Equilibrium and Non-Equilibrium Statistical Thermodynamics

Cover......Page 1
EQUILIBRIUM AND NON-EQUILIBRIUM STATISTICAL THERMODYNAMICS......Page 3
Title......Page 5
Copyright......Page 6
1 Thermostatics......Page 7
2 Statistical entropy and Boltzmann distribution......Page 8
3 Canonical and grand canonical ensembles: applications......Page 9
4 Critical phenomena......Page 10
5 Quantum statistics......Page 11
6 Irreversible processes: macroscopic theory......Page 12
8 Irreversible processes: kinetic theory......Page 13
9 Topics in non-equilibrium statistical mechanics......Page 14
Appendix......Page 16
Preface......Page 17
1.1.1 Microscopic and macroscopic descriptions......Page 19
1.1.2 Walls......Page 21
1.1.3 Work, heat, internal energy......Page 23
1.1.4 Definition of thermal equilibrium......Page 26
1.2.1 Internal constraints......Page 27
1.2.2 Principle of maximum entropy......Page 28
1.2.3 Intensive variables: temperature, pressure, chemical potential......Page 30
1.2.4 Quasi-static and reversible processes......Page 35
1.2.5 Maximum work and heat engines......Page 38
1.3.1 Thermodynamic potentials and Massieu functions......Page 40
1.3.2 Specific heats......Page 42
1.3.3 Gibbs–Duhem relation......Page 44
1.4.1 Concavity of entropy and convexity of energy......Page 45
1.4.2 Stability conditions and their consequences......Page 46
1.5.1 Statement of the third law......Page 49
1.5.2 Application to metastable states......Page 50
1.5.3 Low temperature behaviour of specific heats......Page 51
1.6.2 Internal variable in equilibrium......Page 53
1.6.3 Relations between thermodynamic coefficients......Page 54
1.6.6 Equation of state for a fluid......Page 55
1.7.1 Reversible and irreversible free expansions of an ideal gas......Page 56
1.7.2 van der Waals equation of state......Page 57
1.7.3 Equation of state for a solid......Page 58
1.7.4 Specific heats of a rod......Page 59
1.7.5 Surface tension of a soap film......Page 60
1.7.7 Adiabatic demagnetization of a paramagnetic salt......Page 61
1.8 Further reading......Page 63
2.1.1 Time evolution in quantum mechanics......Page 65
2.1.2 The density operators and their time evolution......Page 67
2.1.3 Quantum phase space......Page 69
2.1.4 (P, V, E) relation for a mono-atomic ideal gas......Page 71
2.2.1 Liouville’s theorem......Page 73
2.2.2 Density in phase space......Page 74
2.3.1 Entropy of a probability distribution......Page 77
2.3.2 Statistical entropy of a mixed quantum state......Page 78
2.3.3 Time evolution of the statistical entropy......Page 81
2.4.1 Postulate of maximum of statistical entropy......Page 82
2.4.2 Equilibrium distribution......Page 83
2.4.3 Legendre transformation......Page 85
2.4.4 Canonical and grand canonical ensembles......Page 86
2.5.1 Heat and work: first law......Page 88
2.5.2 Entropy and temperature: second law......Page 90
2.5.3 Entropy of mixing......Page 92
2.5.4 Pressure and chemical potential......Page 95
2.6.1 Microscopic reversibility and macroscopic irreversibility......Page 97
2.6.2 Physical basis of irreversibility......Page 99
2.6.3 Loss of information and the growth of entropy......Page 101
2.7.1 Density operator for spin-1/2......Page 104
2.7.3 Liouville theorem and continuity equation......Page 106
2.7.6 Heat exchanges between system and reservoir......Page 107
2.7.8 Fluctuation-response theorem......Page 108
2.7.10 Entropy of mixing and osmotic pressure......Page 110
2.8 Further reading......Page 111
3.1.1 Mean values and fluctuations......Page 113
3.1.2 Partition function and thermodynamics of an ideal gas......Page 116
3.1.3 Paramagnetism......Page 119
3.1.4 Ferromagnetism and the Ising model......Page 123
3.1.5 Thermodynamic limit......Page 130
3.2.1 Classical limit......Page 133
3.2.2 Maxwell distribution......Page 134
3.2.3 Equipartition theorem......Page 137
3.2.4 Specific heat of a diatomic ideal gas......Page 139
3.3.1 Qualitative discussion......Page 140
3.3.2 Partition function of a diatomic molecule......Page 143
3.4 From ideal gases to liquids......Page 145
3.4.1 Pair correlation function......Page 147
3.4.2 Measurement of the pair correlation function......Page 150
3.4.3 Pressure and energy......Page 152
3.5.1 Basic formulae......Page 154
3.5.2 Coexistence of phases......Page 155
3.5.3 Equilibrium condition at constant pressure......Page 156
3.5.4 Equilibrium and stability conditions at constant......Page 158
3.5.5 Chemical reactions......Page 160
3.6.1 Grand partition function......Page 164
3.6.2 Mono-atomic ideal gas......Page 167
3.6.3 Thermodynamics and fluctuations......Page 168
3.7.2 Equation of state for the Einstein model of a solid......Page 170
3.7.4 Nuclear specific heat of a metal......Page 171
3.7.5 Solid and liquid vapour pressures......Page 172
3.7.6 Electron trapping in a solid......Page 173
3.8.1 One-dimensional Ising model......Page 174
3.8.2 Negative temperatures......Page 176
3.8.3 Diatomic molecules......Page 178
3.8.4 Models of a boundary surface......Page 179
3.8.5 Debye–Hückel approximation......Page 183
3.8.6 Thin metallic film......Page 184
3.8.7 Beyond the ideal gas: first term of virial expansion......Page 186
3.8.8 Theory of nucleation......Page 189
3.9 Further reading......Page 191
4 Critical phenomena......Page 193
4.1.1 Some exact results for the Ising model......Page 195
4.1.2 Correlation functions......Page 202
4.1.3 Broken symmetry......Page 206
4.1.4 Critical exponents......Page 210
4.2.1 A convexity inequality......Page 212
4.2.2 Fundamental equation of mean field theory......Page 213
4.2.3 Broken symmetry and critical exponents......Page 216
4.3.1 Landau functional......Page 221
4.3.2 Broken continuous symmetry......Page 225
4.3.3 Ginzburg–Landau Hamiltonian......Page 228
4.3.4 Beyond Landau’s theory......Page 230
4.3.5 Ginzburg criterion......Page 232
4.4.1 Spin blocks......Page 235
4.4.2 Critical exponents and scaling transformations......Page 241
4.4.3 Critical manifold and fixed points......Page 245
4.4.4 Limit distributions and correlation functions......Page 251
4.4.5 Magnetization and free energy......Page 254
4.5.1 Gaussian fixed point......Page 257
4.5.2 Non-Gaussian fixed point......Page 260
4.5.3 Critical exponents to order......Page 266
4.5.4 anomalousScalingoperatorsand dimensions......Page 269
4.6.1 High temperature expansion and Kramers–Wannier duality......Page 271
4.6.4 Accuracy of the variational method......Page 273
4.6.5 Shape and energy of an Ising wall......Page 274
4.6.6 The Ginzburg–Landau theory of superconductivity......Page 275
4.6.8 Critical exponents for n 
 1......Page 277
4.6.9 Renormalization of the Gaussian model......Page 279
4.6.11 Critical exponents to order for n = 1......Page 280
4.6.13 Energy–energy correlations......Page 281
4.6.14 ‘Derivation’ of the Ginzburg–Landau Hamiltonian from the Ising model......Page 282
4.7 Further reading......Page 283
5 Quantum statistics......Page 285
5.1 Bose–Einstein and Fermi–Dirac distributions......Page 286
5.1.1 Grand partition function......Page 287
5.1.2 Classical limit: Maxwell–Boltzmann statistics......Page 289
5.1.3 Chemical potential and relativity......Page 290
5.2.1 Ideal Fermi gas at zero temperature......Page 291
5.2.2 Ideal Fermi gas at low temperature......Page 294
5.2.3 Corrections to the ideal Fermi gas......Page 299
5.3.1 Electromagnetic radiation in thermal equilibrium......Page 302
5.3.2 Black body radiation......Page 305
5.4.1 Simple model of vibrations in solids......Page 307
5.4.2 Debye approximation......Page 312
5.4.3 Calculation of thermodynamic functions......Page 314
5.5.1 Bose–Einstein condensation......Page 317
5.5.2 Thermodynamics of the condensed phase......Page 322
5.5.3 Applications: atomic condensates and helium-4......Page 326
5.6.3 Two-dimensional Fermi gas......Page 330
5.6.4 Non-degenerate Fermi gas......Page 331
5.6.6 Phonons and magnons......Page 332
5.6.7 Photon–electron–positron equilibrium in a star......Page 333
5.7.1 Pauli paramagnetism......Page 334
5.7.2 Landau diamagnetism......Page 336
5.7.3 White dwarf stars......Page 337
5.7.4 Quark–gluon plasma......Page 339
5.7.5 Bose–Einstein condensates of atomic gases......Page 341
5.7.6 Solid–liquid equilibrium for helium-3......Page 343
5.7.7 Superfluidity for hardcore bosons......Page 347
5.8 Further reading......Page 352
6 Irreversible processes: macroscopic theory......Page 353
6.1.1 Conservation laws......Page 354
6.1.2 Local equation of state......Page 357
6.1.3 Affinities and transport coefficients......Page 359
Heat diffusion in an insulating solid (or a simple .uid)......Page 360
Particle diffusion......Page 362
6.1.5 Dissipation and entropy production......Page 363
6.2.1 Coupling between thermal and particle diffusion......Page 367
6.2.2 Electrodynamics......Page 368
6.3.1 Conservation laws in a simple fluid......Page 371
Conservation of momentum......Page 372
6.3.2 Derivation of current densities......Page 376
6.3.3 Transport coefficients and the Navier–Stokes equation......Page 378
6.4.3 Relation between viscosity and diffusion......Page 382
6.4.5 Lord Kelvin’s model of Earth cooling......Page 383
6.5.1 Entropy current in hydrodynamics......Page 384
6.5.2 Hydrodynamics of the perfect fluid......Page 386
6.5.3 Thermoelectric effects......Page 387
6.5.4 Isomerization reactions......Page 389
6.6 Further reading......Page 391
7.1 Markov chains, convergence and detailed balance......Page 393
7.2.1 Implementation......Page 397
7.2.2 Measurements......Page 398
7.2.3 Autocorrelation, thermalization and error bars......Page 400
7.3 Critical slowing down and cluster algorithms......Page 402
7.4 Quantum Monte Carlo: bosons......Page 406
7.4.1 Formulation and implementation......Page 407
7.4.2 Measurements......Page 414
7.4.3 Quantum spin-1/2 models......Page 416
7.5 Quantum Monte Carlo: fermions......Page 418
7.6 Finite size scaling......Page 422
7.7 Random number generators......Page 426
7.8.2 Finite size scaling in infinite geometries......Page 428
7.9.1 Two-dimensional Ising model: Metropolis......Page 429
7.9.2 Two-dimensional Ising model: Glauber......Page 431
7.9.3 Two-dimensional clock model......Page 432
7.9.4 Two-dimensional XY model: Kosterlitz–Thouless transition......Page 437
7.9.5 Two-dimensional XY model: superfluidity and critical velocity......Page 441
7.9.6 Simple quantum model: single spin in transverse field......Page 449
7.9.7 One-dimensional Ising model in transverse field: quantum phase transition......Page 451
7.9.8 Quantum anharmonic oscillator: path integrals......Page 453
7.10 Further reading......Page 459
8.1.1 Distribution function......Page 461
8.1.2 Cross section, collision time, mean free path......Page 462
Thermal conductivity or energy transport......Page 467
Viscosity or momentum transport......Page 469
Diffusion or particle transport......Page 470
8.2.1 Spatio-temporal evolution of the distribution function......Page 471
8.2.2 Basic equations of the Boltzmann–Lorentz model......Page 473
8.2.3 Conservation laws and continuity equations......Page 475
8.2.4 Linearization: Chapman–Enskog approximation......Page 476
8.2.5 Currents and transport coefficients......Page 480
8.3.1 Collision term......Page 482
8.3.2 Conservation laws......Page 487
8.3.3 H-theorem......Page 490
8.4.1 Linearization of the Boltzmann equation......Page 494
8.4.2 Variational method......Page 496
8.4.3 Calculation of the viscosity......Page 499
8.5.1 Time distribution of collisions......Page 502
8.5.4 Calculation of the collision time......Page 503
8.5.6 Equilibrium distribution from the Boltzmann equation......Page 504
8.6.1 Thermal diffusion in the Boltzmann–Lorentz model......Page 505
A Introduction......Page 506
B Classical ideal gas......Page 507
C Ideal Fermi gas......Page 509
8.6.3 Photon diffusion and energy transport in the Sun......Page 510
A Preliminaries......Page 511
C Model for the Sun......Page 512
8.6.4 Momentum transfer in a shear flow......Page 513
B Coef.cient of viscosity......Page 514
8.6.5 Electrical conductivity in a magnetic field and quantum Hall effect......Page 515
A Electric conductivity in the presence of a magnetic field......Page 516
B Simplified model and the Hall effect......Page 517
C Quantum Hall effect......Page 518
A Specific heat......Page 520
B The two-fluid model......Page 521
A Static properties......Page 523
B Boltzmann equation......Page 526
8.6.8 Calculation of the coefficient of thermal conductivity......Page 528
8.7 Further reading......Page 530
9 Topics in non-equilibrium statistical mechanics......Page 531
9.1.1 Dynamical susceptibility......Page 532
9.1.2 Nyquist theorem......Page 536
9.1.3 Analyticity properties......Page 538
9.1.4 Spin diffusion......Page 540
9.2.1 Quantum fluctuation response theorem......Page 544
9.2.2 Quantum Kubo function......Page 546
9.2.3 Fluctuation-dissipation theorem......Page 548
9.2.4 Symmetry properties and dissipation......Page 549
9.2.5 Sum rules......Page 551
9.3 Projection method and memory effects......Page 553
9.3.1 Phenomenological introduction to memory effects......Page 554
9.3.2 Projectors......Page 556
9.3.3 Langevin–Mori equation......Page 558
9.3.4 Brownian motion: qualitative description......Page 561
9.3.5 Brownian motion: the m/M → 0 limit......Page 563
9.4.1 Definitions and first properties......Page 565
9.4.2 Ornstein–Uhlenbeck process......Page 567
9.5.1 Derivation of Fokker–Planck from Langevin equation......Page 570
9.5.2 Equilibrium and convergence to equilibrium......Page 572
9.5.3 Space-dependent diffusion coefficient......Page 574
9.6 Numerical integration......Page 576
9.7.1 Linear response: forced harmonic oscillator......Page 580
9.7.2 Force on a Brownian particle......Page 581
9.7.4 Mori’s scalar product......Page 582
9.7.5 Symmetry properties of χ''ij......Page 583
9.7.7 Proof of the f-sum rule in quantum mechanics......Page 584
9.7.8 Diffusion of a Brownian particle......Page 585
9.7.9 Strong friction limit: harmonic oscillator......Page 586
9.7.11 Moments of the Fokker–Planck equation......Page 587
9.7.13 Numerical integration of the Langevin equation......Page 588
9.7.14 Metastable states and escape times......Page 589
9.8.1 Inelastic light scattering from a suspension of particles......Page 590
9.8.2 Light scattering by a simple fluid......Page 594
A Preliminary results......Page 598
B The model......Page 599
9.8.4 Itô versus Stratonovitch dilemma......Page 600
9.8.5 Kramers equation......Page 602
9.9 Further reading......Page 603
A.1.1 Legendre transform with one variable......Page 605
A.1.2 Multivariate Legendre transform......Page 606
A.2 Lagrange multipliers......Page 607
A.3.1 Traces......Page 609
A.3.2 Tensor products......Page 610
A.4.1 Rotations......Page 611
A.4.2 Tensors......Page 614
A.5.1 Gaussian integrals......Page 616
A.5.2 Integrals of quantum statistics......Page 618
A.6 Functional derivatives......Page 619
A.7 Units and physical constants......Page 622
References......Page 623
Index......Page 629