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1 | (10) |
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1 | (1) |
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1.2 What are the Questions? |
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2 | (1) |
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2 | (2) |
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1.4 Basic Concepts and Assumptions |
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4 | (2) |
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6 | (5) |
Part I Entropy |
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2 The Classical Ideal Gas |
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11 | (5) |
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11 | (1) |
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2.2 Phase Space of a Classical Gas |
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12 | (1) |
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13 | (1) |
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13 | (1) |
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2.5 Boltzmann's Definition of the Entropy |
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14 | (1) |
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14 | (1) |
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2.7 Independence of Positions and Momenta |
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15 | (1) |
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15 | (1) |
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3 Discrete Probability Theory |
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16 | (27) |
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16 | (2) |
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3.2 Discrete Random Variables and Probabilities |
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18 | (1) |
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3.3 Probability Theory for Multiple Random Variables |
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19 | (2) |
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3.4 Random Numbers and Functions of Random Variables |
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21 | (3) |
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3.5 Mean, Variance, and Standard Deviation |
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24 | (1) |
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25 | (1) |
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3.7 Sets of Independent Random Numbers |
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25 | (2) |
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3.8 Binomial Distribution |
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27 | (2) |
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3.9 Gaussian Approximation to the Binomial Distribution |
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29 | (1) |
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3.10 A Digression on Gaussian Integrals |
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30 | (1) |
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3.11 Stirling's Approximation for N! |
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31 | (3) |
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3.12 Binomial Distribution with Stirling's Approximation |
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34 | (1) |
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3.13 Multinomial Distribution |
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35 | (1) |
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36 | (7) |
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4 The Classical Ideal Gas: Configurational Entropy |
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43 | (11) |
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4.1 Separation of Entropy into Two Parts |
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43 | (1) |
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4.2 Probability Distribution of Particles |
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44 | (1) |
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4.3 Distribution of Particles between Two Isolated Systems that were Previously in Equilibrium |
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45 | (1) |
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4.4 Consequences of the Binomial Distribution |
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46 | (1) |
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4.5 Actual Number versus Average Number |
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47 | (1) |
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4.6 The 'Thermodynamic Limit' |
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48 | (1) |
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4.7 Probability and Entropy |
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48 | (3) |
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4.8 The Generalization to M > or = to 2 Systems |
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51 | (1) |
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4.9 An Analytic Approximation for the Configurational Entropy |
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52 | (1) |
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53 | (1) |
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5 Continuous Random Numbers |
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54 | (16) |
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5.1 Continuous Dice and Probability Densities |
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54 | (1) |
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5.2 Probability Densities |
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55 | (2) |
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5.3 Dirac Delta Functions |
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57 | (4) |
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5.4 Transformations of Continuous Random Variables |
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61 | (2) |
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63 | (2) |
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65 | (5) |
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6 The Classical Ideal Gas: Energy Dependence of Entropy |
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70 | (11) |
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6.1 Distribution for the Energy between Two Subsystems |
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70 | (2) |
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6.2 Evaluation of Ωp |
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72 | (3) |
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6.3 Distribution of Energy between Two Isolated Subsystems that were Previously in Equilibrium |
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75 | (1) |
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6.4 Probability Distribution for Large N |
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76 | (2) |
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6.5 The Logarithm of the Probability Distribution and the Energy-Dependent Terms in the Entropy |
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78 | (1) |
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6.6 The Generalization to M > or = to 2 systems |
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79 | (2) |
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7 Classical Gases: Ideal and Otherwise |
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81 | (18) |
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7.1 Entropy of a Composite System of Classical Ideal Gases |
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81 | (1) |
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7.2 Equilibrium Conditions for the Ideal Gas |
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82 | (3) |
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7.3 The Volume-Dependence of the Entropy |
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85 | (2) |
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87 | (1) |
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7.5 Indistinguishable Particles |
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87 | (2) |
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7.6 Entropy of a Composite System of Interacting Particles |
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89 | (6) |
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7.7 The Second Law of Thermodynamics |
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95 | (1) |
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7.8 Equilibrium between Subsystems |
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95 | (2) |
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7.9 The Zeroth Law of Thermodynamics |
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97 | (1) |
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97 | (2) |
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8 Temperature, Pressure, Chemical Potential, and All That |
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99 | (16) |
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99 | (1) |
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8.2 What do we Mean by 'Temperature'? |
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100 | (1) |
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8.3 Derivation of the Ideal Gas Law |
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101 | (4) |
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105 | (1) |
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8.5 The Pressure and the Entropy |
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106 | (1) |
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8.6 The Temperature and the Entropy |
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106 | (1) |
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8.7 Equilibrium with Asymmetric Pistons, Revisited |
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107 | (1) |
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8.8 The Entropy and the Chemical Potential |
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108 | (1) |
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8.9 The Fundamental Relation and Equations of State |
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109 | (1) |
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8.10 The Differential Form of the Fundamental Relation |
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109 | (1) |
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8.11 Thermometers and Pressure Gauges |
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110 | (1) |
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110 | (1) |
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111 | (4) |
Part II Thermodynamics |
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9 The Postulates and Laws of Thermodynamics |
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115 | (9) |
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115 | (2) |
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9.2 Microscopic and Macroscopic States |
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117 | (1) |
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9.3 Macroscopic Equilibrium States |
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117 | (1) |
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118 | (1) |
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9.5 Properties and Descriptions |
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118 | (1) |
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9.6 The Essential Postulates of Thermodynamics |
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118 | (2) |
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9.7 Optional Postulates of Thermodynamics |
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120 | (3) |
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9.8 The Laws of Thermodynamics |
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123 | (1) |
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10 Perturbations of Thermodynamic State Functions |
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124 | (8) |
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10.1 Small Changes in State Functions |
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124 | (1) |
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10.2 Conservation of Energy |
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125 | (1) |
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10.3 Mathematical Digression on Exact and Inexact Differentials |
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125 | (3) |
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10.4 Conservation of Energy Revisited |
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128 | (1) |
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10.5 An Equation to Remember |
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129 | (1) |
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130 | (2) |
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11 Thermodynamic Processes |
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132 | (16) |
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11.1 Irreversible, Reversible, and Quasi-Static Processes |
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132 | (1) |
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133 | (2) |
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135 | (1) |
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11.4 Refrigerators and Air Conditioners |
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136 | (1) |
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137 | (1) |
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137 | (2) |
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11.7 Alternative Formulations of the Second Law |
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139 | (1) |
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11.8 Positive and Negative Temperatures |
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140 | (6) |
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146 | (2) |
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12 Thermodynamic Potentials |
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148 | (11) |
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12.1 Mathematical Digression: The Legendre Transform |
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148 | (4) |
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12.2 Helmholtz Free Energy |
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152 | (1) |
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153 | (2) |
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155 | (1) |
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12.5 Other Thermodynamic Potentials |
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155 | (1) |
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156 | (1) |
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12.7 Summary of Legendre Transforms |
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156 | (1) |
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157 | (2) |
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13 The Consequences of Extensivity |
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159 | (6) |
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160 | (1) |
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13.2 The Gibbs-Duhem Relation |
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161 | (1) |
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13.3 Reconstructing the Fundamental Relation |
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162 | (1) |
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13.4 Thermodynamic Potentials |
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163 | (2) |
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14 Thermodynamic Identities |
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165 | (19) |
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14.1 Small Changes and Partial Derivatives |
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165 | (1) |
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14.2 A Warning about Partial Derivatives |
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165 | (1) |
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14.3 First and Second Derivatives |
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166 | (2) |
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14.4 Standard Set of Second Derivatives |
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168 | (1) |
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169 | (1) |
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14.6 Manipulating Partial Derivatives |
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170 | (4) |
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14.7 Working with Jacobians |
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174 | (2) |
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14.8 Examples of Identity Derivations |
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176 | (3) |
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179 | (1) |
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180 | (4) |
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184 | (11) |
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15.1 Energy Minimum Principle |
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185 | (3) |
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15.2 Minimum Principle for the Helmholtz Free Energy |
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188 | (2) |
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15.3 Minimum Principle for the Enthalpy |
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190 | (1) |
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15.4 Minimum Principle for the Gibbs Free Energy |
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191 | (1) |
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192 | (1) |
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15.6 Maximum Principle for Massieu Functions |
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193 | (1) |
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194 | (1) |
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194 | (1) |
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195 | (10) |
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195 | (1) |
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16.2 Stability Criteria based on the Energy Minimum Principle |
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196 | (3) |
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16.3 Stability Criteria based on the Helmholtz Free Energy Minimum Principle |
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199 | (1) |
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16.4 Stability Criteria based on the Enthalpy Minimization Principle |
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200 | (1) |
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16.5 Inequalities for Compressibilities and Specific Heats |
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201 | (1) |
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16.6 Other Stability Criteria |
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201 | (2) |
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203 | (2) |
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205 | (18) |
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17.1 The van der Waals Fluid |
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206 | (1) |
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17.2 Derivation of the van der Waals Equation |
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206 | (1) |
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17.3 Behavior of the van der Waals Fluid |
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207 | (1) |
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208 | (2) |
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17.5 The Liquid-Gas Phase Transition |
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210 | (2) |
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17.6 Maxwell Construction |
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212 | (1) |
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212 | (1) |
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213 | (1) |
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17.9 Helmholtz Free Energy |
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214 | (2) |
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216 | (1) |
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17.11 The Clausius-Clapeyron Equation |
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217 | (1) |
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218 | (1) |
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219 | (4) |
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18 The Nernst Postulate: The Third Law of Thermodynamics |
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223 | (8) |
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18.1 Classical Ideal Gas Violates the Nernst Postulate |
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223 | (1) |
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18.2 Planck's Form of the Nernst Postulate |
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224 | (1) |
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18.3 Consequences of the Nernst Postulate |
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224 | (1) |
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18.4 Coefficient of Thermal Expansion at Low Temperatures |
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225 | (1) |
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18.5 The Impossibility of Attaining a Temperature of Absolute Zero |
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226 | (1) |
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18.6 Summary and Signposts |
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226 | (1) |
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227 | (4) |
Part III Classical Statistical Mechanics |
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19 Ensembles in Classical Statistical Mechanics |
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231 | (27) |
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19.1 Microcanonical Ensemble |
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232 | (1) |
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19.2 Molecular Dynamics: Computer Simulations |
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232 | (2) |
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234 | (3) |
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19.4 The Partition Function as an Integral over Phase Space |
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237 | (1) |
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19.5 The Liouville Theorem |
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238 | (2) |
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19.6 Consequences of the Canonical Distribution |
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240 | (1) |
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19.7 The Helmholtz Free Energy |
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241 | (1) |
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19.8 Thermodynamic Identities |
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242 | (1) |
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19.9 Beyond Thermodynamic Identities |
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243 | (1) |
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19.10 Integration over the Momenta |
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244 | (1) |
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19.11 Monte Carlo Computer Simulations |
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245 | (4) |
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19.12 Factorization of the Partition Function: The Best Trick in Statistical Mechanics |
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249 | (1) |
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19.13 Simple Harmonic Oscillator |
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250 | (2) |
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252 | (6) |
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20 Classical Ensembles: Grand and Otherwise |
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258 | (8) |
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20.1 Grand Canonical Ensemble |
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258 | (1) |
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20.2 Grand Canonical Probability Distribution |
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259 | (2) |
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20.3 Importance of the Grand Canonical Partition Function |
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261 | (1) |
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20.4 Z(T,V,µ) for the Ideal Gas |
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262 | (1) |
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20.5 Summary of the Most Important Ensembles |
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263 | (1) |
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20.6 Other Classical Ensembles |
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264 | (1) |
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264 | (2) |
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21 Refining the Definition of Entropy |
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266 | (5) |
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21.1 The Canonical Entropy |
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266 | (2) |
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21.2 The Grand Canonical Entropy |
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268 | (2) |
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270 | (1) |
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271 | (14) |
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22.1 What Needs to be Explained? |
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271 | (1) |
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22.2 Trivial Form of Irreversibility |
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272 | (1) |
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22.3 Boltzmann's H-Theorem |
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272 | (1) |
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22.4 Loschmidt's Umkehreinwand |
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272 | (1) |
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22.5 Zermelo's Wiederkehreinwand |
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273 | (1) |
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22.6 Free Expansion of a Classical Ideal Gas |
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273 | (5) |
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22.7 Zermelo's Wiederkehreinwand Revisited |
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278 | (1) |
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22.8 Loschmidt's Umkehreinwand Revisited |
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278 | (1) |
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22.9 What is 'Equilibrium'? |
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279 | (1) |
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279 | (2) |
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22.11 Interacting Particles |
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281 | (4) |
Part IV Quantum Statistical Mechanics |
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285 | (12) |
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23.1 Basic Quantum Mechanics |
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286 | (1) |
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287 | (3) |
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290 | (1) |
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23.4 Two Types of Probability |
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290 | (3) |
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293 | (1) |
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23.6 Uniqueness of the Ensemble |
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294 | (1) |
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295 | (1) |
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23.8 The Quantum Microcanonical Ensemble |
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296 | (1) |
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24 Quantum Canonical Ensemble |
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297 | (25) |
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24.1 Derivation of the QM Canonical Ensemble |
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297 | (2) |
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24.2 Thermal Averages and the Average Energy |
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299 | (1) |
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24.3 The Quantum Mechanical Partition Function |
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299 | (3) |
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24.4 The Quantum Mechanical Entropy |
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302 | (1) |
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24.5 The Origin of the Third Law of Thermodynamics |
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303 | (2) |
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24.6 Derivatives of Thermal Averages |
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305 | (1) |
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24.7 Factorization of the Partition Function |
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306 | (2) |
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308 | (1) |
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309 | (2) |
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24.10 Simple Harmonic Oscillator |
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311 | (2) |
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24.11 Einstein Model of a Crystal |
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313 | (2) |
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315 | (7) |
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322 | (9) |
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322 | (1) |
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25.2 Universal Frequency Spectrum |
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322 | (1) |
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323 | (1) |
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25.4 Two Types of Quantization |
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323 | (2) |
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25.5 Black-Body Energy Spectrum |
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325 | (3) |
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328 | (1) |
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25.7 Total Black-Body Radiation |
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329 | (1) |
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25.8 Significance of Black-Body Radiation |
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329 | (1) |
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330 | (1) |
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331 | (19) |
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26.1 Model of a Harmonic Solid |
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331 | (1) |
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332 | (4) |
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26.3 Transformation of the Energy |
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336 | (2) |
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26.4 The Frequency Spectrum |
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338 | (2) |
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26.5 Alternate Derivation: Equations of Motion |
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340 | (1) |
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26.6 The Energy in the Classical Model |
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341 | (1) |
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26.7 The Quantum Harmonic Crystal |
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342 | (1) |
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343 | (5) |
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348 | (2) |
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350 | (19) |
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27.1 Single-Particle Quantum States |
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350 | (2) |
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27.2 Density of Single-Particle States |
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352 | (1) |
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27.3 Many-Particle Quantum States |
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353 | (2) |
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27.4 Quantum Canonical Ensemble |
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355 | (1) |
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27.5 Grand Canonical Ensemble |
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355 | (1) |
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27.6 A New Notation for Energy Levels |
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356 | (1) |
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27.7 Exchanging Sums and Products |
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357 | (1) |
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27.8 Grand Canonical Partition Function for Independent Particles |
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358 | (1) |
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27.9 Distinguishable Quantum Particles |
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359 | (1) |
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27.10 Sneaky Derivation of PV =NkBT |
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360 | (1) |
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27.11 Equations for U = (E) and (N) |
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360 | (2) |
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362 | (1) |
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362 | (1) |
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27.14 Summary of Equations for Fermions and Bosons |
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363 | (1) |
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27.15 Integral Form of Equations for N and U |
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364 | (1) |
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27.16 Basic Strategy for Fermions and Bosons |
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365 | (1) |
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365 | (2) |
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367 | (2) |
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28 Bose-Einstein Statistics |
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369 | (17) |
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28.1 Basic Equations for Bosons |
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369 | (1) |
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369 | (1) |
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370 | (1) |
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28.4 Low-Temperature Behavior of µ |
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371 | (2) |
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28.5 Bose-Einstein Condensation |
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373 | (1) |
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28.6 Below the Einstein Temperature |
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373 | (2) |
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28.7 Energy of an Ideal Gas of Bosons |
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375 | (1) |
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28.8 What About the Second-Lowest Energy State? |
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376 | (1) |
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28.9 The Pressure below T < TE |
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377 | (1) |
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28.10 Transition Line in P-V Plot |
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378 | (1) |
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28.11 A Numerical Approach to Bose-Einstein Statistics |
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378 | (2) |
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380 | (6) |
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29 Fermi-Dirac Statistics |
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386 | (18) |
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29.1 Basic Equations for Fermions |
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386 | (1) |
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29.2 The Fermi Function and the Fermi Energy |
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387 | (1) |
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388 | (1) |
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29.4 Systems with a Discrete Energy Spectrum |
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389 | (1) |
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29.5 Systems with Continuous Energy Spectra |
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390 | (1) |
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391 | (1) |
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391 | (1) |
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29.8 Compressibility of Metals |
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392 | (1) |
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29.9 Sommerfeld Expansion |
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393 | (3) |
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29.10 General Fermi Gas at Low Temperatures |
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396 | (2) |
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29.11 Ideal Fermi Gas at Low Temperatures |
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398 | (1) |
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399 | (5) |
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30 Insulators and Semiconductors |
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404 | (19) |
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30.1 Tight-Binding Approximation |
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404 | (2) |
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406 | (2) |
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30.3 Nearly-Free Electrons |
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408 | (2) |
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30.4 Energy Bands and Energy Gaps |
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410 | (1) |
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30.5 Where is the Fermi Energy? |
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411 | (1) |
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30.6 Fermi Energy in a Band (Metals) |
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412 | (1) |
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30.7 Fermi Energy in a Gap |
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412 | (4) |
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30.8 Intrinsic Semiconductors |
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416 | (1) |
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30.9 Extrinsic Semiconductors |
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416 | (2) |
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30.10 Semiconductor Statistics |
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418 | (4) |
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30.11 Semiconductor Physics |
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422 | (1) |
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31 Phase Transitions and the Ising Model |
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423 | (24) |
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424 | (1) |
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31.2 The Ising Chain in a Magnetic Field (J = 0) |
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425 | (1) |
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31.3 The Ising Chain with h = 0, but F not = to 0 |
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426 | (2) |
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31.4 The Ising Chain with both F not = to 0 and h not = to 0 |
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428 | (4) |
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31.5 Mean-Field Approximation |
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432 | (4) |
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436 | (1) |
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31.7 Mean-Field Exponents |
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437 | (1) |
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31.8 Analogy with the van der Waals Approximation |
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438 | (1) |
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439 | (1) |
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31.10 Beyond Landau Theory |
|
|
440 | (1) |
|
|
441 | (6) |
Appendix: Computer Calculations and Python |
|
447 | (10) |
|
|
447 | (2) |
|
|
449 | (1) |
|
|
449 | (1) |
|
A.4 The First Python Program |
|
|
450 | (1) |
|
|
451 | (1) |
|
|
452 | (1) |
|
A.7 Reporting Python Results |
|
|
453 | (2) |
|
|
455 | (1) |
|
|
455 | (1) |
|
|
456 | (1) |
Index |
|
457 | |