3 d rB rB 3 d rA rA Statistical Mechanics Daniel F. Styer November 2019 Statistical Mechanics Dan Styer Schiffer Professor of Physics Oberlin College Oberlin, Ohio 44074-1088 [email protected] http://www.oberlin.edu/physics/dstyer c 6 November 2019 Daniel F. Styer The copyright holder grants the freedom to copy, modify, convey, adapt, and/or redistribute this work under the terms of the Creative Commons Attribution Share Alike 4.0 International License. A copy of that license is available at https://creativecommons.org/licenses/by-sa/4.0/legalcode. Although, as a matter of history, statistical mechanics owes its origin to investigations in thermodynamics, it seems eminently worthy of an independent development, both on account of the elegance and simplicity of its principles, and because it yields new results and places old truths in a new light. | J. Willard Gibbs Elementary Principles in Statistical Mechanics Contents 0 Preface 1 1 The Properties of Matter in Bulk 4 1.1 What is Statistical Mechanics About? . 4 1.2 Outline of Book . 5 1.3 Fluid Statics . 6 1.4 Phase Diagrams . 7 1.5 Additional Problems . 8 2 Principles of Statistical Mechanics 11 2.1 Microscopic Description of a Classical System . 11 2.2 Macroscopic Description of a Large Equilibrium System . 14 2.3 Fundamental Assumption . 16 2.4 Statistical Definition of Entropy . 18 2.5 Entropy of a Monatomic Ideal Gas . 20 2.6 Qualitative Features of Entropy . 26 2.7 Using Entropy to Find (Define) Temperature and Pressure . 37 2.8 Additional Problems . 47 3 Thermodynamics 49 3.1 Heat and Work . 49 3.2 Entropy . 51 i ii CONTENTS 3.3 Heat Engines . 55 3.4 Multivariate Calculus . 57 3.5 Thermodynamic Quantities . 64 3.6 The Thermodynamic Dance . 67 3.7 Non-fluid Systems . 74 3.8 Thermodynamics Applied to Fluids . 75 3.9 Thermodynamics Applied to Phase Transitions . 83 3.10 Thermodynamics Applied to Chemical Reactions . 86 3.11 Thermodynamics Applied to Light . 90 3.12 Additional Problems . 95 4 Ensembles 99 4.1 The Canonical Ensemble . 99 4.2 Meaning of the Term \Ensemble" . 101 4.3 Classical Monatomic Ideal Gas . 101 4.4 Energy Dispersion in the Canonical Ensemble . 104 4.5 Temperature as a Control Variable for Energy (Canonical Ensemble) . 107 4.6 The Equivalence of Canonical and Microcanonical Ensembles . 108 4.7 The Grand Canonical Ensemble . 108 4.8 The Grand Canonical Ensemble in the Thermodynamic Limit . 109 4.9 Summary of Major Ensembles . 111 4.10 Quantal Statistical Mechanics . 112 4.11 Ensemble Problems I . 114 4.12 Ensemble Problems II . 121 5 Classical Ideal Gases 123 5.1 Classical Monatomic Ideal Gases . 123 5.2 Classical Diatomic Ideal Gases . 126 5.3 Heat Capacity of an Ideal Gas . 126 5.4 Specific Heat of a Hetero-nuclear Diatomic Ideal Gas . 131 5.5 Chemical Reactions Between Gases . 134 5.6 Problems . 134 CONTENTS iii 6 Quantal Ideal Gases 138 6.1 Introduction . 138 6.2 The Interchange Rule . 138 6.3 Quantum Mechanics of Independent Identical Particles . 139 6.4 Statistical Mechanics of Independent Identical Particles . 146 6.5 Quantum Mechanics of Free Particles . 151 6.6 Fermi-Dirac Statistics . 152 6.7 Bose-Einstein Statistics . 154 6.8 Specific Heat of the Ideal Fermion Gas . 159 6.9 Additional Problems . 167 7 Harmonic Lattice Vibrations 170 7.1 The Problem . 170 7.2 Statistical Mechanics of the Problem . 170 7.3 Normal Modes for a One-dimensional Chain . 171 7.4 Normal Modes in Three Dimensions . 171 7.5 Low-temperature Heat Capacity . 171 7.6 More Realistic Models . 173 7.7 What is a Phonon? . 173 7.8 Additional Problems . 173 8 Interacting Classical Fluids 175 8.1 Introduction . 175 8.2 Perturbation Theory . 177 8.3 Variational Methods . 181 8.4 Distribution Functions . 182 8.5 Correlations and Scattering . 185 8.6 The Hard Sphere Fluid . 185 iv CONTENTS 9 Strongly Interacting Systems and Phase Transitions 188 9.1 Introduction to Magnetic Systems and Models . 188 9.2 Free Energy of the One-Dimensional Ising Model . 188 9.3 The Mean-Field Approximation . 193 9.4 Correlation Functions in the Ising Model . 193 9.5 Computer Simulation . 199 9.6 Additional Problems . 208 A Series and Integrals 214 B Evaluating the Gaussian Integral 215 C Clinic on the Gamma Function 216 D Volume of a Sphere in d Dimensions 218 E Stirling's Approximation 221 F The Euler-MacLaurin Formula and Asymptotic Series 223 G Ramblings on the Riemann Zeta Function 224 H Tutorial on Matrix Diagonalization 230 H.1 What's in a name? . 230 H.2 Vectors in two dimensions . 231 H.3 Tensors in two dimensions . 233 H.4 Tensors in three dimensions . 236 H.5 Tensors in d dimensions . 237 H.6 Linear transformations in two dimensions . 238 H.7 What does \eigen" mean? . 239 H.8 How to diagonalize a symmetric matrix . 240 H.9 A glance at computer algorithms . 246 H.10 A glance at non-symmetric matrices and the Jordan form . ..