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Thermal and Statistical Physics – Lecture Notes

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Thermal and Statistical Physics – Lecture Notes PHYS20352 Thermal and statistical physics Tobias Galla May 7, 2014 Contents 1 Introduction 5 1.1 What this course is about ............................. 5 1.2 Why is this course interesting? ........................... 7 1.3 A few practical things ............................... 8 1.4 A few basic concepts ................................ 9 1.4.1 Isolated system ............................... 9 1.4.2 Equilibrium state .............................. 10 1.4.3 Thermodynamic variables ......................... 10 1.4.4 Equation of state .............................. 10 1.4.5 Quasi-static process ............................. 10 1.4.6 Indicator diagram .............................. 10 1.4.7 Reversible process .............................. 10 2 The first law of thermodynamics 12 2.1 A bit of history ................................... 12 2.2 The first law of thermodynamics .......................... 15 2.2.1 The first law ................................. 15 2.2.2 The first law for cycles ........................... 16 2.3 The form ofdW ¯ ................................... 16 2.3.1 Fluid systems (or gases) .......................... 16 2.3.2 Work done in changing the length of a wire ............... 17 2.3.3 Work done in changing the area of a surface film ............. 17 2.3.4 Work done on magnetic materials by changing the field ......... 18 2.3.5 Work done and the indicator diagram ................... 19 2.4 Specific heats .................................... 20 2.4.1 Example 1: y = V ............................. 20 2.4.2 Example 2: y = P ............................. 21 2.4.3 Special case: ideal gas ........................... 21 2.5 Real and Ideal gases: A Review .......................... 22 2.5.1 Ideal gas law ................................ 22 2.5.2 Heat capacity and further properties of ideal gases ............ 22 2.5.3 Real gases .................................. 23 2.6 The zeroth law of thermodynamics and temperature* .............. 24 2.6.1 The zeroth law ............................... 24 2.6.2 Thermometers and temperature scales .................. 25 1 3 The second law of thermodynamics 27 3.1 Introduction ..................................... 27 3.2 Heat engines and refrigerators ........................... 28 3.2.1 Heat engines ................................. 28 3.2.2 Refrigerators ................................ 30 3.3 The second law of thermodynamics ........................ 31 3.4 Carnot cycles and Carnot engines ......................... 33 3.4.1 Background ................................. 33 3.4.2 Carnot engines and Carnot’s theorem ................... 34 3.4.3 Calculating the efficiency of a Carnot cycle ................ 36 3.5 The thermodynamic temperature scale* ...................... 38 3.6 Entropy and maximum-entropy principle ..................... 41 3.6.1 Clausius’ Theorem ............................. 41 3.6.2 Entropy and the maximum-entropy principle ............... 44 3.6.3 Examples involving entropy changes .................... 46 3.7 The arrow of time .................................. 50 4 The mathematical structure of the theory of thermodynamics 53 4.1 Mathematical tools ................................. 53 4.1.1 Differentials ................................. 53 4.1.2 Legendre transformation .......................... 58 4.2 The postulates of thermodynamics* ........................ 60 4.3 The fundamental thermodynamic relation .................... 61 4.4 Thermodynamic Potentials ............................. 62 4.4.1 Introduction and definition of the potentials ............... 62 4.4.2 Thermodynamic potentials and Legendre transforms .......... 64 4.4.3 Energy picture and entropy picture* ................... 64 4.5 Available work and interpretation of the potentials ............... 67 4.5.1 Available work ............................... 67 4.5.2 The physical meaning of the potentials .................. 69 4.6 The approach to equilibrium ............................ 73 4.7 The Maxwell relations ............................... 74 4.7.1 Derivation .................................. 74 4.7.2 An application ............................... 75 4.8 Heat capacities and calculating entropy ...................... 76 4.9 Open systems and phase equilibrium conditions ................. 78 4.9.1 Open systems and chemical potential ................... 78 4.9.2 Equilibria between phases ......................... 80 4.10 The Clausius-Clapeyron equation* ........................ 80 4.11 Limitations of classical thermodynamics ..................... 83 5 The statistical basis of thermodynamics 84 5.1 Introduction ..................................... 84 5.2 The distinction between macrostates and microstates .............. 85 5.2.1 Macrostates ................................. 85 5.2.2 Microstates ................................. 85 5.2.3 Example ................................... 86 2 5.3 A crash course in probability theory ........................ 87 5.3.1 Motivation .................................. 87 5.3.2 Models with discrete microstates ..................... 87 5.3.3 Models with continuous states ....................... 89 5.4 Ensemble view and the ergodic hypothesis .................... 90 5.4.1 Time average and ensemble average .................... 90 5.4.2 Microcanonical ensemble and postulate of equal a-priori probabilities . 91 5.4.3 First Example ................................ 92 5.4.4 Second example ............................... 92 5.5 The statistical basis of entropy ........................... 94 5.6 Example the ideal spin-half paramagnet ..................... 96 5.6.1 Simple model - basic setup ......................... 96 5.6.2 Microstates and macrostates ........................ 96 5.6.3 Stirling approximation ........................... 97 5.6.4 Paramagnet - the thermodynamic limit .................. 98 5.6.5 Spin-1/2 paramagnet ............................ 100 5.7 Information entropy and the principle of maximal ignorance .......... 102 5.7.1 The method of Lagrange multipliers .................... 102 5.7.2 Shannon entropy .............................. 104 5.7.3 The microcanonical ensemble ....................... 104 5.8 Density of states .................................. 106 5.8.1 Qualitative argument: ........................... 106 5.8.2 Quantum mechanical density of states .................. 106 6 The canonical ensemble 108 6.1 Recap: constant-energy distribution (a.k.a. microcanonical ensemble) ..... 108 6.2 Derivation of the Boltzmann distribution ..................... 109 6.3 Derivation from maximum-entropy principle ................... 111 6.4 The partition function ............................... 113 6.4.1 Expected internal energy .......................... 113 6.4.2 Entropy ................................... 113 6.4.3 Energy fluctuations ............................. 114 6.4.4 The connection with thermodynamics ................... 115 6.4.5 Further comments* ............................. 117 6.5 The independent-particle approximation: one-body partition function ..... 119 6.5.1 Systems with independent particles .................... 119 6.5.2 Distinguishable and indistinguishable particles .............. 120 6.5.3 Example ................................... 121 6.6 Further simple examples of partition function calculations ........... 123 6.6.1 The ideal spin-1/2 paramagnet ...................... 123 6.6.2 A simple model for a one-dimensional solid ................ 124 6.6.3 Classical ideal gas of N particles in a volume V ............. 124 6.6.4 Einstein model of a one-dimensional solid ................. 125 6.7 Example: The classical ideal gas .......................... 126 6.8 Translational energy of molecules: Quantum treatment ............. 127 6.9 Example: The ideal spin-1/2 paramagnet ..................... 129 3 6.9.1 General results ............................... 129 6.9.2 Low-temperature and high-temperature limits of the energy ...... 130 6.9.3 High-temperature and low-temperature limits of CB ........... 131 6.9.4 Entropy and magnetic moment ...................... 131 6.9.5 The third law of thermodynamics ..................... 133 6.9.6 Adiabatic demagnetization and the third law of thermodynamics* . 133 6.10 Vibrational and rotational energy of diatomic molecules ............. 135 6.10.1 Vibrational energy contribution ...................... 136 6.10.2 Rotational energy contribution ...................... 136 6.10.3 Translational energy contribution ..................... 138 6.11 The equipartition theorem ............................. 138 6.12 The Maxwell-Boltzmann velocity distribution .................. 140 7 What’s next?* 144 7.1 Interacting systems ................................. 144 7.2 Monte Carlo methods ................................ 146 7.3 Systems with variable particle numbers ...................... 147 7.4 Quantum systems .................................. 148 7.5 Non-equilibrium dynamics ............................. 149 7.6 Interdisciplinary applications of statistical physics: complex systems ...... 150 4 Chapter 1 Introduction “A theory is the more impressive the greater the simplicity of its premises is, the more different kinds of things it relates, and the more extended is its area of ap- plicability. Therefore the deep impression which classical thermodynamics made upon me. It is the only physical theory of universal content concerning which I am convinced that within the framework of the applicability of its basic concepts, it will never be overthrown.” (Albert Einstein) “The laws of thermodynamics, as empirically determined,
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