Equilibrium and non-equilibrium statistical mechanics/ Carolyne M. Van Vliet

PREFACE vii
E Q U I L I B R I U M S T A T I S T I C A L M E C H A N I C S
PART A. GENERAL PRINCIPLES OF MANY-PARTICLE SYSTEMS
Chapter I. Introduction to the State of Large Systems and some Probability
Concepts 5
1.1 Purpose of statistical mechanics for classical and quantum systems 5
1.2 Gamma-space and its quantum equivalent 8
1.3 The thermodynamic state 12
1.3.1 Macroscopic thermodynamics; Callen's entropy 12
1.3.2 Statistical mechanical state functions; Gibbs' entropy 15
1.4 Various ensembles and their main state functions 17
1.5 Fluctuating state variables; the a- space 18
1.6 Some mathematical distribution functions 20
1.6.1 The binomial distribution and random walk 20
1.6.2 The multinomial distribution function 24
1.6.3 The Poisson distribution, Gauss distribution and normal distribution 25
1.6.4 Multivariate distributions, the Maxwell¿Boltzmann distribution and
the virial theorem 28
1.7 Transforms of probability functions 33
1.7.1 Characteristic functions 33
1.7.2 The generating function according to Laplace 37
1.7.3 The factorial moment generating function, Fowler transform and
cumulants 38
1.7.4 The Mellin transform 42
1.8 Problems to Chapter I 43
Chapter II. Statistics of Closed Systems 46
2.1 Liouville's theorem and the microcanonical density function 46
2.2 The ergodic hypothesis 48
2.3 Von Neumann's theorem and the microcanonical density operator 50
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2.4 Macro-probability in classical and quantum statistics 53
2.5 Examples of extension in phase space and accessible number of quantum
states 56
2.5.1 Ideal gas 56
2.5.2 An assembly of oscillators; 58
2.5.3 A general form for ??(E,V,N) in the microcanonical ensemble 61
2.6 Problems to Chapter II 63
Chapter III. Thermodynamics in the Microcanonical Ensemble, Classical
and Quantal 65
3.1 Gibbs' form of the ergodic density; entropy for classical systems 65
3.2 Entropy for quantum systems 69
3.3 The equipartition law 71
3.4 Various forms for the entropy in closed and in open systems; 73
3.5 Properties of entropy 74
3.5.1 Elementary entropies, Sackur¿Tetrode formula 74
3.5.2 Homogeneous entropy form and Gibbs¿Duhem relation 77
3.5.3 Maxwell relations 78
3.5.4 Nernst¿s law and statistical mechanics 79
3.6 Equilibrium and local stability requirements 81
3.7 Entropy and probability 83
3.7.1 Boltzmann¿s principle 84
3.7.2 The Boltzmann-Einstein principle 84
3.8 The Gibbs entropy function 85
3.8.1 Gibbs¿ entropy function for a nonequilibrium state 85
3.8.2 Failure for the second law of thermod. in precisely defined microstate 86
3.8.3 Coarse-graining and the second law 87
3.8.4 Fluctuations of extensive and intensive variables 88
3.9 Problems to Chapter III 91
Chapter IV. Ensembles in the Presence of Reservoirs: The Canonical
and Grand-Canonical Ensemble 93
1. GENERAL FORMALISM AND SOME QUANTUM APPLICATIONS
4.1 The canonical ensemble for systems in contact with a heat bath 93
4.2 The grand-canonical ensemble for systems with energy / particle exchange 96
4.3 Quantum Illustrations of the canonical and grand-canonical distribution 102
4.3.1 The Fermi¿Dirac and Bose¿Einstein results 102
4.3.2 The one-dimensional Ising model 105
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2. DENSE CLASSICAL GASES AND FURTHER APPLICATIONS
4.4 Second virial coefficient for a classical gas and van der Waals¿ law 108
4.4.1 Ornstein¿s method as elaborated by van Kampen 109
4.4.2 Van der Waals¿ equation; fluctuations and pair correlation function 112
4.4.3 Cluster-integral method 117
4.5 Tonk¿s hard-core gas and Takahashi¿s nearest neighbour gas 120
4.6 The method of steepest descent 124
4.7 Dense gases and the virial coefficients via the grand-canonical ensemble 127
4.7.1 Cluster expansion 127
4.7.2 Cumulant expansion 133
4.8 Mean-field theories 135
4.8.1 The Ising Hamiltonian and the Weiss molecular field 135
4.8.2 The Bragg¿Williams method. Order-disorder transitions 137
4.9 Landau¿Ginzburg theory for phase transitions; ?-points 145
4.9.1 General procedure 145
4.9.2 Classification of phase transitions 147
4.10 Ionized gases ¿ plasmas ¿ or electron-hole gases in condensed matter 150
4.10.1 Coulomb interactions. The Debye¿Hückel theory 150
4.10.2 Coulomb interactions via the pair-dist. function; BBGK hierarchy 154
4.11 Distribution functions, correlation functions and covariance functions 157
4.12 Problems to Chapter IV 163
Chapter V. Generalized Canonical Ensembles 170
5.1 Formal results 170
5.1.1 The generalized ensemble probability 170
5.1.2 Thermodynamic functions 173
5.1.3 The macroscopic thermodynamic distribution 174
5.2 Transformation theory using the Fowler generating function 175
5.3 Fluctuations: general results 177
5.3.1 Extensive variables 177
5.3.2 Intensive variables 179
5.3.3 Examples and conclusions 181
5.4 Carrier-density fluctuations in a solid 183
5.4.1 Microscopic occupancy fluctuations 183
5.4.2 Macroscopic occupancy fluctuations 185
5.4.3 Examples for non-degenerate and degenerate semiconductors 186
5.5 Fluctuations in systems interacting with finite reservoirs 189
5.6 Alternate Fermi¿Dirac distributions 191
5.7 Problems to Chapter V 195
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PART B. CLASSICAL AND QUANTUM FORMALISMS. THE
BOLTZMANN GAS, THE PERFECT BOSE GAS AND FERMI GAS
Chapter VI. The Boltzmann Distribution and Chemical Applications 201
6.1 Aspects of molecular distributions 201
6.2 The Darwin¿Fowler procedure 204
6.3 Thermodynamic functions and standard forms 206
6.4 Fluctuations of the distribution function 208
6.5 Classical Illustrations 210
6.5.1 Effect of a magnetic field; Bohr-van Leeuwen theorem 210
6.5.2 Generalized Sackur¿Tetrode formula and the equations of state 211
6.6 Oscillators 213
6.6.1 The Planck oscillator 213
6.6.2 The Fermi oscillator 214
6.7 Rotators 215
6.8 Dielectric and paramagnetic dipoles 221
6.9 Chemical equilibrium and the mass-action law 223
6.10 Problems to Chapter VI 226
Chapter VII. The Occupation-Number State Formalism; Spin and 228
Statistics
7.1 Symmetrization of states 228
7.2 Systems of bosons. Creation and annihilation operators 231
7.3 Many-body boson operators 234
7.3.1 Expressions in terms of , ¿ k k a a 234
7.3.2 Field operators and local variables 238
7.4 On the quantization of fields 242
7.5 Fermion operators; anticommutation rules 246
7.6 Many-body fermion operators 250
7.6.1 Expressions in terms of , ¿ k k c c 250
7.6.2 Field operators; spin 252
7.7 The boson-fermion dichotomy: general remarks 253
7.8 The Hartree¿Fock equation* 254
7.9 Problems to Chapter VII 258
Chapter VIII. Ideal Quantum Gases and Elementary Excitations in Solids 260
8.1 Bose Einstein statistics for zero-restmass particles; blackbody radiation 260
8.1.1 Planck¿s law; original considerations 260
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8.1.2 Quantization of the electromagnetic field; photons 262
8.2 The perfect Bose gas 265
8.3 Bose¿Einstein condensation 269
8.3.1 The P-v¿ and the P-Tdiagram 269
8.3.2 Coexisting phases and thermodynamic functions 272
8.4 The perfect Fermi gas 275
8.5 Lattice vibrations; phonons 280
8.5.1 Continuum description. Einstein and Debye specific heat 280
8.5.2 Normal modes; running-wave boson operators 282
8.6 Elements of electron-phonon interaction 288
8.7 Problems to Chapter VIII 295
PART C. QUANTUM SYSTEMS WITH STRONG INTERACTIONS
Chapter IX. Critical Phenomena: General Features of Phase Transitions 301
9.1 Introductory remarks 301
9.2 Critical fluctuations 302
9.2.1 Elements of functional theory 302
9.2.2 Landau-Ginzburg density functionals 304
9.2.3 Spatial correlation function 306
9.3 Critical exponents and scaling relations 309
9.4 Thermodynamic inequalities 313
9.4.1 Magnetic systems; the Curie-point transition 313
9.4.2 Vapour-liquid transition and the coexistence region 315
9.5 Dimensional analysis 318
9.6 Other scaling hypotheses 319
9.6.1 Widom scaling 319
9.6.2 Kadanoff scaling 321
9.7 Other topics in phase transitions 324
9.7.1 Symmetry breaking and order parameters 324
9.7.2 The tricritical point 326
9.7.3 The Ginzburg criterion 330
9.7.4 The Kosterlitz¿Thouless transition 331
9.8 Problems to Chapter IX 334
Chapter X. Renormalization: Theory and Examples 338
10.1 Objective of renormalization 338
10.2 The linear spin chain revisited 339
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10.3 The renormalization group 343
10.3.1 Fixed points, infinitesimal transformations and scaling fields 343
10.3.2 Connection with Widom¿s scaling function; critical exponents 347
10.4 Niemeijer¿van Leeuwen cumulant expansion for triangular lattice 348
10.4.1 First-order results 350
10.4.2 Higher-order results 353
10.5 The ¿classical spin¿ Gaussian model 356
10.6 Elements of the S4 model and the epsilon expansion 361
10.7 Problems to Chapter X 370
Chapter XI. The Two-dimensional Ising Model and Spin Waves 372
11.1 Historical notes. Review of the 1D model 372
11.2 The transfer matrix for the rectangular lattice 376
11.2.1 Procedure 376
11.2.2 Transformation to an interacting fermion problem 378
11.2.3 Running-wave fermion operators 381
11.2.4 Bogoliubov¿Valatin transformation 386
11.3 The critical temperature and the thermodynamic functions 387
11.4 The spontaneous magnetization 395
11.4.1 Spin-spin correlation function 395
11.4.2 Evaluation of the Toeplitz determinant; Onsager¿s result 400
11.5 Ferromagnetism as excitation of magnons 403
11.6 Bose¿Einstein statistics for magnons 406
11.7 The Heisenberg antiferromagnet 408
11.8 Problems to Chapter XI 412
Chapter XII. Aspects of Quantum Fluids 415
1. VARIOUS SPECIAL THEORIES
12.1 Superfluidity; general features 415
12.2 Elements of Feynman¿s theory 421
12.2.1 The ground state and single-quantum excited state 421
12.2.2 The excitation spectrum for T > 0 423
12.3 Bogoliubov¿s theory of excitations in 4He 426
12.3.1 The grand Hamiltonian 426
12.3.2 The chemical potential 431
12.4 Gaseous atomic Bose¿Einstein condensates 433
12.4.1 Quantum equations for the near-perfect B¿E gas 434
12.4.2 Properties and solutions of the Gross¿Pitaevskii equation 436
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12.5 Superconductivity 440
12.5.1 The Fröhlich Hamiltonian 440
12.5.2 Cooper pairs 443
12.6 The BCS Hamiltonian 445
12.6.1 The ground state 445
12.6.2 Finite temperature results 450
12.7 Excitations in Fermi liquids; 3He 455
12.7.1 Original Fermi liquid theory (Landau) and some empirical data 455
12.7.2 The ground state and pair-correlation function 465
12.8 Modern developments of 3He 467
12.8.1 Other excitations 467
12.8.2 Balian¿Werthamer (BW) Hamiltonian for the superfluid phases 470
2. FORMAL THEORY; DIAGRAMMATIC METHODS
12.9 Perturbation expansion of the grand-canonical partition function 474
12.9.1 The interaction picture; expansion of the evolution operator 474
12.9.2 Generalized Wick¿s theorem 478
12.10 Momentum-space diagrams 481
12.10.1 Feynman diagrams 482
12.10.2 Hugenholtz diagrams 487
12.10.3 Fourier-transformed frequency diagrams 490
12.11 Full Propagators (or Green¿s functions) for normal quantum fluids 494
12.11.1 Spatial and momentum forms 494
12.11.2 Cumulant expansion of the Green¿s function in free propagators 499
12.12 Self-energy and Dyson¿s equation 502
12.13 Fermi liquids revisited 506
12.13.1 The Hartree¿Fock approximation 506
12.13.2 The ring approximation (RPA) 513
(a) Classical electron gas with positive charge background 518
(b) Quantum electron gas near T = 0 520
12.14 Problems to Chapter XII 523
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N O N - E Q U I L I B R I U M S T A T I S T I S T I C A L
M E C H A N I C S
PART D. CLASSICAL TRANSPORT THEORY
Chapter XIII. The Boltzmann Equation and Boltzmann¿s ?-Theorem 531
13.1 Introduction to Boltzmann theory 531
13.2 The Boltzmann equation in velocity-position space 533
13.3 The Boltzmann equation for solids with extended states 538
13.4 Connection with the cross section; examples of ?(?) 543
13.4.1 Matrix element squared ??cross section 543
13.4.2 Classical and quantum mechanical examples of ?(?) 545
13.5 Boltzmann¿s H-theorem 550
13.5.1 Derivation 550
13.5.2 Further discussion of Boltzmann¿s H-theorem 554
13.6 The equilibrium solutions 556
13.6.1 The classical gas 556
13.6.2 Quantum gases 556
13.7 The equilibrium entropy 558
13.8 Problems to Chapter XIII 561
Chapter XIV. Hydrodynamic Equations and Conservation Theorems,
Barycentric Flow 563
14.1 Conservation theorems 563
14.1.1 Full theorems 563
14.1.2 Zero-order or Eulerian conservation theorems 569
14.2 The phenomenological equations in classical systems 571
14.2.1 The basis of the flow problem 571
14.2.2 Relaxation-time model 574
14.2.3 Computation of the vector and tensor flow averages in systems
with barycentric flow 576
14.3 The hydrodynamic equations 581
14.4 Computation of the entropy production 584
14.5 Problems to Chapter XIV 588
Chapter XV. Further Applications 589
1. NEAR-EQUILIBRIUM TRANSPORT
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15.1 Electron gas in metals: the perturbation description 589
15.2 The streaming-vector method 592
15.2.1 Fluxes in the absence of a magnetic field 592
15.2.2 Incorporation of a magnetic field 596
15.3 Entropy production and heat flux 599
15.4 The phenomenological equations for solids 601
15.4.1 General scheme 601
15.4.2 Galvanomagnetic and thermomagnetic effects 603
15.5 Mobility computations* 607
15.5.1 Resistivity of metals; Bloch¿s formula 607
15.5.2 Acoustic phonon scattering in nondegenerate semiconductors 612
2. TRANSPORT FAR FROM EQUILIBRIUM; STEADY-STATE DISTRIBUTIONS AND FLOW
15.6 The coupled Boltzmann equations in the v-language; expansion in
spherical polynomials 615
15.7 The zero-order and first-order collision integrals in a binary plasma 619
15.8 Electron heating in plasmas: the Druyvesteyn distribution 621
15.9 Coupled Boltzmann equations for hot electrons in semiconductors 623
15.10 The steady-state distribution for a hot electron gas 624
15.11 Transport in hot electron systems 630
15.12 Problems to Chapter XV 632
PART E. LINEAR RESPONSE THEORY AND QUANTUM TRANSPORT
Chapter XVI. Linear Response Theory, Reduced Operators and
Convergent Forms 637
1. THE ORIGINAL KUBO¿GREEN FORMALISM
16.1 Introduction to linear response theory 637
16.2 The response function and the relaxation function 639
16.3 The frequency domain; various forms 644
16.3.1 The commutator form 644
16.3.2 The Kubo form and the Fujita form 646
16.3.3 The correlation form 648
16.3.4 The fluctuation-dissipation theorem 650
16.4 The Wiener¿Khintchine theorem 654
16.5 Density-density correlations and the dynamic structure factor 657
16.5.1 General considerations 657
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16.5.2 Another form of the fluctuation-dissipation theorem 660
16.5.3 Thermodynamics and sum-rules 661
16.6 A return to quantum liquids 664
16.6.1 Self-consistent field approximation 664
16.6.2 Excitations in the Bose liquid 666
16.6.3 Fermi liquids 668
16.6.4 Real time Green¿s functions and the diagrammatic evaluation 672
16.7 Kubo-theory conductivity computations 674
16.8 Criticism of linear response theory 677
16.8.1 Van Kampen¿s objections 677
16.8.2 Our criticism 678
2. REDUCED OPERATORS AND CONVERGENT FORMS
16.9 The master operator in Liouville space 679
16.9.1 Results for small times 681
16.9.2 Results for large times 685
16.10 Irreversible transport equations via projector operators 686
16.10.1 Some theorems 686
16.10.2 Reduction of the Heisenberg equation of motion; diagonal part 689
16.10.3 The full reduced Heisenberg equation and the current operator 692
16.10.4 Consequences for the many-body response formulae 696
16.11 The Pauli¿Van Hove master equation 698
16.12 The full master equation (FME) 701
16.13 Approach to equilibrium 704
16.14 The Onsager¿Casimir reciprocity relations 705
16.14.1 The diagonal correlation functions 705
16.14.2 The nondiagonal correlation functions 708
16.14.3 Some lemmas 710
16.15 An exact response result: Cohen¿Van Vliet 711
16.16 Problems to Chapter XVI 715
Chapter XVII. The Quantum Boltzmann Equation and Some Applications 719
1. THE QUANTUM BOLTZMANN EQUATION: SCOPE AND ESSENCE
17.1 From the master equation to the quantum Boltzmann equation 719
17.1.1 The quantum Boltzmann equation for binary interactions 720
17.1.2 The quantum Boltzmann equation for electron-phonon interaction 724
17.2 Discussion of the equilibrium and steady-state distribution 725
17.3 Extended states. Recovery of the BTE via the Wigner formalism 727
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17.4 Generalized Calecki equation for the nonlinear current flux 731
17.5 The linearized quantum Boltzmann equation* 732
17.6 Electrical Conductivities in the linear regime* 735
17.6.1 Ponderomotive conductivities 736
17.6.2 The Argyres¿Roth formula for the collisional conductivity 738
17.7 Localised states: a direct perturbation treatment 739
17.8 Diagonal and nondiagonal conductivities from modified LRT 741
2. SOME APPLICATIONS OF MODIFIED LINEAR RESPONSE THEORY
17.9 Landau states: 3D applications 744
17.9.1 The ordinary Hall effect 744
17.9.2 Transverse magnetoresistance 747
17.10 Landau States: 2D and 1D applications 749
17.10.1 The quantum Hall effect 749
17.10.2 Magnetophonon resonances 752
17.11 Slightly disordered metals. The Aharonov¿Bohm effect 760
17.11.1 Landauer¿Büttiker models 763
17.11.2 Diagrammatic methods 764
17.11.3 Modified LRT results 765
3. THE MASTER HIERARCHY
17.12 Kinetic Equations for quantum systems with binary interactions 771
17.12.1 Fermion moment equations and Fokker¿Planck moments 771
17.12.2 Boson moment equations 777
17.13 Problems to Chapter XVII 778
PART F. STOCHASTIC PHENOMENA
Chapter XVIII. Brownian Motion and the Mesoscopic Master Equation 781
18.1 Introduction to fluctuations and stochastic phenomena 781
1. THE MESOSCOPIC MASTER EQUATION AND THE MOMENT EQUATIONS
18.2 Probabilistic description of Ornstein and Burger 783
18.2.1 Purely random processes 783
18.2.2 Markovian random processes 784
18.3 Derivation of the mesoscopic master equation 786
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18.4 The Kramers¿Moyal expansion and the Fokker¿Planck equation 789
18.5 The phenomenological equations and the fluctuation-relaxation theorem 790
18.5.1 One-variable master equation; birth-death rate processes 795
18.5.2 Multivariate gain-loss processes 798
18.5.3 Electronic fluctuations out of equilibrium 800
18.5.4 Fluctuations about the hydrodynamic state; Brillouin scattering 804
18.6 The Langevin equation 805
18.6-1 General procedure 805
18.6.2 The sources of gain-loss processes and of G-R noise 809
2. BROWNIAN MOTION PROPER. VELOCITY FLUCTUATIONS AND DIFFUSION
18.7 Diffusion and random walk 811
18.7.1 Einstein¿s result 811
18.7.2 Langevin approach of Uhlenbeck and Ornstein 812
18.7.3 Fokker¿Planck solution for the bivariate process{v(t),r(t)} 813
18.7.4 Harmonically bound Brownian particle 816
18.8 Velocity fluctuations and diffusion in condensed matter 817
3. SPECTRAL ANALYSIS
18.9 Overview and Wiener¿Khintchine theorem 821
18.10 The short-time average. MacDonald¿s theorem and Milatz¿ theorem 822
18.10.1 Application to shot noise and similar phenomena 824
18.10.2 Modulated emission noise and wave-interaction noise 826
18.11 Method of elementary events. Campbell¿s and Carson¿s theorems 829
18.12 The Allan-variance theorem 833
18.12.1 Inversion of the Allen variance theorem 836
18.12.2 Counting experiments; non-adjacent sampling 840
18.13 On the origin of 1/f-like noise 842
18.13 The spectra of G-R noise 843
18.13.1 Three-level systems 843
18.14.2 General structure of multi-level G-R noise 850
18.14 Problems to Chapter XVIII 854
Chapter XIX. Branching Processes and Continuous Stochastic Processes 857
1. THE COMPOUNDING THEOREM AND APPLICATIONS
19.1 The compounding theorem, variance theorem and addition theorem 857
19.2 Bernoulli and geometric compounding 860
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2. METHOD OF RECURRENT GENERATING FUNCTIONS
19.3 Preamble 864
19.4 Singly-incited branching processes. One-carrier avalanche 865
19.5 Doubly-incited branching processes. Two-carrier avalanche 869
3. TRANSPORT FLUCTUATIONS
19.6 On the two Green¿s function procedures for transport noise 877
19.6.1 The correlation method and uniqueness 878
19.6.2 The response form or Langevin form 883
19.7 Applications: Rayleigh diffusion and ambipolar sweep-out 885
19.8 Inhomogeneous systems, effect of boundary terms, examples 890
19.9 Problems to Chapter XIX 896
Chapter XX. Stochastic Optical Signals and Photon Fluctuations 898
20.1 Introductory remarks 898
20.2 Analytic signals and coherence 900
20.3 The quantum field 905
20.4 The pseudo-classical field 907
20.4.1 Sudarshan¿Glauber transform of the statistical density operator 907
20.4.2 Pseudo-classical form of the coherence tensors 908
20.5 Examples for thermal and non-thermal radiation fields 910
20.6 Photon counting. Theory and some experimental results 917
20.7 Problems to Chapter XX 922
Appendix A. The Schrödinger, Heisenberg and Interaction Pictures 923
A.1 Schrödinger form 923
A.2 The Heisenberg picture and connection with classical mechanics 924
A.3 The interaction picture 926
Appendix B. Spin and Statistics 929
B.1 Generalities on fields 930
B.2 Statistics for a scalar spin-zero field 933
B.3 The connection for a field of general spin 935
AUTHOR INDEX 941
SUBJECT INDEX 947