Statistical Mechanics, Third Edition

Cover......Page 1
Title page......Page 4
Edition notice......Page 5
Contents......Page 6
Preface to the Third Edition......Page 14
Preface to the Second Edition......Page 18
Preface to the First Edition......Page 20
Historical Introduction......Page 22
1.1 The macroscopic and the microscopic states......Page 28
1.2 Contact between statistics and thermodynamics: physical significance of the number Ω(N, V, E)......Page 30
1.3 Further contact between statistics and thermodynamics......Page 33
1.4 The classical ideal gas......Page 36
1.5 The entropy of mixing and the Gibbs paradox......Page 43
1.6 The “correct” enumeration of the microstates......Page 47
Problems......Page 49
2.1 Phase space of a classical system......Page 51
2.2 Liouville\'s theorem and its consequences......Page 53
2.3 The microcanonical ensemble......Page 56
2.4 Examples......Page 58
2.5 Quantum states and the phase space......Page 61
Problems......Page 63
3 The Canonical Ensemble......Page 65
3.1 Equilibrium between a system and a heat reservoir......Page 66
3.2 A system in the canonical ensemble......Page 67
3.3 Physical significance of the various statistical quantities in the canonical ensemble......Page 76
3.4 Alternative expressions for the partition function......Page 78
3.5 The classical systems......Page 80
3.6 Energy fluctuations in the canonical ensemble: correspondence with the microcanonical ensemble......Page 84
3.7 Two theorems — the “equipartition” and the “virial”......Page 87
3.8 A system of harmonic oscillators......Page 91
3.9 The statistics of paramagnetism......Page 96
3.10 Thermodynamics of magnetic systems: negative temperatures......Page 103
Problems......Page 109
4.1 Equilibrium between a system and a particle-energy reservoir......Page 117
4.2 A system in the grand canonical ensemble......Page 119
4.3 Physical significance of the various statistical quantities......Page 121
4.4 Examples......Page 124
4.5 Density and energy fluctuations in the grand canonical ensemble: correspondence with other ensembles......Page 129
4.6 Thermodynamic phase diagrams......Page 131
4.7 Phase equilibrium and the Clausius–Clapeyron equation......Page 135
Problems......Page 137
5.1 Quantum-mechanical ensemble theory: the density matrix......Page 140
5.2.A The microcanonical ensemble......Page 144
5.2.B The canonical ensemble......Page 146
5.3.A An electron in a magnetic field......Page 147
5.3.B A free particle in a box......Page 148
5.3.C A linear harmonic oscillator......Page 150
5.4 Systems composed of indistinguishable particles......Page 153
5.5 The density matrix and the partition function of a system of free particles......Page 158
Problems......Page 164
6.1 An ideal gas in a quantum-mechanical microcanonical ensemble......Page 166
6.2 An ideal gas in other quantum-mechanical ensembles......Page 171
6.3 Statistics of the occupation numbers......Page 174
6.4 Kinetic considerations......Page 177
6.5 Gaseous systems composed of molecules with internal motion......Page 180
6.5.A Monatomic molecules......Page 182
6.5.B Diatomic molecules......Page 183
6.5.C Polyatomic molecules......Page 193
6.6 Chemical equilibrium......Page 195
Problems......Page 198
7 Ideal Bose Systems......Page 204
7.1 Thermodynamic behavior of an ideal Bose gas......Page 205
7.2 Bose–Einstein condensation in ultracold atomic gases......Page 216
7.2.A Detection of the Bose–Einstein condensate......Page 218
7.2.B Thermodynamic properties of the Bose–Einstein condensate......Page 221
7.3 Thermodynamics of the blackbody radiation......Page 225
7.4 The field of sound waves......Page 230
7.5 Inertial density of the sound field......Page 237
7.6 Elementary excitations in liquid helium II......Page 240
Problems......Page 248
8.1 Thermodynamic behavior of an ideal Fermi gas......Page 255
8.2 Magnetic behavior of an ideal Fermi gas......Page 262
8.2.A Pauli paramagnetism......Page 263
8.2.B Landau diamagnetism......Page 267
8.3 The electron gas in metals......Page 271
8.3.A Thermionic emission (the Richardson effect)......Page 275
8.3.B Photoelectric emission (the Hallwachs effect)......Page 279
8.4 Ultracold atomic Fermi gases......Page 282
8.5 Statistical equilibrium of white dwarf stars......Page 283
8.6 Statistical model of the atom......Page 288
Problems......Page 293
9.1 Observational evidence of the Big Bang......Page 298
9.2 Evolution of the temperature of the universe......Page 303
9.3 Relativistic electrons, positrons, and neutrinos......Page 305
9.4 Neutron fraction......Page 308
9.5 Annihilation of the positrons and electrons......Page 310
9.6 Neutrino temperature......Page 312
9.7 Primordial nucleosynthesis......Page 313
9.8 Recombination......Page 316
9.9 Epilogue......Page 318
Problems......Page 319
10.1 Cluster expansion for a classical gas......Page 321
10.2 Virial expansion of the equation of state......Page 329
10.3 Evaluation of the virial coefficients......Page 331
10.4 General remarks on cluster expansions......Page 337
10.5 Exact treatment of the second  virial coefficient......Page 342
10.6 Cluster expansion for a quantum-mechanical system......Page 347
10.7 Correlations and scattering......Page 353
10.7.A Static structure factor......Page 357
10.7.B Scattering from crystalline solids......Page 360
Problems......Page 362
11.1 The formalism of second quantization......Page 366
11.2 Low-temperature behavior of an imperfect Bose gas......Page 376
11.2.A Effects of interactions on ultracold atomic Bose–Einstein condensates......Page 379
11.3 Low-lying states of an imperfect Bose gas......Page 382
11.4 Energy spectrum of a Bose liquid......Page 387
11.5 States with quantized circulation......Page 391
11.6 Quantized vortex rings and the breakdown of superfluidity......Page 397
11.7 Low-lying states of an imperfect Fermi gas......Page 400
11.8 Energy spectrum of a Fermi liquid: Landau\'s phenomenological theory......Page 406
11.9 Condensation in Fermi systems......Page 413
Problems......Page 415
12 Phase Transitions: Criticality, Universality, and Scaling......Page 422
12.1 General remarks on the problem of condensation......Page 423
12.2 Condensation of a van der Waals gas......Page 428
12.3 A dynamical model of phase transitions......Page 432
12.4 The lattice gas and the binary alloy......Page 438
12.5 Ising model in the zeroth approximation......Page 441
12.6 Ising model in the first approximation......Page 448
12.7 The critical exponents......Page 456
12.8 Thermodynamic inequalities......Page 459
12.9 Landau\'s phenomenological theory......Page 463
12.10 Scaling hypothesis for thermodynamic functions......Page 467
12.11 The role of correlations and fluctuations......Page 470
12.12 The critical exponents ν and η......Page 477
12.13 A final look at the mean field theory......Page 481
Problems......Page 484
13.1 One-dimensional fluid models......Page 491
13.1.A Hard spheres on a ring......Page 492
13.1.B Isobaric ensemble of a one-dimensional fluid......Page 493
13.2 The Ising model in one dimension......Page 496
13.3 The n-vector models in one dimension......Page 502
13.4 The Ising model in two dimensions......Page 508
13.4.A The two-dimensional Ising model on a finite lattice......Page 520
13.5 The spherical model in arbitrary dimensions......Page 528
13.6 The ideal Bose gas in arbitrary dimensions......Page 539
13.7 Other models......Page 546
Problems......Page 550
14 Phase Transitions: The Renormalization Group Approach......Page 558
14.1 The conceptual basis of scaling......Page 559
14.2.A The Ising model in one dimension......Page 562
14.2.B The spherical model in one dimension......Page 566
14.2.C The Ising model in two dimensions......Page 568
14.3 The renormalization group: general formulation......Page 571
14.4.A The Ising model in one dimension......Page 578
14.4.C The Ising model in two dimensions......Page 579
14.4.D The ε-expansion......Page 582
Dimension d ≲ 4, so that ε is a small positive number......Page 584
14.4.E The 1/n expansion......Page 586
14.4.F Other topics......Page 587
14.5 Finite-size scaling......Page 589
Problems......Page 598
15 Fluctuations and Nonequilibrium Statistical Mechanics......Page 601
15.1 Equilibrium thermodynamic fluctuations......Page 602
15.2 The Einstein–Smoluchowski theory of the Brownian motion......Page 605
15.3 The Langevin theory of the Brownian motion......Page 611
15.3.A Brownian motion of a harmonic oscillator......Page 619
15.4 Approach to equilibrium: the Fokker–Planck equation......Page 621
15.5 Spectral analysis of fluctuations: the Wiener–Khintchine theorem......Page 627
15.6 The fluctuation–dissipation theorem......Page 635
15.6.A Derivation of the fluctuation–dissipation theorem from linear response theory......Page 639
15.6.B Inelastic scattering......Page 642
15.7 The Onsager relations......Page 644
Problems......Page 650
16.1 Introduction and statistics......Page 654
16.2 Monte Carlo simulations......Page 657
16.2.A Metropolis Monte Carlo algorithm......Page 658
16.3 Molecular dynamics......Page 660
16.3.A Molecular dynamics algorithm......Page 662
16.4 Particle simulations......Page 663
16.4.A Simulations of hard spheres......Page 664
16.5 Computer simulation caveats......Page 667
Problems......Page 668
Appendix A Influence of boundary conditions on the distribution of quantum states......Page 670
Appendix B Certain mathematical functions......Page 672
Appendix C “Volume” and “surface area” of an n-dimensional sphere of radius R......Page 679
Appendix D On Bose–Einstein functions......Page 681
Appendix E On Fermi–Dirac functions......Page 684
Appendix F A rigorous analysis of the ideal Bose gas and the onset of Bose–Einstein condensation......Page 687
Appendix G On Watson functions......Page 692
Appendix H Thermodynamic relationships......Page 693
Entropy S(N, V, U) and the microcanonical ensemble......Page 694
Helmholtz free energy A(N, V, T) = U – TS and the canonical ensemble......Page 695
Thermodynamic potential π(μ, V, T) = –A + μN = PV and the grand canonical ensemble......Page 696
Gibbs free energy G(N, P, T) = A + PV = U – TS + PV = μN and the isobaric ensemble......Page 697
Enthalpy H(N, P, S) = U + PV......Page 698
Convexity and variances......Page 699
Appendix I Pseudorandom numbers......Page 700
Bibliography......Page 703
B......Page 722
C......Page 723
D......Page 724
F......Page 725
H......Page 726
L......Page 727
M......Page 728
P......Page 729
R......Page 730
S......Page 731
V......Page 732
Z......Page 733