Statistical and Thermal Physics

Preface xiChapter 1: From Microscopic to Macroscopic Behavior 11.1 Introduction 11.2 Some Qualitative Observations 21.3 Doing Work and the Quality of Energy 41.4 Some Simple Simulations 51.5 Measuring the Pressure and Temperature 151.6 Work, Heating, and the First Law of Thermodynamics 191.7 *The Fundamental Need for a Statistical Approach 201.8 *Time and Ensemble Averages 221.9 Models of Matter 221.9.1 The ideal gas 231.9.2 Interparticle potentials 231.9.3 Lattice models 231.10 Importance of Simulations 241.11 Dimensionless Quantities 241.12 Summary 251.13 Supplementary Notes 271.13.1 Approach to equilibrium 271.13.2 Mathematics refresher 28Vocabulary 28Additional Problems 29Suggestions for Further Reading 30Chapter 2: Thermodynamic Concepts and Processes 322.1 Introduction 322.2 The System 332.3 Thermodynamic Equilibrium 342.4 Temperature 352.5 Pressure Equation of State 382.6 Some Thermodynamic Processes 392.7 Work 402.8 The First Law of Thermodynamics 442.9 Energy Equation of State 472.10 Heat Capacities and Enthalpy 482.11 Quasistatic Adiabatic Processes 512.12 The Second Law of Thermodynamics 552.13 The Thermodynamic Temperature 582.14 The Second Law and Heat Engines 602.15 Entropy Changes 672.16 Equivalence of Thermodynamic and Ideal Gas Scale Temperatures 742.17 The Thermodynamic Pressure 752.18 The Fundamental Thermodynamic Relation 762.19 The Entropy of an Ideal Classical Gas 772.20 The Third Law of Thermodynamics 782.21 Free Energies 792.22 Thermodynamic Derivatives 842.23 *Applications to Irreversible Processes 902.23.1 Joule or free expansion process 902.23.2 Joule-Thomson process 912.24 Supplementary Notes 942.24.1 The mathematics of thermodynamics 942.24.2 Thermodynamic potentials and Legendre transforms 97Vocabulary 99Additional Problems 100Suggestions for Further Reading 108Chapter 3: Concepts of Probability 1113.1 Probability in Everyday Life 1113.2 The Rules of Probability 1143.3 Mean Values 1193.4 The Meaning of Probability 1213.4.1 Information and uncertainty 1243.4.2 *Bayesian inference 1283.5 Bernoulli Processes and the Binomial Distribution 1343.6 Continuous Probability Distributions 1473.7 The Central Limit Theorem (or Why ThermodynamicsIs Possible) 1513.8 *The Poisson Distribution or Should You Fly? 1553.9 *Traffic Flow and the Exponential Distribution 1563.10 *Are All Probability Distributions Gaussian? 1593.11 Supplementary Notes 1613.11.1 Method of undetermined multipliers 1613.11.2 Derivation of the central limit theorem 163Vocabulary 167Additional Problems 168Suggestions for Further Reading 177Chapter 4: The Methodology of Statistical Mechanics 1804.1 Introduction 1804.2 A Simple Example of a Thermal Interaction 1824.3 Counting Microstates 1924.3.1 Noninteracting spins 1924.3.2 A particle in a one-dimensional box 1934.3.3 One-dimensional harmonic oscillator 1964.3.4 One particle in a two-dimensional box 1974.3.5 One particle in a three-dimensional box 1984.3.6 Two noninteracting identical particles and thesemiclassical limit 1994.4 The Number of States of Many Noninteracting Particles:Semiclassical Limit 2014.5 The Microcanonical Ensemble (Fixed E, V, and N) 2034.6 The Canonical Ensemble (Fixed T, V, and N) 2094.7 Connection between Thermodynamics and Statistical Mechanicsin the Canonical Ensemble 2164.8 Simple Applications of the Canonical Ensemble 2184.9 An Ideal Thermometer 2224.10 Simulation of the Microcanonical Ensemble 2254.11 Simulation of the Canonical Ensemble 2264.12 Grand Canonical Ensemble (Fixed T, V, and ?) 2274.13 *Entropy Is Not a Measure of Disorder 2294.14 Supplementary Notes 2314.14.1 The volume of a hypersphere 2314.14.2 Fluctuations in the canonical ensemble 232Vocabulary 233Additional Problems 234Suggestions for Further Reading 239Chapter 5: Magnetic Systems 2415.1 Paramagnetism 2415.2 Noninteracting Magnetic Moments 2425.3 Thermodynamics of Magnetism 2465.4 The Ising Model 2485.5 The Ising Chain 2495.5.1 Exact enumeration 2505.5.2 Spin-spin correlation function 2535.5.3 Simulations of the Ising chain 2565.5.4 *Transfer matrix 2575.5.5 Absence of a phase transition in one dimension 2605.6 The Two-Dimensional Ising Model 2615.6.1 Onsager solution 2625.6.2 Computer simulation of the two-dimensional Ising model 2675.7 Mean-Field Theory 2705.7.1 *Phase diagram of the Ising model 2765.8 *Simulation of the Density of States 2795.9 *Lattice Gas 2825.10 Supplementary Notes 2865.10.1 The Heisenberg model of magnetism 2865.10.2 Low temperature expansion 2885.10.3 High temperature expansion 2905.10.4 Bethe approximation 2925.10.5 Fully connected Ising model 2955.10.6 Metastability and nucleation 297Vocabulary 300Additional Problems 300Suggestions for Further Reading 306Chapter 6: Many-Particle Systems 3086.1 The Ideal Gas in the Semiclassical Limit 3086.2 Classical Statistical Mechanics 3186.2.1 The equipartition theorem 3186.2.2 The Maxwell velocity distribution 3216.2.3 The Maxwell speed distribution 3236.3 Occupation Numbers and Bose and Fermi Statistics 3256.4 Distribution Functions of Ideal Bose and Fermi Gases 3276.5 Single Particle Density of States 3296.5.1 Photons 3316.5.2 Nonrelativistic particles 3326.6 The Equation of State of an Ideal Classical Gas: Applicationof the Grand Canonical Ensemble 3346.7 Blackbody Radiation 3376.8 The Ideal Fermi Gas 3416.8.1 Ground state properties 3426.8.2 Low temperature properties 3456.9 The Heat Capacity of a Crystalline Solid 3516.9.1 The Einstein model 3516.9.2 Debye theory 3526.10 The Ideal Bose Gas and Bose Condensation 3546.11 Supplementary Notes 3606.11.1 Fluctuations in the number of particles 3606.11.2 Low temperature expansion of an ideal Fermi gas 363Vocabulary 365Additional Problems 366Suggestions for Further Reading 374Chapter 7: The Chemical Potential and Phase Equilibria 3767.1 Meaning of the Chemical Potential 3767.2 Measuring the Chemical Potential in Simulations 3807.2.1 The Widom insertion method 3807.2.2 The chemical demon algorithm 3827.3 Phase Equilibria 3857.3.1 Equilibrium conditions 3867.3.2 Simple phase diagrams 3877.3.3 Clausius-Clapeyron equation 3897.4 The van der Waals Equation of State 3937.4.1 Maxwell construction 3937.4.2 *The van der Waals critical point 4007.5 *Chemical Reactions 403Vocabulary 407Additional Problems 407Suggestions for Further Reading 408Chapter 8: Classical Gases and Liquids 4108.1 Introduction 4108.2 Density Expansion 4108.3 The Second Virial Coefficient 4148.4 *Diagrammatic Expansions 4198.4.1 Cumulants 4208.4.2 High temperature expansion 4218.4.3 Density expansion 4268.4.4 Higher order virial coefficients for hard spheres 4288.5 The Radial Distribution Function 4308.6 Perturbation Theory of Liquids 4378.6.1 The van der Waals equation 4398.7 *The Ornstein-Zernike Equation and Integral Equations for g(r ) 4418.8 *One-Component Plasma 4458.9 Supplementary Notes 4498.9.1 The third virial coefficient for hard spheres 4498.9.2 Definition of g(r ) in terms of the local particle density 4508.9.3 X-ray scattering and the static structure function 451Vocabulary 455Additional Problems 456Suggestions for Further Reading 458Chapter 9: Critical Phenomena: Landau Theory and the Renormalization Group Method 4599.1 Landau Theory of Phase Transitions 4599.2 Universality and Scaling Relations 4679.3 A Geometrical Phase Transition 4699.4 Renormalization Group Method for Percolation 4759.5 The Renormalization Group Method and the One-Dimensional Ising Model 4799.6 ?The Renormalization Group Method and the Two-Dimensional Ising Model 484Vocabulary 490Additional Problems 491Suggestions for Further Reading 492Appendix: Physical Constants and Mathematical Relations 495A.1 Physical Constants and Conversion Factors 495A.2 Hyperbolic Functions 496A.3 Approximations 496A.4 Euler-Maclaurin Formula 497A.5 Gaussian Integrals 497A.6 Stirling's Approximation 498A.7 Bernoulli Numbers 500A.8 Probability Distributions 500A.9 Fourier Transforms 500A.10 The Delta Function 501A.11 Convolution Integrals 502A.12 Fermi and Bose Integrals 503Index 505