Lectures on Kinetic Theory of Gases and Statistical Physics

1 1. Lectures on Kinetic Theory of Gases and Statistical Physics (Oxford Physics Paper A1). Alexander A. Schekochihin . The Rudolf Peierls Centre for Theoretical Physics , University of Oxford, Oxford OX1 3NP, UK. Merton College, Oxford OX1 4JD, UK. (compiled on 30 November 2020). These are the notes for my Lectures on Kinetic Theory and Statistical Physics , being part of the 2nd -year course (Paper A1) at Oxford. I taught the course in 2011-18, jointly with Professors Andrew Boothroyd (2011-15) and Julien Devriendt (2015-18). Only my part of the course is covered in these notes.

2 I will be grateful for any feedback from students, tutors or sympathisers. CONTENTS. [Part I. Basic Thermodynamics] 5. Part II. Kinetic Theory 5. 1. Statistical Description of a Gas 5. Introduction 5. Energy 8. Thermodynamic Limit 9. Kinetic Calculation of Pressure 11. Isotropic Distributions 13. 2. Classical Ideal Gas in Equilibrium 15. Maxwell's Distribution 16. Equation of State and Temperature 19. Validity of the Classical Limit 21. Nonrelativistic Limit 21. No Quantum Correlations 21. 3. Effusion 21. 4. Collisions 25. Cross-section 25.

3 Collision Time 26. Mean Free Path 26. Relative Speed 26. 5. From Local to Global Equilibrium (Transport Equations) 28. Inhomogeneous Distributions 28. Local Maxwellian Equilibrium 29. Conservation Laws 30. Temperature 31. Velocity 33. Thermal Conductivity and Viscosity 35. E-mail: 2 A. A. Schekochihin Transport Equations 37. Relaxation to Global Equilibrium 37. Initial-Value Problem: Fourier Decomposition 38. Dimensional Estimate of Transport Coefficients 39. Separation of Scales 39. Sources, Sinks and Boundaries 39. Steady-State Solutions 40.

4 Time-Periodic Solutions 41. Diffusion 43. Derivation of the Diffusion Equation 43. Random-Walk Model 43. Diffusive Spreading 44. 6. Kinetic Calculation of Transport Coefficients 45. A Nice but Dodgy Derivation 45. Viscosity 45. Thermal Conductivity 47. Why This Derivation is Dodgy 47. Kinetic Expressions for Fluxes 48. Kinetic Equation 49. Conservation Laws and Fluid Equations 50. Number Density 50. Momentum Density 51. Energy Density 52. Collision Operator 55. Solution of the Kinetic Equation 56. Calculation of Fluxes 57. Momentum Flux 58.

5 Heat Flux 58. Calculation of Fluxes in 3D 60. Kinetic Theory of Brownian Particles 61. Langevin Equation 61. Diffusion in Velocity Space 62. Brownian Motion 63. Kinetic Equation for Brownian Particles 64. Diffusion in Position Space 65. Part III. Foundations of Statistical Mechanics 66. 7. From Microphysics to Macrophysics 66. What Are We Trying to Do? 66. The System and Its States 67. Pressure 67. 8. Principle of Maximum Entropy 68. Quantifying Ignorance 68. Complete Ignorance 69. Some Knowledge 69. Assignment of Likelihoods 69. Some properties of Gibbs Shannon Entropy 71.

6 Shannon's Theorem 72. Method of Lagrange Multipliers 74. Test of the Method: Isolated System 76. Oxford Physics Lectures : Kinetic Theory and Statistical Physics 3. 9. Canonical Ensemble 77. Gibbs Distribution 77. Construction of Thermodynamics 78. Some Mathematical Niceties 79. Third Law 80. Part I Obviated, Road Ahead Clear 81. 10. Thermodynamic Equilibria and Stability 84. Additivity of Entropy 84. Thermal Equilibrium 85. Physical Interpretation of the Canonical Ensemble 86. Mechanical and Dynamical Equilibria 87. Thermal Equilibrium 89.

7 Mechanical Equilibrium 89. Dynamical Equilibrium 89. Stability 90. Thermal Stability 90. Dynamical Stability 91. 11. Statistical Mechanics of Classical Monatomic Ideal Gas 92. Single-Particle States 93. Down the Garden Path.. 93. Single-Particle Partition Function 94. Digression: Density of States 94. Disaster Strikes 95. Gibbs Paradox 95. Distinguishability 96. Correct Partition Function 96. Thermodynamics of Classical Ideal Gas 98. Maxwell's Distribution 99. 12. Entropy, Ensembles and the Meaning of Probabilities 100. Boltzmann Entropy and the Ensembles 100.

8 Boltzmann's Formula 100. Microcanonical Ensemble 101. Alternative (Original) Construction of the Canonical Ensemble 104. Gibbs vs. Boltzmann and the Meaning of Probabilities 105. Whose Uncertainty? 107. Second Law 107. 13. Density Matrix and Entropy in Quantum Mechanics 109. Statistical and Quantum Uncertainty 109. Density Matrix 110. Quantum Entropy and Canonical Ensemble 111. Time Evolution and the Second Law 111. How Information Is Lost 112. [Part IV. Statistical Mechanics of Simple Systems] 113. Part V. Open Systems 114. 14. Grand Canonical Ensemble 114.

9 Grand Canonical Distribution 114. Thermodynamics of Open Systems and the Meaning of Chemical Potential116. Particle Equilibrium 118. 4 A. A. Schekochihin Grand Partition Function and Chemical Potential of Classical Ideal Gas 119. Equilibria of Inhomogeneous Systems 121. Chemical Potential and Thermodynamic Potentials 123. Free Energy 123. Gibbs Free Energy 124. Meaning of Grand Potential 124. 15. Multi-Species (Multi-Component) Systems 127. Generalisation of the Grand Canonical Formalism to Many Species 127. Gibbs Free Energy vs. s 128. Fractional Concentrations 128.

10 Particle Equilibrium and Gibbs Phase Rule 129. Chemical Equilibrium 129. Chemical Equilibrium in a Mixture of Classical Ideal Gases : Law of Mass Action 130. Part VI. Quantum Gases 132. 16. Quantum Ideal Gases 132. Fermions and Bosons 133. Partition Function 134. Occupation Number Statistics and Thermodynamics 134. Calculations in Continuum Limit 136. From Sums to Integrals 136. Chemical Potential of a Quantum Ideal Gas 137. Classical Limit 138. Mean Energy of a Quantum Ideal Gas 139. Grand Potential of a Quantum Ideal Gas 139. Equation of State of a Quantum Ideal Gas 140.