Mathematical Foundations of Thermodynamics (eBook)

Front Cover 1
Mathematical Foundations of Thermodynamics 6
Copyright Page 7
Table of Contents 8
Preface 12
Introduction 16
CHAPTER 1. FUNDAMENTAL CONCEPTS 24
1.1. The Structure of a Physical Theory 24
1.2. Primitive Observers 27
1.3. The Classical Formulations of Thermodynamics 29
1.4. Systems and States 31
1.5. Relations between States 37
1.6. The Axioms 42
CHAPTER 2. FORMAL PROCESSES 45
2.1. Definitions and Axioms 45
2.2. Addition of Processes 47
2.3. Ordering of Processes 47
2.4. Further Properties of Processes 49
CHAPTER 3. COMPONENTS OF CONTENT 52
3.1. Definition 52
3.2. Existence of Components of Content 54
CHAPTER 4. IRREVERSIBILITY 56
4.1. Irreversibility Functions 56
4.2. The Construction of an Irreversibility Function 58
4.3. Irreversibility and Entropy 60
CHAPTER 5. MECHANICAL SYSTEMS AND ADIABATIC PROCESSES 63
5.1. Physical Considerations 63
5.2. Mechanical States and Processes 67
5.3. Adiabatic Processes 68
CHAPTER 6. ENTROPY 71
6.1. Entropy Functions 71
6.2. The Construction of an Entropy Function 72
CHAPTER 7. TOPOLOGICAL CONSIDERATIONS 75
7.1. Components of Content 75
7.2. Entropy 79
CHAPTER 8. THERMODYNAMIC SPACE 87
8.1. Definitions 87
8.2. The Case of Finite Dimension 89
8.3. Mathematical Commentary 92
CHAPTER 9. EQUILIBRIUM STATES AND POTENTIAL 97
9.1. Equilibrium States 97
9.2. Components of Potential 99
CHAPTER 10. PERFECT EQUILIBRIUM STATES 105
10.1. Motivation 105
10.2. Properties of Perfect Equilibrium States 106
10.3. Perfect Thermodynamic Systems 108
CHAPTER 11. THERMODYNAMICS OF A RIGIDLY ENCLOSED SYSTEM 111
11.1. General Discussion 111
11.2. A Pathological Example 116
11.3. The Construction of an Energy Function 120
11.4. The Construction of an Entropy Function 123
CHAPTER 12. SYSTEMS OF VARIABLE VOLUME 127
12.1. Volume and Pressure 127
12.2. Simple Systems 131
CHAPTER 13. ELECTRIC AND MAGNETIC SYSTEMS 134
13.1. Electrostatic Systems 134
13.2. Magnetic Systems 139
13.3. Hysteresis 141
CHAPTER 14. GALILEAN THERMODYNAMICS 145
14.1. The Components of Content 146
14.2. Galilean Transformations 149
14.3. The Equilibrium Surface 150
14.4. Properties of Equilibrium States 151
14.5. Thermodynamic Particles 152
] 4.6. Local Properties in an Eqilibrium State 153
14.7. Some Special Cases 156
14.8. The Centrifugal Effect 159
CHAPTER 15. SYMMETRY IN THERMODYNAMICS 164
15.1. Introduction 164
15.2. The Principle of Equivalence 167
15.3. An Example 168
15.4. The Symmetry Group 170
15.5. The Transformation of States 172
15.6. The Transformation of Functions of State 174
CHAPTER 16. SPECIAL RELATIVISTIC THERMODYNAMICS 178
16.1. The Inhomogeneous Lorentz Group 179
16.2. The Components of Content 183
16.3. Rest Mass and Spin 186
16.4. The Representation of States in Space-Time 187
16.5. Centre of Mass and Spin Angular Momentum 188
16.6. The Transformation of Entropy 190
16.7. Equilibrium States and Temperature 192
16.8. Local Properties of an Equilibrium State 195
16.9. Conclusion 203
APPENDIX A. THE FORMAL THEORY 206
A . l . Notation 207
A.2. States and Processes 207
A. 3. Components of Content 210
A.4. Quasi-Entropy 214
A.5. The Duality Principle 220
A.6. Boundedness 221
A.7. Equilibrium States 223
A.8. Potentials 225
A.9. Absolute Entropy 226
APPENDIX B. SUBADDITIVE FUNCTIONS ON A GROUP 230
B.l. Partially Ordered Sets 230
B.2. Subadditive Functions 231
B.3. The Extension Theorem 234
APPENDIX C. THE PHYSICAL BASIS FOR THE ADJOINT REPRESENTATION 238
C.l. The Case of Special Relativity 238
C.2. The General Case 241
REFERENCES 244
INDEX 246
SYMBOL INDEX 252