Statistical Mechanics (eBook)

Front Cover 1
Statistical Mechanics 4
Copyright Page 5
Contents 6
Preface to the Second Edition 12
Preface to the First Edition 14
Historical Introduction 16
Notes 21
Chapter 1. The Statistical Basis of Thermodynamics 24
1.1. The macroscopic and the microscopic states 24
1.2. Contact between statistics and thermodynamics: physical significance of the number O(N,V, E) 26
1.3. Further contact between statistics and thermodynamics 28
1.4. The classical ideal gas 30
1.5. The entropy of mixing and the Gibbs paradox 37
1.6. The "correct" enumeration of the microstates 40
Problems 41
Notes 43
Chapter 2. Elements of Ensemble Theory 45
2.1. Phase space of a classical system 45
2.2. Liouville's theorem and its consequences 47
2.3. The microcanonical ensemble 49
2.4. Examples 51
2.5. Quantum states and the phase space 54
Problems 55
Notes 56
Chapter 3. The Canonical Ensemble 58
3.1. Equilibrium between a system and a heat reservoir 59
3.2. A system in the canonical ensemble 60
3.3. Physical significance of the various statistical quantities in the canonical ensemble 68
3.4. Alternative expressions for the partition function 70
3.5. The classical systems 71
3.6. Energy fluctuations in the canonical ensemble: correspondence with the microcanonical ensemble 75
3.7. Two theorems–the "equipartition" and the "virial" 78
3.8. A system of harmonic oscillators 81
3.9. The statistics of paramagnetism 86
3.10. Thermodynamics of magnetic systems: negative temperatures 92
Problems 98
Notes 104
Chapter 4. The Grand Canonical Ensemble 105
4.1. Equilibrium between a system and a particle–energy reservoir 105
4.2. A system in the grand canonical ensemble 107
4.3. Physical significance of the various statistical quantities 108
4.4. Examples 111
4.5. Density and energy fluctuations in the grand canonical ensemble: correspondence with other ensembles 115
Problems 117
Notes 118
Chapter 5. Formulation of Quantum Statistics 119
5.1. Quantum-mechanical ensemble theory: the density matrix 119
5.2. Statistics of the various ensembles 122
5.3. Examples 126
5.4. Systems composed of indistinguishable particles 130
5.5. The density matrix and the partition function of a system of free particles 134
Problems 139
Notes 140
Chapter 6. The Theory of Simple Gases 142
6.1. An ideal gas in a quantum-mechanical microcanonical ensemble 142
6.2. An ideal gas in other quantum-mechanical ensembles 146
6.3. Statistics of the occupation numbers 149
6.4. Kinetic considerations 152
6.5. Gaseous systems composed of molecules with internal motion 155
Problems 167
Notes 170
Chapter 7. Ideal Bose Systems 172
7.1. Thermodynamic behavior of an ideal Bose gas 173
7.2. Thermodynamics of the black-body radiation 183
7.3. The field of sound waves 187
7.4. Inertial density of the sound field 194
7.5. Elementary excitations in liquid helium II 197
Problems 203
Notes 208
Chapter 8. Ideal Fermi Systems 210
8.1. Thermodynamic behavior of an ideal Fermi gas 210
8.2. Magnetic behavior of an ideal Fermi gas 216
8.3. The electron gas in metals 224
8.4. Statistical equilibrium of white dwarf stars 234
8.5. Statistical model of the atom 238
Problems 242
Notes 246
Chapter 9. Statistical Mechanics of Interacting Systems: The Method of Cluster Expansions 247
9.1. Cluster expansion for a classical gas 247
9.2. Virial expansion of the equation of state 254
9.3. Evaluation of the virial coefficients 255
9.4. General remarks on cluster expansions 260
9.5. Exact treatment of the second virial coefficient 264
9.6. Cluster expansion for a quantum-mechanical system 269
Problems 274
Notes 276
Chapter 10. Statistical Mechanics of Interacting Systems: The Method of Quantized Fields 277
10.1. The formalism of second quantization 277
10.2. Low-temperature behavior of an imperfect Bose gas 285
10.3. Low-lying states of an imperfect Bose gas 288
10.4. Energy spectrum of a Bose liquid 292
10.5. States with quantized circulation 295
10.6. Quantized vortex rings and the breakdown of superfluidity 300
10.7. Low-lying states of an imperfect Fermi gas 304
10.8. Energy spectrum of a Fermi liquid: Landau's phenomenological theory 308
Problems 314
Notes 317
Chapter 11. Phase Transitions: Criticality, Universality and Scaling 320
11.1. General remarks on the problem of condensation 321
11.2. Condensation of a van der Waals gas 325
11.3. A dynamical model of phase transitions 329
11.4. The lattice gas and the binary alloy 334
11.5. Ising model in the zeroth approximation 336
11.6. Ising model in the first approximation 343
11.7. The critical exponents 349
11.8. Thermodynamic inequalities 353
11.9. Landau's phenomenological theory 356
11.10. Scaling hypothesis for thermodynamic functions 359
11.11. The role of correlations and fluctuations 363
11.12. The critical exponents v and . 
368 
11.13. A final look at the mean, field theory 371
Problems 374
Notes 379
Chapter 12. Phase Transitions: Exact (or Almost Exact) Results for the Various Models 381
12.1. The Ising model in one dimension 381
12.2. The n-vector models in one dimension 387
12.3. The Ising model in two dimensions 392
12.4. The spherical model in arbitrary dimensions 404
12.5. The ideal Bose gas in arbitrary dimensions 413
12.6. Other models 419
Problems 422
Notes 428
Chapter 13. Phase Transitions: The Renormalization Group Approach 429
13.1. The conceptual basis of scaling 430
13.2. Some simple examples of renormalization 433
13.3. The renormalization group: general formulation 441
13.4. Applications of the renormalization group 446
13.5. Finite-size scaling 456
Problems 464
Notes 466
Chapter 14. Fluctuations 467
14.1. Thermodynamic fluctuations 468
14.2. Spatial correlations in a fluid 471
14.3. The Einstein–Smoluchowski theory of the Brownian motion 474
14.4. The Langevin theory of the Brownian motion 479
14.5. Approach to equilibrium: the Fokker– 
484 
14.6. Spectral analysis of fluctuations: the Wiener– 
489 
14.7. The fluctuation– 
496 
14.8. The Onsager relations 499
Problems 504
Notes 507
Appendixes 510
A. Influence of boundary conditions on the distribution of quantum states 510
B. Certain mathematical functions 512
C. "Volume" and "surface area" of an n-dimensional sphere of radius R 519
D. On Bose–Einstein functions 521
E. On Fermi-Dirac functions 523
F. On Watson functions 525
Notes 527
Bibliography 528
Index 538