteristictime, the so-called“relaxationtime”, with
a range of values which depends on the considered system (the study of off-
equilibriumphenomena,inparticulartransportphenomena,isanotherbranch
xi