Statistical Mechanics, Third Edition

✍ Scribed by R K Pathria, Paul D. Beale

• Publisher: Academic Press
• Year: 2011
• Tongue: English
• Leaves: 722
• Edition: 3
• Category: Library

✦ Synopsis

This third edition includes new sections on Bose-Einstein condensation and degenerate Fermi behavior of ultracold atomic gases, and two new chapters on computer simulation methods and the thermodynamics of the early universe. We have also added new sections on chemical and phase equilibrium, and expanded our discussions of correlations and scattering, quantized fields, finite-size effects, and the fluctuation-dissipation theorem. We hope this new edition will continue to provide new generations of students with a solid training in the methods of statistical physics. Bose-Einstein condensation in atomic gases Thermodynamics of the early universe -Computer simulations: Monte Carlo and molecular dynamics Correlation functions and scattering Fluctuation-dissipation theorem and the dynamical structure factor Chemical equilibrium Exact solution of the two-dimensional Ising model for finite systems Degenerate atomic Fermi gases Exact solutions of one-dimensional fluid models Interactions in ultracold Bose and Fermi gases Brownian motion of anisotropic particles and harmonic oscillators About this Edition This third edition includes new sections on Bose-Einstein condensation and degenerate Fermi behavior of ultracold atomic gases, and two new chapters on computer simulation methods and the thermodynamics of the early universe. We have also added new sections on chemical and phase equilibrium, and expanded our discussions of correlations and scattering, quantized fields, finite-size effects and the fluctuation-dissipation theorem. We hope this new edition will continue to provide new generations of students with a solid training in the methods of statistical physics. New this Edition Bose–Einstein condensation and degenerate Fermi gas behavior in ultracold atomic gases Finite-size scaling behavior of Bose-Einstein condensates Thermodynamics of the early universe Chemical equilibrium Monte Carlo and molecular dynamics simulations Correlation functions and scattering Fluctuation-dissipation theorem and the dynamical structure factor Phase equilibrium and the Clausius-Clapeyron equation Exact solutions of one-dimensional fluid models Exact solution of the two-dimensional Ising model on a finite lattice Summary of thermodynamic assemblies and associated statistical ensembles Pseudorandom number generators Dozens of new homework problems Read a sample chapter from Statistical Mechanics.

✦ Table of Contents

Statistical Mechanics......Page 3
Copyright......Page 4
Preface to the Third Edition......Page 5
Preface to the Second Edition......Page 8
Preface to the First Edition......Page 9
Historical Introduction......Page 11
The macroscopic and the microscopic states......Page 17
Contact between statistics and thermodynamics: physical significance of the number Ω (N, V, E)......Page 19
Further contact between statistics and thermodynamics......Page 22
The classical ideal gas......Page 25
The entropy of mixing and the Gibbs paradox......Page 32
The “correct” enumeration of the microstates......Page 36
Problems......Page 38
Phase space of a classical system......Page 40
Liouville's theorem and its consequences......Page 42
The microcanonical ensemble......Page 45
Examples......Page 47
Quantum states and the phase space......Page 50
Problems......Page 52
The Canonical Ensemble......Page 54
Equilibrium between a system and a heat reservoir......Page 55
A system in the canonical ensemble......Page 56
Physical significance of the various statistical quantities in the canonical ensemble......Page 65
Alternative expressions for the partition function......Page 67
The classical systems......Page 69
Energy fluctuations in the canonical ensemble: correspondence with the microcanonical ensemble......Page 73
Two theorems — the “equipartition” and the “virial”......Page 76
A system of harmonic oscillators......Page 80
The statistics of paramagnetism......Page 85
Thermodynamics of magnetic systems: negative temperatures......Page 92
Problems......Page 98
Equilibrium between a system and a particle-energy reservoir......Page 106
A system in the grand canonical ensemble......Page 108
Physical significance of the various statistical quantities......Page 110
Examples......Page 113
Density and energy fluctuations in the grand canonical ensemble: correspondence with other ensembles......Page 118
Thermodynamic phase diagrams......Page 120
Phase equilibrium and the Clausius–Clapeyron equation......Page 124
Problems......Page 126
Quantum-mechanical ensemble theory: the density matrix......Page 129
The microcanonical ensemble......Page 133
The canonical ensemble......Page 135
An electron in a magnetic field......Page 136
A free particle in a box......Page 137
A linear harmonic oscillator......Page 139
Systems composed of indistinguishable particles......Page 142
The density matrix and the partition function of a system of free particles......Page 147
Problems......Page 153
An ideal gas in a quantum-mechanical microcanonical ensemble......Page 155
An ideal gas in other quantum-mechanical ensembles......Page 160
Statistics of the occupation numbers......Page 163
Kinetic considerations......Page 166
Gaseous systems composed of molecules with internal motion......Page 169
Monatomic molecules......Page 171
Diatomic molecules......Page 172
Polyatomic molecules......Page 182
Chemical equilibrium......Page 184
Problems......Page 187
Ideal Bose Systems......Page 193
Thermodynamic behavior of an ideal Bose gas......Page 194
Bose–Einstein condensation in ultracold atomic gases......Page 205
Detection of the Bose–Einstein condensate......Page 207
Thermodynamic properties of the Bose–Einstein condensate......Page 210
Thermodynamics of the blackbody radiation......Page 214
The field of sound waves......Page 219
Inertial density of the sound field......Page 226
Elementary excitations in liquid helium II......Page 229
Problems......Page 237
Thermodynamic behavior of an ideal Fermi gas......Page 244
Magnetic behavior of an ideal Fermi gas......Page 251
Pauli paramagnetism......Page 252
Landau diamagnetism......Page 256
The electron gas in metals......Page 260
Thermionic emission (the Richardson effect)......Page 264
Photoelectric emission (the Hallwachs effect)......Page 268
Ultracold atomic Fermi gases......Page 271
Statistical equilibrium of white dwarf stars......Page 272
Statistical model of the atom......Page 277
Problems......Page 282
Observational evidence of the Big Bang......Page 287
Evolution of the temperature of the universe......Page 292
Relativistic electrons, positrons, and neutrinos......Page 294
Neutron fraction......Page 297
Annihilation of the positrons and electrons......Page 299
Neutrino temperature......Page 301
Primordial nucleosynthesis......Page 302
Recombination......Page 305
Epilogue......Page 307
Problems......Page 308
Cluster expansion for a classical gas......Page 310
Virial expansion of the equation of state......Page 318
Evaluation of the virial coefficients......Page 320
General remarks on cluster expansions......Page 326
Exact treatment of the second virial coefficient......Page 331
Cluster expansion for a quantum-mechanical system......Page 336
Correlations and scattering......Page 342
Static structure factor......Page 346
Scattering from crystalline solids......Page 349
Problems......Page 351
The formalism of second quantization......Page 355
Low-temperature behavior of an imperfect Bose gas......Page 365
Effects of interactions on ultracold atomic Bose–Einstein condensates......Page 368
Low-lying states of an imperfect Bose gas......Page 371
Energy spectrum of a Bose liquid......Page 376
States with quantized circulation......Page 380
Quantized vortex rings and the breakdown of superfluidity......Page 386
Low-lying states of an imperfect Fermi gas......Page 389
Energy spectrum of a Fermi liquid: Landau's phenomenological theory
......Page 395
Condensation in Fermi systems......Page 402
Problems......Page 404
Phase Transitions: Criticality, Universality, and Scaling......Page 411
General remarks on the problem of condensation......Page 412
Condensation of a van der Waals gas......Page 417
A dynamical model of phase transitions......Page 421
The lattice gas and the binary alloy......Page 427
Ising model in the zeroth approximation......Page 430
Ising model in the first approximation......Page 437
The critical exponents......Page 445
Thermodynamic inequalities......Page 448
Landau's phenomenological theory......Page 452
Scaling hypothesis for thermodynamic functions......Page 456
The role of correlations and fluctuations......Page 459
The critical exponents ν and η
......Page 466
A final look at the mean field theory......Page 470
Problems......Page 473
One-dimensional fluid models......Page 480
Hard spheres on a ring......Page 481
Isobaric ensemble of a one-dimensional fluid......Page 482
The Ising model in one dimension......Page 485
The n-vector models in one dimension......Page 491
The Ising model in two dimensions......Page 497
The two-dimensional Ising model on a finite lattice......Page 509
The spherical model in arbitrary dimensions......Page 517
The ideal Bose gas in arbitrary dimensions......Page 528
Other models......Page 535
Problems......Page 539
Phase Transitions: The Renormalization Group Approach......Page 547
The conceptual basis of scaling......Page 548
The Ising model in one dimension......Page 551
The spherical model in one dimension......Page 555
The Ising model in two dimensions......Page 557
The renormalization group: general formulation......Page 560
The Ising model in one dimension......Page 567
The Ising model in two dimensions......Page 568
The ε-expansion......Page 571
Dimension d ≲ 4, so that ε is a small positive number......Page 573
The 1/n expansion......Page 575
Other topics......Page 576
Finite-size scaling......Page 578
Problems......Page 587
Fluctuations and Nonequilibrium Statistical Mechanics......Page 590
Equilibrium thermodynamic fluctuations......Page 591
The Einstein–Smoluchowski theory of the Brownian motion......Page 594
The Langevin theory of the Brownian motion......Page 600
Brownian motion of a harmonic oscillator......Page 608
Approach to equilibrium: the Fokker–Planck equation......Page 610
Spectral analysis of fluctuations: the Wiener–Khintchine theorem......Page 616
The fluctuation–dissipation theorem......Page 624
Derivation of the fluctuation–dissipation theorem from linear response theory......Page 628
Inelastic scattering......Page 631
The Onsager relations......Page 633
Problems......Page 639
Introduction and statistics......Page 643
Monte Carlo simulations......Page 646
Metropolis Monte Carlo algorithm......Page 647
Molecular dynamics......Page 649
Molecular dynamics algorithm......Page 651
Particle simulations......Page 652
Simulations of hard spheres......Page 653
Computer simulation caveats......Page 656
Problems......Page 657
Influence of boundary conditions on the distribution of quantum states......Page 659
Certain mathematical functions......Page 661
“Volume” and “surface area” of an n-dimensional sphere of radius R......Page 668
On Bose–Einstein functions......Page 670
On Fermi–Dirac functions......Page 673
A rigorous analysis of the ideal Bose gas and the onset of Bose–Einstein condensation......Page 676
On Watson functions......Page 681
Thermodynamic relationships......Page 682
Entropy S(N, V, U) and the microcanonical ensemble......Page 683
Helmholtz free energy A(N, V, T) = U – TS and the canonical ensemble......Page 684
Thermodynamic potential π(μ, V, T) = –A + μN = PV and the grand canonical ensemble......Page 685
Gibbs free energy G(N, P, T) = A + PV = U – TS + PV = μN and the isobaric ensemble......Page 686
Enthalpy H(N, P, S) = U + PV......Page 687
Convexity and variances......Page 688
Pseudorandom numbers......Page 689
Bibliography......Page 692
B......Page 711
C......Page 712
D......Page 713
F......Page 714
H......Page 715
L......Page 716
M......Page 717
P......Page 718
R......Page 719
S......Page 720
V......Page 721
Z......Page 722