Statistical and Thermal Physics

Preface xi Chapter 1: From Microscopic to Macroscopic Behavior 1 1.1 Introduction 1 1.2 Some Qualitative Observations 2 1.3 Doing Work and the Quality of Energy 4 1.4 Some Simple Simulations 5 1.5 Measuring the Pressure and Temperature 15 1.6 Work, Heating, and the First Law of Thermodynamics 19 1.7 *The Fundamental Need for a Statistical Approach 20 1.8 *Time and Ensemble Averages 22 1.9 Models of Matter 22 1.9.1 The ideal gas 23 1.9.2 Interparticle potentials 23 1.9.3 Lattice models 23 1.10 Importance of Simulations 24 1.11 Dimensionless Quantities 24 1.12 Summary 25 1.13 Supplementary Notes 27 1.13.1 Approach to equilibrium 27 1.13.2 Mathematics refresher 28 Vocabulary 28 Additional Problems 29 Suggestions for Further Reading 30 Chapter 2: Thermodynamic Concepts and Processes 32 2.1 Introduction 32 2.2 The System 33 2.3 Thermodynamic Equilibrium 34 2.4 Temperature 35 2.5 Pressure Equation of State 38 2.6 Some Thermodynamic Processes 39 2.7 Work 40 2.8 The First Law of Thermodynamics 44 2.9 Energy Equation of State 47 2.10 Heat Capacities and Enthalpy 48 2.11 Quasistatic Adiabatic Processes 51 2.12 The Second Law of Thermodynamics 55 2.13 The Thermodynamic Temperature 58 2.14 The Second Law and Heat Engines 60 2.15 Entropy Changes 67 2.16 Equivalence of Thermodynamic and Ideal Gas Scale Temperatures 74 2.17 The Thermodynamic Pressure 75 2.18 The Fundamental Thermodynamic Relation 76 2.19 The Entropy of an Ideal Classical Gas 77 2.20 The Third Law of Thermodynamics 78 2.21 Free Energies 79 2.22 Thermodynamic Derivatives 84 2.23 *Applications to Irreversible Processes 90 2.23.1 Joule or free expansion process 90 2.23.2 Joule-Thomson process 91 2.24 Supplementary Notes 94 2.24.1 The mathematics of thermodynamics 94 2.24.2 Thermodynamic potentials and Legendre transforms 97 Vocabulary 99 Additional Problems 100 Suggestions for Further Reading 108 Chapter 3: Concepts of Probability 111 3.1 Probability in Everyday Life 111 3.2 The Rules of Probability 114 3.3 Mean Values 119 3.4 The Meaning of Probability 121 3.4.1 Information and uncertainty 124 3.4.2 *Bayesian inference 128 3.5 Bernoulli Processes and the Binomial Distribution 134 3.6 Continuous Probability Distributions 147 3.7 The Central Limit Theorem (or Why Thermodynamics Is Possible) 151 3.8 *The Poisson Distribution or Should You Fly? 155 3.9 *Traffic Flow and the Exponential Distribution 156 3.10 *Are All Probability Distributions Gaussian? 159 3.11 Supplementary Notes 161 3.11.1 Method of undetermined multipliers 161 3.11.2 Derivation of the central limit theorem 163 Vocabulary 167 Additional Problems 168 Suggestions for Further Reading 177 Chapter 4: The Methodology of Statistical Mechanics 180 4.1 Introduction 180 4.2 A Simple Example of a Thermal Interaction 182 4.3 Counting Microstates 192 4.3.1 Noninteracting spins 192 4.3.2 A particle in a one-dimensional box 193 4.3.3 One-dimensional harmonic oscillator 196 4.3.4 One particle in a two-dimensional box 197 4.3.5 One particle in a three-dimensional box 198 4.3.6 Two noninteracting identical particles and the semiclassical limit 199 4.4 The Number of States of Many Noninteracting Particles: Semiclassical Limit 201 4.5 The Microcanonical Ensemble (Fixed E, V, and N) 203 4.6 The Canonical Ensemble (Fixed T, V, and N) 209 4.7 Connection between Thermodynamics and Statistical Mechanics in the Canonical Ensemble 216 4.8 Simple Applications of the Canonical Ensemble 218 4.9 An Ideal Thermometer 222 4.10 Simulation of the Microcanonical Ensemble 225 4.11 Simulation of the Canonical Ensemble 226 4.12 Grand Canonical Ensemble (Fixed T, V, and ?) 227 4.13 *Entropy Is Not a Measure of Disorder 229 4.14 Supplementary Notes 231 4.14.1 The volume of a hypersphere 231 4.14.2 Fluctuations in the canonical ensemble 232 Vocabulary 233 Additional Problems 234 Suggestions for Further Reading 239 Chapter 5: Magnetic Systems 241 5.1 Paramagnetism 241 5.2 Noninteracting Magnetic Moments 242 5.3 Thermodynamics of Magnetism 246 5.4 The Ising Model 248 5.5 The Ising Chain 249 5.5.1 Exact enumeration 250 5.5.2 Spin-spin correlation function 253 5.5.3 Simulations of the Ising chain 256 5.5.4 *Transfer matrix 257 5.5.5 Absence of a phase transition in one dimension 260 5.6 The Two-Dimensional Ising Model 261 5.6.1 Onsager solution 262 5.6.2 Computer simulation of the two-dimensional Ising model 267 5.7 Mean-Field Theory 270 5.7.1 *Phase diagram of the Ising model 276 5.8 *Simulation of the Density of States 279 5.9 *Lattice Gas 282 5.10 Supplementary Notes 286 5.10.1 The Heisenberg model of magnetism 286 5.10.2 Low temperature expansion 288 5.10.3 High temperature expansion 290 5.10.4 Bethe approximation 292 5.10.5 Fully connected Ising model 295 5.10.6 Metastability and nucleation 297 Vocabulary 300 Additional Problems 300 Suggestions for Further Reading 306 Chapter 6: Many-Particle Systems 308 6.1 The Ideal Gas in the Semiclassical Limit 308 6.2 Classical Statistical Mechanics 318 6.2.1 The equipartition theorem 318 6.2.2 The Maxwell velocity distribution 321 6.2.3 The Maxwell speed distribution 323 6.3 Occupation Numbers and Bose and Fermi Statistics 325 6.4 Distribution Functions of Ideal Bose and Fermi Gases 327 6.5 Single Particle Density of States 329 6.5.1 Photons 331 6.5.2 Nonrelativistic particles 332 6.6 The Equation of State of an Ideal Classical Gas: Application of the Grand Canonical Ensemble 334 6.7 Blackbody Radiation 337 6.8 The Ideal Fermi Gas 341 6.8.1 Ground state properties 342 6.8.2 Low temperature properties 345 6.9 The Heat Capacity of a Crystalline Solid 351 6.9.1 The Einstein model 351 6.9.2 Debye theory 352 6.10 The Ideal Bose Gas and Bose Condensation 354 6.11 Supplementary Notes 360 6.11.1 Fluctuations in the number of particles 360 6.11.2 Low temperature expansion of an ideal Fermi gas 363 Vocabulary 365 Additional Problems 366 Suggestions for Further Reading 374 Chapter 7: The Chemical Potential and Phase Equilibria 376 7.1 Meaning of the Chemical Potential 376 7.2 Measuring the Chemical Potential in Simulations 380 7.2.1 The Widom insertion method 380 7.2.2 The chemical demon algorithm 382 7.3 Phase Equilibria 385 7.3.1 Equilibrium conditions 386 7.3.2 Simple phase diagrams 387 7.3.3 Clausius-Clapeyron equation 389 7.4 The van der Waals Equation of State 393 7.4.1 Maxwell construction 393 7.4.2 *The van der Waals critical point 400 7.5 *Chemical Reactions 403 Vocabulary 407 Additional Problems 407 Suggestions for Further Reading 408 Chapter 8: Classical Gases and Liquids 410 8.1 Introduction 410 8.2 Density Expansion 410 8.3 The Second Virial Coefficient 414 8.4 *Diagrammatic Expansions 419 8.4.1 Cumulants 420 8.4.2 High temperature expansion 421 8.4.3 Density expansion 426 8.4.4 Higher order virial coefficients for hard spheres 428 8.5 The Radial Distribution Function 430 8.6 Perturbation Theory of Liquids 437 8.6.1 The van der Waals equation 439 8.7 *The Ornstein-Zernike Equation and Integral Equations for g(r ) 441 8.8 *One-Component Plasma 445 8.9 Supplementary Notes 449 8.9.1 The third virial coefficient for hard spheres 449 8.9.2 Definition of g(r ) in terms of the local particle density 450 8.9.3 X-ray scattering and the static structure function 451 Vocabulary 455 Additional Problems 456 Suggestions for Further Reading 458 Chapter 9: Critical Phenomena: Landau Theory and the Renormalization Group Method 459 9.1 Landau Theory of Phase Transitions 459 9.2 Universality and Scaling Relations 467 9.3 A Geometrical Phase Transition 469 9.4 Renormalization Group Method for Percolation 475 9.5 The Renormalization Group Method and the One-Dimensional Ising Model 479 9.6 ?The Renormalization Group Method and the Two-Dimensional Ising Model 484 Vocabulary 490 Additional Problems 491 Suggestions for Further Reading 492 Appendix: Physical Constants and Mathematical Relations 495 A.1 Physical Constants and Conversion Factors 495 A.2 Hyperbolic Functions 496 A.3 Approximations 496 A.4 Euler-Maclaurin Formula 497 A.5 Gaussian Integrals 497 A.6 Stirling's Approximation 498 A.7 Bernoulli Numbers 500 A.8 Probability Distributions 500 A.9 Fourier Transforms 500 A.10 The Delta Function 501 A.11 Convolution Integrals 502 A.12 Fermi and Bose Integrals 503 Index 505