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Thermal and Statistical Physics

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Thermal and Statistical Physics Contents 1 From Microscopic to Macroscopic Behavior 1 1.1 Introduction . 1 1.2 Some qualitative observations . 3 1.3 Doing work . 4 1.4 Quality of energy . 5 1.5 Some simple simulations . 6 1.6 Work, heating, and the first law of thermodynamics . 14 1.7 Measuring the pressure and temperature . 15 1.8 *The fundamental need for a statistical approach . 18 1.9 *Time and ensemble averages . 20 1.10 *Models of matter . 20 1.10.1 The ideal gas . 21 1.10.2 Interparticle potentials . 21 1.10.3 Lattice models . 21 1.11 Importance of simulations . 22 1.12 Summary . 23 Additional Problems . 24 Suggestions for Further Reading . 25 2 Thermodynamic Concepts 26 2.1 Introduction . 26 2.2 The system . 27 2.3 Thermodynamic Equilibrium . 27 2.4 Temperature . 28 2.5 Pressure Equation of State . 31 2.6 Some Thermodynamic Processes . 33 2.7 Work . 34 i CONTENTS ii 2.8 The First Law of Thermodynamics . 37 2.9 Energy Equation of State . 39 2.10 Heat Capacities and Enthalpy . 40 2.11 Adiabatic Processes . 43 2.12 The Second Law of Thermodynamics . 47 2.13 The Thermodynamic Temperature . 49 2.14 The Second Law and Heat Engines . 51 2.15 Entropy Changes . 56 2.16 Equivalence of Thermodynamic and Ideal Gas Scale Temperatures . 60 2.17 The Thermodynamic Pressure . 61 2.18 The Fundamental Thermodynamic Relation . 62 2.19 The Entropy of an Ideal Gas . 63 2.20 The Third Law of Thermodynamics . 64 2.21 Free Energies . 65 Appendix 2B: Mathematics of Thermodynamics . 70 Additional Problems . 73 Suggestions for Further Reading . 80 3 Concepts of Probability 82 3.1 Probability in everyday life . 82 3.2 The rules of probability . 84 3.3 Mean values . 89 3.4 The meaning of probability . 91 3.4.1 Information and uncertainty . 93 3.4.2 *Bayesian inference . 97 3.5 Bernoulli processes and the binomial distribution . 99 3.6 Continuous probability distributions . 109 3.7 The Gaussian distribution as a limit of the binomial distribution . 111 3.8 The central limit theorem or why is thermodynamics possible? . 113 3.9 The Poisson distribution and should you fly in airplanes? . 116 3.10 *Traffic flow and the exponential distribution . 117 3.11 *Are all probability distributions Gaussian? . 119 Additional Problems . 128 Suggestions for Further Reading . 136 CONTENTS iii 4 Statistical Mechanics 138 4.1 Introduction . 138 4.2 A simple example of a thermal interaction . 140 4.3 Counting microstates . 150 4.3.1 Noninteracting spins . 150 4.3.2 *One-dimensional Ising model . 150 4.3.3 A particle in a one-dimensional box . 151 4.3.4 One-dimensional harmonic oscillator . 153 4.3.5 One particle in a two-dimensional box . 154 4.3.6 One particle in a three-dimensional box . 156 4.3.7 Two noninteracting identical particles and the semiclassical limit . 156 4.4 The number of states of N noninteracting particles: Semiclassical limit . 158 4.5 The microcanonical ensemble (fixed E, V, and N) . 160 4.6 Systems in contact with a heat bath: The canonical ensemble (fixed T, V, and N) 165 4.7 Connection between statistical mechanics and thermodynamics . 170 4.8 Simple applications of the canonical ensemble . 172 4.9 A simple thermometer . 175 4.10 Simulations of the microcanonical ensemble . 177 4.11 Simulations of the canonical ensemble . 178 4.12 Grand canonical ensemble (fixed T, V, and µ) . 179 4.13 Entropy and disorder . 181 Appendix 4A: The Volume of a Hypersphere . 183 Appendix 4B: Fluctuations in the Canonical Ensemble . 184 Additional Problems . 185 Suggestions for Further Reading . 188 5 Magnetic Systems 190 5.1 Paramagnetism . 190 5.2 Thermodynamics of magnetism . 194 5.3 The Ising model . 194 5.4 The Ising Chain . 195 5.4.1 Exact enumeration . 195 5.4.2 ∗Spin-spin correlation function . 199 5.4.3 Simulations of the Ising chain . 201 5.4.4 *Transfer matrix . 203 5.4.5 Absence of a phase transition in one dimension . 205 5.5 The Two-Dimensional Ising Model . 206 CONTENTS iv 5.5.1 Onsager solution . 206 5.5.2 Computer simulation of the two-dimensional Ising model . 211 5.6 Mean-Field Theory . 211 5.7 *Infinite-range interactions . 216 Additional Problems . 224 Suggestions for Further Reading . 228 6 Noninteracting Particle Systems 230 6.1 Introduction . 230 6.2 The Classical Ideal Gas . 230 6.3 Classical Systems and the Equipartition Theorem . 238 6.4 Maxwell Velocity Distribution . 240 6.5 Occupation Numbers and Bose and Fermi Statistics . 243 6.6 Distribution Functions of Ideal Bose and Fermi Gases . 245 6.7 Single Particle Density of States . 247 6.7.1 Photons . 249 6.7.2 Electrons . ..
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