A Modern Course in Statistical Physics

Transcription

1 A Modern Course in Statistical Physics 2nd Edition L. E. REICHL A Wiley-Interscience Publication JOHN WILEY & SONS, INC. New York Chichester Weinheim Brisbane Singapore Toronto

2 CONTENTS Preface xix 1. Introduction 1 1.A. Overview 1 l.b. Plan of Book 2 l.c. Use as a Textbook 5 PART ONE THERMODYNAMICS 2. Introduction to Thermodynamics 9 2.A. Introductory Remarks 9 2.B. 2.C. State Variables and Exact Differentials Some Mechanical Equations of State C1. 2.C.2. 2.C.3. 2.C.4. 2.C.5. 2.C.6. 2.C.7. 2.C.8. Ideal Gas Law Virial Expansion Van der Waals Equation of State Solids Elastic Wire or Rod Surface Tension Electric Polarization Curie's Law D. The Laws of Thermodynamics 21 2.D.I. Zeroth Law 22 2.D.2. First Law 22 2.D.3. Second Law 23 2.D.4. Third Law 31 2.E. Fundamental Equation of Thermodynamics 33 2.F. Thermodynamic Potentials 36 2.F.I. Internal Energy 37 2.F.2. Enthalpy 40 2.F.3. Helmholz Free Energy 42 2.F.4. Gibbs Free Energy 45 2.F.5. Grand Potential 48

3 viii CONTENTS 2.G. Response Functions 50 2.G. 1. Thermal Response Functions (Heat Capacity) 50 2.G.2. Mechanical Response Functions 53 2.H. Stability of the Equilibrium State 55 2.H. 1. Conditions for Local Equilibrium in a PVT System 55 2.H.2. Conditions for Local Stability in a PVT System 57 2.H.3. Implications of the Stability Requirements for the Free Energies 63 S2.A. Cooling and Liquefactions of Gases 66 S2.A.1. The Joule Effect: Free Expansion 66 S2.A.2. The Joule-Kelvin Effect: Throttling 68 S2.B. Entropy of Mixing and the Gibbs Paradox 72 S2.C. Osmotic Pressure in Dilute Solutions 74 S2.D. The Thermodynamics of Chemical Reactions 78 S2.D.1. The Affinity 78 S2.D.2. Stability 82 S2.E. The Thermodynamics of Electrolytes 86 References 89 Problems The Thermodynamics of Phase Transitions 96 3.A. Introductory Remarks 96 3.B. Coexistence of Phases: Gibbs Phase Rule 98 3.C. Classification of Phase Transitions D. Pure PVT Systems D.I. Phase Diagrams D.2. Coexistence Curves: Clausius-Clapyron Equation D.3. Liquid-Vapor Coexistence Region D.4. The van der Waals Equation E. Superconductors F. The Helium Liquids F.I. Liquid He F.2. Liquid He F.3. Liquid He 3 -He 4 Mixtures G. Landau Theory G.I. Continuous Phase Transitions G.2. First-Order Transitions 134

4 CONTENTS ix 3.H. Critical Exponents H.I. Definition of Critical Exponents H.2. The Critical Exponents for Pure PVT Systems 137 S3.A. Surface Tension 142 S3.B. Thermomechanical Effect 146 S3.C. The Critical Exponents for the Curie Point 149 S3.D. Tricritical Points 151 S3.E. Binary Mixtures 153 S3.E.1. Stability Conditions 154 S3.E.2. Equilibrium Conditions 155 S3.E.3. Coexistence Curve 160 S3.F. The Ginzburg-Landau Theory of Superconductors 162 References 166 Problems 167 PART TWO CONCEPTS FROM PROBABILITY THEORY 4. Elementary Probability Theory and Limit Theorems A. Introduction B. Permutations and Combinations C. Definition of Probability D. Stochastic Variables and Probability D.I. Distribution Functions D.2. Moments D.3. Characteristic Functions D.4. Jointly Distributed Stochastic Variables E. Binomial Distributions E.I. The Binomial Distribution E.2. The Gaussian (For Normal) Distribution E.3. The Poisson Distribution E.4. Binomial Random Walk F. A Central Limit Theorem and Law of Large Numbers F.I. A Central Limit Theorem F.2. The Law of Large Numbers 198 S4.A. Lattice Random Walk 199 S4.A.1. One-Dimensional Lattice 200 S4.A.2. Random Walk in Higher Dimension 203

5 x CONTENTS S4.B. Infinitely Divisible Distributions 207 S4.B.1. Gaussian Distribution 208 S4.B.2. Poisson Distribution 209 S4.B.3. Cauchy Distribution 209 S4.B.4. Levy Distribution 210 S4.C. The Central Limit Theorem 211 S4.C.1. Useful Inequalities 212 S4.C.2. Convergence to a Gaussian 213 S4.D. Weierstrass Random Walk 214 S4.D.1. Discrete One-Dimensional Random Walk 215 S4.D.2. Continuum Limit of One-Dimensional Discrete Random Walk 217 S4.D.3. Two-Dimensional Discrete Random Walk (Levy Flight) 218 S4.E. General Form of Infinitely Divisible Distributions 221 S4.E.1. Levy-Khintchine Formula 222 S4.E.2. Kolmogorov Formula 223 References 225 Problems Stochastic Dynamics and Brownian Motion A. Introduction B. General Theory C. Markov Chains C1. Spectral Properties C.2. Random Walk D. The Master Equation D.I. Derivation of the Master Equation D.2. Detailed Balance D.3. Mean First Passage Time E. Brownian Motion E.I.. Langevin Equation E.2. The Spectral Density (Power Spectrum) 254 S5.A. Time Periodic Markov Chain 258 S5.B. Master Equation for Birth-Death Processes 260 S5.B. 1. The Master Equation 260 S5.B.2. Linear Birth-Death Processes 261 S5.B.3. Nonlinear Birth-Death Processes 265

6 CONTENTS xi S5.C. The Fokker-Planck Equation 266 S5.C.1. Probability Flow in Phase Space 266 S5.C.2. Probability Flow for Brownian Particle 267 S5.C.3. The Strong Friction Limit 270 S5.C.4. Solution of Fokker-Planck Equations with One Variable 271 S5.D. Approximations to the Master Equation 276 References 278 Problems The Foundations of Statistical Mechanics A. Introduction B. The Liouville Equation of Motion C. Ergodic Theory and the Foundation of Statistical Mechanics D. The Quantum Probability Density Operator 303 S6.A. Reduced Probability Densities and the BBGKY Hierarchy 310 S6.B. Reduced Density Matrices and the Wigner Distribution 314 S6.C. Microscopic Balance Equations 319 S6.D. Mixing Flow 321 S6.E. Anharmonic Oscillator Systems 326 S6.F. Newtonian Dynamics and Irreversibility 334 References 335 Problems 336 PART THREE EQUILIBRIUM STATISTICAL MECHANICS 7. Equilibrium Statistical Mechanics A. Introduction B. The Microcanonical Ensemble C. Einstein Fluctuation Theory C.I. General Discussion C.2. Fluid Systems D. The Canonical Ensemble D.I. Probability Density Operator D.2. Systems of Indistinguishable Particles D.3. Systems of Distinguishable Particles E. Heat Capacity of a Debye Solid 364

7 xii CONTENTS 7.F. Order-Disorder Transitions F.I. Exact Solution for a One-Dimensional Lattice F.2. Mean Field Theory for a d-dimensional Lattice G. The Grand Canonical Ensemble H. Ideal Quantum Gases H.I. Bose-Einstein Gases H.2. Fermi-Dirac Ideal Gases 392 S7.A. Heat Capacity of Lattice Vibrations on a One- Dimensional Lattice Exact Solution 401 S7.A.1. Exact Expression Large N 404 S7.A.2. Continuum Approximation Large N 406 S7.B. Momentum Condensation in an Interacting Fermi Fluid 407 S7.C. The Yang-Lee Theory of Phase Transitions 418 References 422 Problems Order-Disorder Transitions and Renormalization Theory A. Introduction B. Static Correlation Functions and Response Functions B.I. General Relations ~ B.2. Application to the Ising Lattice C. Scaling C.I. Homogeneous Functions C.2. Widom Scaling C.3. Kadanoff Scaling D. Microscopic Calculation of Critical Exponents 440 S8.A. Critical Exponents for the 5"* Model 448 S8.B. Exact Solution of the Two-Dimensional Ising Model 462 S8.B.1. Partition Function 462 S8.B.2. Antisymmetric Matrices and Dimer Graphs 466 S8.B.3. Closed Graphs and Mixed Dimer Graphs 469 S8.B.4. Partition Function for Infinite Planar Lattice 475 References 485 Problems Interacting Fluids A. Introduction B. Thermodynamics and the Radial Distribution Function 489

8 CONTENTS xiii 9.C. Virial Expansion of the Equation of State C1. Virial Expansions and Cluster Functions C.2. The Second Virial Coefficient C.3. Higher-Order Virial Coefficients 506 S9.A. The Pressure and Compressibility Equations 507 S9.A.1. The Pressure Equation 508 S9.A.2. The Compressibility Equation 509 S9.B. Ornstein-Zernicke Equation 510 S9.C. Third Virial Coefficient 513 S9.C.1. Square-Well Potential 514 S9.C.2. Lennard-Jones 6-12 Potential 515 S9.D. Virial Coefficients for Quantum Gases 517 References 526 Problems 527 PART FOUR NONEQUILD3RIUM STATISTICAL MECHANICS 10. Hydrodynamic Processes Near Equilibrium A. Introduction B. Navier-Stokes Hydrodynamic Equations B.1. Balance Equations B.2. Entropy Source and Entropy Current B.3. Transport Coefficients C. Linearized Hydrodynamic Equations C.1. Linearization of the Hydrodynamic Equations C.2. Transverse Hydrodynamic Modes C.3. Longitudinal Hydrodynamic Modes D. Dynamic Equilibrium Fluctuations and Transport Processes D.1. Onsager's Relations D.2. Weiner-Khintchine Theorem E. Linear Response Theory and the Fluctuation-Dissipation Theorem E.1. The Response Matrix E.2. Causality E.3. The Fluctuation-Dissipation Theorem E.4. Power Absorption F. Transport Properties of Mixtures O.F.I. Entropy Production in Multicomponent Systems 574

9 xiv CONTENTS 10.F.2. Fick's Law for Diffusion F.3. Thermal Diffusion F.4. Electrical Conductivity and Diffusion in Fluids 583 S10.A. Onsager's Relations When a Magnetic Field is Present 586 S10.B. Microscopic Linear Response Theory 589 S10.C. Light Scattering 592 S10.C.1. Scattered Electric Field 594 S10.C.2. Intensity of Scattered Light 597 S10.D. Thermoelectricity 600 S10.D.1. The Peltier Effect 601 S10.D.2. The Seebeck Effect 603 S10.D.3. Thomson Heat 605 S10.E. Entropy Production in Discontinuous Systems 605 SI O.E.I. Volume Flow Across a Membrane 606 S10.E.2. Ion Transport Across a Membrane 610 S10.F. Stochastic Hydrodynamics 612 SI O.F.I. Stochastic Hydrodynamic Equations 613 S10.F.2. Properties of Equilibrium Correlation Functions 614 S10.F.3. Random Current Correlation Functions 617 S10.G. Long-Time Tails 620 S10.G.1. Fluid Flow Around the Brownian Particle 621 S10.G.2. Drag Force on the Brownian Particle 623 S10.G.3. Velocity Autocorrelation Function 624 S10.H. Superfluid Hydrodynamics 631 S10.H.1. Superfluid Hydrodynamic Equations 631 S10.H.2. Sound Modes 635 SI0.1. General Definition of Hydrodynamic Modes Projection Operators Conserved Quantities 642 S Hydrodynamic Modes Due to Broken Symmetry 644 References 649 Problems Transport Theory A. Introduction 656 ll.b. Elementary Transport Theory B.I. The Maxwell-Boltzmann Distribution B.2. The Mean Free Path 658

10 CONTENTS xv 11.B.3. The Collision Frequency 659 ll.b.4. Self-Diffusion B.5. The Coefficients of Viscosity and Thermal Conductivity 664 ll.b.6. The Rate of Reaction 666 ll.c. The Boltzmann Equation C.I. Two-Body Scattering C.2. Derivation of the Boltzmann Equation C.3. Boltzmann's H Theorem 680 ll.d. Linearized Boltzmann and Lorentz-Boltzmann Equations 682 ll.d.l. Kinetic Equations for a Two-Component Gas D.2. Collision Operators 684 ll.e. Coefficient of Self-Diffusion E.I. Derivation of the Diffusion Equation E.2. Eigenfrequencies of the Lorentz- Boltzmann Equation 690 ll.f. Coefficients of Viscosity and Thermal Conductivity F. 1. Derivation of the Hydrodynamic Equations F.2. Eigenfrequencies of the Boltzmann Equation F.3. Shear Viscosity and Thermal Conductivity 700 ll.g. Computation of Transport Coefficients G.I. Sonine Polynomials G.2. Diffusion Coefficient G.3. Thermal Conductivity G.4. Shear Viscosity 708 Sll.A. Beyond the Boltzmann Equation 710 References 717 Problems Nonequilibrium Phase Transitions A. Introduction B. Nonequilibrium Stability Criteria B.1. Stability Conditions Near Equilibrium B.2. Stability Conditions Far From Equilibrium C. The Schlogl Model D. The Brusselator D.1. The Brusselator A Nonlinear Chemical Model D.2. Boundary Conditions 737

11 xvi CONTENTS 12.D.3. Linear Stability Analysis E. The Rayleigh-Benard Instability E.1. Hydrodynamic Equations and Boundary Conditions E.2. Linear Stability Analysis 747 SI2.A. Fluctuations Near a Nonequilibrium Phase Transition 753 S12.A.1. Fluctuations in the Rayleigh-Benard System 753 S12.A.2. Fluctuations in the Brusselator 760 S12.A.3. The Time-Dependent Ginzburg-Landau Equation 764 References 765 Problems 767 APPENDICES A. Balance Equations 768 A.I. General Fluid Flow 768 A.2. General Balance Equation 771 References 773 B. Systems of Identical Particles 774 B.I. Position and Momentum Eigenstates 774 B.I.I. Free Particle 775 B.I.2. Particle in a Box 776 B.2. Symmetrized A?-Particle Position and Momentum Eigenstates 777 B.2.1. Symmetrized Momentum Eigenstates for Bose-Einstein Particles 778 B.2.2. Antisymmetrized Momentum Eigenstates for Fermi-Dirac Particles B.2.3. Partition Functions and Expectation Values B.3. The Number Representation B.3.1. The Number Representation for Bosons B.3.2. The Number Representation for Fermions B.3.3. Field Operators References C. Stability of Solutions to Nonlinear Equations C.I. Linear Stability Theory

12 CONTENTS xvii C.2. Limit Cycles 795 C.3. Liapounov Functions and Global Stability 796 References 798 Author Index 799 Subject Index 804