Statistical Mechanics, Third 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. 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. 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 This classic text, first published in 1972, is designed for graduate physics courses in statistical mechanics. The second edition, published in 1996, incorporated three comprehensive chapters on phase transitions and critical phenomena.

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

This classic text, first published in 1972, is designed for graduate physics courses in statistical mechanics. The second edition, published in 1996, incorporated three comprehensive chapters on phase transitions and critical phenomena. 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 Main subject categories: • Statistical mechanics • Thermodynamics • Mathematical physics • Mathematical ModellingThis classic text, first published in 1972, is designed for graduate physics courses in statistical mechanics. The second edition, published in 1996, incorporated three comprehensive chapters on phase transitions and critical phenomena.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. The statistical basis of thermodynamics Elements of the ensemble theory The canonical ensemble The grand canonical ensemble Formulation of quantum statistics The theory of simple gases Ideal Bose systems Ideal Fermi systems Thermodynamics of the early universe Statistical mechanics of interacting systems : the method of cluster expansions Statistical mechanics of interacting systems : the method of quantized fields Phase transitions : criticality, universality, and scaling Phase transitions : exact (or almost exact) results for models Phase transitions : the renormalization group approach Fluctuations and nonequilibrium statistical mechanics Computer simulations.