Statistical mechanics/ Pathria,R.K.

1. The Statistical Basis of Thermodynamics 1
1.1. The macroscopic and the microscopic states 1
1.2. Contact between statistics and thermodynamics:
physical significance of the number Q(N, V,E) 3
1.3. Further contact between statistics and thermodynamics 6
1.4. The classical ideal gas 9
1.5. The entropy of mixing and the Gibbs paradox 16
1.6. The "correct" enumeration of the microstates 20
Problems 22
2. Elements of Ensemble Theory 25
2.1. Phase space of a classical system 25
2.2. Liouville's theorem and its consequences 27
2.3. The microcanonical ensemble 30
2.4. Examples 32
2.5. Quantum states and the phase space 35
Problems 37
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3. The Canonical Ensemble 39
3.1. Equilibrium between a system and a heat reservoir 40
3.2. A System in the canonical ensemble 41
3.3. Physical significance of the various statistical quantities
in the canonical ensemble 50
3.4. Alternative expressions for the partition function 52
3.5. The classical systems 54
3.6. Energy Fluctuations in the canonical ensemble:
correspondence with the microcanonical ensemble 58
3.7. TWo theorems —the "equipartition" and the "virial" 61
3.8. A System of harmonic oscillators 65
3.9. The statistics of paramagnetism 70
3.10. Thermodynamics of magnetic systems:
negative temperatures 77
Problems q3
4. The Grand Canonical Ensemble 91
4.1. Equilibrium between system and a particle-energy
reservoir gj
4.2. A System in the grand canonical ensemble 93
4.3. Physical significance of the various statistical quantities 95
4.4. Examples 9g
4.5. Density and energy fluctuations in the grand canonical
ensemble: correspondence with other ensembles 103
4.6. Thermodynamic phase diagrams IO5
4.7. Phase equilibrium and the Clausius-Clapeyron equation 109
Problems
5. Formulation of Quantum Statistics
5.1. Quantum-mechanical ensemble theory:
the density matrix
5.2. Statistics of the various ensembles

5.3. Examples 122
5.4. Systems composed of indistinguishable particles 128
5.5. The density matrix and the partition function of a
system of free particles 133
Problems 139
6. The Theory of Simple Gases 141
6.1. An ideal gas in a quantum-mechanical
microcanonical ensemble 141
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
6.6. Chemical equilibrium 170
Problems 173
7. Ideal Bose Systems 179
7.1. Thermodynamic behavior of an ideal Bose gas i80
7.2. Bose-Einstein condensation in ultracold atomic gases 191
^ 7.3. Thermodynamics of the blackbody radiation 200
7.4. The field of sound waves 205
7.5. Inertial density of the sound field 212
7.6. Elementary excitations in liquid helium II 215
Problems 223
8. Ideal Fermi Systems 231
8.1. Thermodynamic behavior of an ideal Fermi gas 231
8.2. Magnetic behavior of an ideal Fermi Gas 238
8.3. The electron gas in metals 247
8.4. Ultracold atomic Fermi gases 258
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8.5. Statistical equilibrium of white dwarf stars 259
8.6. Statistical model ofthe atom 264
Problems 269
9. Thermodynamics of the Early Universe 275
9.1. Observational evidence of the Big Bang 275
9.2. Evolution of the temperature of the universe 280
9.3. Relativistic electrons, positrons, and neutrinos 282
9.4. Neutron fraction 285
9.5. Annihilation of the positrons and electrons 287
9.6. Neutrino temperature 289
9.7. Primordial nucleosynthesis 290
9.8. Recombination 293
9.9. Epilogue 295
Problems 296
10. Statistical Mechanics of Interacting Systems:
The Method of Cluster Expansions 299
10.1. Cluster expansion for a classical gas 299
10.2. Virial expansion of the equation of state 307
10.3. Evaluation of the virial coefficients 309
10.4. General remarks on cluster expansions 315
10.5. Exact treatment of the second virisd coefficient 320
10.6. Cluster expansion for a quantum-mechanical system 325
10.7. Correlations and scattering 331
Problems 340
11. Statistical Mechanics of Interacting Systems:
The Method of Quantized Fields 345
11.1. The~formalism of second quantization 345
11.2. Low-temperature behavior of
an imperfect Bose gas 355
11.3. Low-lying states of an imperfect Bose gas 361
11.4. Energy spectrum of a Bose liquid 366
11.5. States with quantized circulation 370
11.6. Quantized vortex rings and the breakdown
of superfluidity 376
11.7. Low-lying states of an imperfect Fermi gas 379
11.8. Energy spectrum of a Fermi liquid: Landau
phenomenological theory 385
11.9. Condensation in Fermi systems 392
Problems 394
12. Phase Transitions: Criticality, Universality, and Scaling 401
12.1. General remarks on the problem of condensation 402
12.2.
Condensation ofa van der Waals gas 407
12.3. A Dynamical model of phase transitions 411
12.4. The lattice gas and the binary alloy 417
12.5. Ising model in the zeroth approximation 420
12.6. Ising model in the first approximation 427
12.7. The critical exponents 435
12.8.
Thermodynamic inequalities 438
12.9. Landau's phenomenological theory 442
12.10. Scaling hypothesis for thermodynamic functions 446
12.11. The role of correlations £ind fluctuations 449
12.12. The critical exponents Vand 77 456
12.13. Afinal look at the mean field theory 460
Problems 463
13. Phase Transitions: Exact (or Almost Exact) Results
for Various Models 471
13.1. One-dimensional fluid models 471
13.2. The Ising model in one dimension 476
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13.3. The n-vector models in one dimension 482
13.4. The Ising model in two dimensions 488
13.5. The spherical model in arbitrary dimensions 508
13.6. The ideal Bose gas in arbitrary dimensions 519
13.7. Other models 526
Problems 530
14. Phase Transitions: The Renormalization Group Approach 539
14.1. The conceptual basis of scaling 540
14.2. Some simple examples of renormalization 543
14.3. The renormalization group: general formulation 552
14.4. Applications of the renormalization group 559
14.5. Finite-size scaling 570
Problems
15. Fluctuations and Nonequilibrium Statistical Mechanics 583
15.1. Equilibrium thermodynamic fluctuations 584
15.2. The Einstein-Smoluchowski theory of the
Brownian motion
15.3. The Langevin theory of the Brownian motion 593
15.4. Approach to equilibrium: tiie Fokker-Planck equation 603
15.5. Spectral £analysis of fluctuations: the
Wiener-Khintchine theorem 609
15.6. The fluctuation-dissipation theorem 617
15.7. The Onsager relations 626
Problems 632
16. Computer Simulations 637
16.1. Introduction and statistics 637
16.2. Monte Carlo simulations 640
16.3. Molecular dynamics 643
16.4. Particle simulations 646
16.5. Computer simulation caveats 650