Statistical Mechanics (eBook)

Front Cover 1
Statistical Mechanics 4
Copyright 5
Table of Contents 6
Preface to the Third Edition 14
Preface to the Second Edition 18
Preface to the First Edition 20
Historical Introduction 22
Chapter 1. The Statistical Basis of Thermodynamics 28
1.1 The macroscopic and the microscopic states 28
1.2 Contact between statistics and thermodynamics: physical significance of the number O(N, V, E) 30
1.3 Further contact between statistics and thermodynamics 33
1.4 The classical ideal gas 36
1.5 The entropy of mixing and the Gibbs paradox 43
1.6 The "correct" enumeration of the microstates 47
Problems 49
Chapter 2. Elements of Ensemble Theory 52
2.1 Phase space of a classical system 52
2.2 Liouville's theorem and its consequences 54
2.3 The microcanonical ensemble 57
2.4 Examples 59
2.5 Quantum states and the phase space 62
Problems 64
Chapter 3. The Canonical Ensemble 66
3.1 Equilibrium between a system and a heat reservoir 67
3.2 A system in the canonical ensemble 68
3.3 Physical significance of the various statistical quantities in the canonical ensemble 77
3.4 Alternative expressions for the partition function 79
3.5 The classical systems 81
3.6 Energy fluctuations in the canonical ensemble: correspondence with the microcanonical ensemble 85
3.7 Two theorems — the "equipartition" and the "virial" 88
3.8 A system of harmonic oscillators 92
3.9 The statistics of paramagnetism 97
3.10 Thermodynamics of magnetic systems: negative temperatures 104
Problems 110
Chapter 4. The Grand Canonical Ensemble 118
4.1 Equilibrium between a system and a particle-energy reservoir 118
4.2 A system in the grand canonical ensemble 120
4.3 Physical significance of the various statistical quantities 122
4.4 Examples 125
4.5 Density and energy fluctuations in the grand canonical ensemble: correspondence with other ensembles 130
4.6 Thermodynamic phase diagrams 132
4.7 Phase equilibrium and the Clausius–Clapeyron equation 136
Problems 138
Chapter 5. Formulation of Quantum Statistics 142
5.1 Quantum-mechanical ensemble theory: the density matrix 142
5.2 Statistics of the various ensembles 146
5.3 Examples 149
5.4 Systems composed of indistinguishable particles 155
5.5 The density matrix and the partition function of a system of free particles 160
Problems 166
Chapter 6. The Theory of Simple Gases 168
6.1 An ideal gas in a quantum-mechanical microcanonical ensemble 168
6.2 An ideal gas in other quantum-mechanical ensembles 173
6.3 Statistics of the occupation numbers 176
6.4 Kinetic considerations 179
6.5 Gaseous systems composed of molecules with internal motion 182
6.6 Chemical equilibrium 197
Problems 200
Chapter 7. Ideal Bose Systems 206
7.1 Thermodynamic behavior of an ideal Bose gas 207
7.2 Bose–Einstein condensation in ultracold atomic gases 218
7.3 Thermodynamics of the blackbody radiation 227
7.4 The field of sound waves 232
7.5 Inertial density of the sound field 239
7.6 Elementary excitations in liquid helium II 242
Problems 250
Chapter 8. Ideal Fermi Systems 258
8.1 Thermodynamic behavior of an ideal Fermi gas 258
8.2 Magnetic behavior of an ideal Fermi gas 265
8.3 The electron gas in metals 274
8.4 Ultracold atomic Fermi gases 285
8.5 Statistical equilibrium of white dwarf stars 286
8.6 Statistical model of the atom 291
Problems 296
Chapter 9. Thermodynamics of the Early Universe 302
9.1 Observational evidence of the Big Bang 302
9.2 Evolution of the temperature of the universe 307
9.3 Relativistic electrons, positrons, and neutrinos 309
9.4 Neutron fraction 312
9.5 Annihilation of the positrons and electrons 314
9.6 Neutrino temperature 316
9.7 Primordial nucleosynthesis 317
9.8 Recombination 320
9.9 Epilogue 322
Problems 323
Chapter 10. Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions 326
10.1 Cluster expansion for a classical gas 326
10.2 Virial expansion of the equation of state 334
10.3 Evaluation of the virial coefficients 336
10.4 General remarks on cluster expansions 342
10.5 Exact treatment of the second virial coefficient 347
10.6 Cluster expansion for a quantum-mechanical system 352
10.7 Correlations and scattering 358
Problems 367
Chapter 11. Statistical Mechanics of Interacting Systems: The Method of Quantized Fields 372
11.1 The formalism of second quantization 372
11.2 Low-temperature behavior of an imperfect Bose gas 382
11.3 Low-lying states of an imperfect Bose gas 388
11.4 Energy spectrum of a Bose liquid 393
11.5 States with quantized circulation 397
11.6 Quantized vortex rings and the breakdown of superfluidity 403
11.7 Low-lying states of an imperfect Fermi gas 406
11.8 Energy spectrum of a Fermi liquid: Landau's phenomenological theory 412
11.9 Condensation in Fermi systems 419
Problems 421
Chapter 12. Phase Transitions: Criticality, Universality, and Scaling 428
12.1 General remarks on the problem of condensation 429
12.2 Condensation of a van der Waals gas 434
12.3 A dynamical model of phase transitions 438
12.4 The lattice gas and the binary alloy 444
12.5 Ising model in the zeroth approximation 447
12.6 Ising model in the first approximation 454
12.7 The critical exponents 462
12.8 Thermodynamic inequalities 465
12.9 Landau's phenomenological theory 469
12.10 Scaling hypothesis for thermodynamic functions 473
12.11 The role of correlations and fluctuations 476
12.12 The critical exponents v and . 483
12.13 A final look at the mean field theory 487
Problems 490
Chapter 13. Phase Transitions: Exact (or Almost Exact) Results for Various Models 498
13.1 One-dimensional fluid models 498
13.2 The Ising model in one dimension 503
13.3 The n-vector models in one dimension 509
13.4 The Ising model in two dimensions 515
13.5 The spherical model in arbitrary dimensions 535
13.6 The ideal Bose gas in arbitrary dimensions 546
13.7 Other models 553
Problems 557
Chapter 14. Phase Transitions: The Renormalization Group Approach 566
14.1 The conceptual basis of scaling 567
14.2 Some simple examples of renormalization 570
14.3 The renormalization group: general formulation 579
14.4 Applications of the renormalization group 586
14.5 Finite-size scaling 597
Problems 606
Chapter 15. Fluctuations and Nonequilibrium Statistical Mechanics 610
15.1 Equilibrium thermodynamic fluctuations 611
15.2 The Einstein–Smoluchowski theory of the Brownian motion 614
15.3 The Langevin theory of the Brownian motion 620
15.4 Approach to equilibrium: the Fokker–Planck equation 630
15.5 Spectral analysis of fluctuations: the Wiener–Khintchine theorem 636
15.6 The fluctuation–dissipation theorem 644
15.7 The Onsager relations 653
Problems 659
Chapter 16. Computer Simulations 664
16.1 Introduction and statistics 664
16.2 Monte Carlo simulations 667
16.3 Molecular dynamics 670
16.4 Particle simulations 673
16.5 Computer simulation caveats 677
Problems 678
Appendices 680
A. Influence of boundary conditions on the distribution of quantum states 680
B. Certain mathematical functions 682
C. "Volume" and "surface area" of an n-dimensional sphere of radius R 689
D. On Bose–Einstein functions 691
E. On Fermi–Dirac functions 694
F. A rigorous analysis of the ideal Bose gas and the onset of Bose–Einstein condensation 697
G. On Watson functions 702
H. Thermodynamic relationships 703
I. Pseudorandom numbers 710
Bibliography 714
Index 734