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- Title Pages
- To my daughter Anna, who helped me through Boltzmann’s dense German
- FOREWORD
- PREFACE
- FIGURE ACKNOWLEDGEMENTS
- Introduction
- 1 A short biography of Ludwig Boltzmann
- 2 Physics before Boltzmann
- 3 Kinetic theory before Boltzmann
- 4 The Boltzmann equation
- 5 Time irreversibility and the H-theorem
- 6 Boltzmann’s relation and the statistical interpretation of entropy
- 7 Boltzmann, Gibbs, and equilibrium statistical mechanics
- 8 The problem of polyatomic molecules
- 9 Boltzmann’s contributions to other branches of physics
- 10 Boltzmann as a philosopher
- 11 Boltzmann and his contemporaries
- 12 The influence of Boltzmann’s ideas on the science and technology of the twentieth century
- EPILOGUE
- CHRONOLOGY
- “A German professor’s journey into Eldorado”*
- Appendix 3.1 Calculation of pressure in a rarefied gas
- Appendix 4.1 The Liouville equation
- Appendix 4.2 Calculation of the effect of collisions of one particle with another
- Appendix 4.3 The BBGKY hierarchy
- Appendix 4.4 The Boltzmann hierarchy and its relation to the Boltzmann equation
- Appendix 4.5 The Boltzmann equation in the homogeneous isotropic case
- Appendix 5.1 Collision-invariants
- Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution
- Appendix 5.3 The H-theorem
- Appendix 5.4 The hourglass model
- Appendix 6.1 Likelihood of a distribution
- Appendix 7.1 The canonical distribution for equilibrium states
- Appendix 8.1 The H-theorem for classical polyatomic molecules
- Appendix 8.2 The equipartition problem
- Appendix 9.1 The Stefan–Boltzmann law
- Appendix 9.2 Wien’s law
- Chapter 1
- Chapter 2
- Chapter 3
- Chapter 4
- Chapter 5
- Chapter 6
- Chapter 7
- Chapter 8
- Chapter 9
- Chapter 10
- Chapter 11
- Chapter 12
- Epilogue
- Index

# (p.255) Appendix 4.1 The Liouville equation

# (p.255) Appendix 4.1 The Liouville equation

- Source:
- Ludwig Boltzmann
- Publisher:
- Oxford University Press

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- Title Pages
- To my daughter Anna, who helped me through Boltzmann’s dense German
- FOREWORD
- PREFACE
- FIGURE ACKNOWLEDGEMENTS
- Introduction
- 1 A short biography of Ludwig Boltzmann
- 2 Physics before Boltzmann
- 3 Kinetic theory before Boltzmann
- 4 The Boltzmann equation
- 5 Time irreversibility and the H-theorem
- 6 Boltzmann’s relation and the statistical interpretation of entropy
- 7 Boltzmann, Gibbs, and equilibrium statistical mechanics
- 8 The problem of polyatomic molecules
- 9 Boltzmann’s contributions to other branches of physics
- 10 Boltzmann as a philosopher
- 11 Boltzmann and his contemporaries
- 12 The influence of Boltzmann’s ideas on the science and technology of the twentieth century
- EPILOGUE
- CHRONOLOGY
- “A German professor’s journey into Eldorado”*
- Appendix 3.1 Calculation of pressure in a rarefied gas
- Appendix 4.1 The Liouville equation
- Appendix 4.2 Calculation of the effect of collisions of one particle with another
- Appendix 4.3 The BBGKY hierarchy
- Appendix 4.4 The Boltzmann hierarchy and its relation to the Boltzmann equation
- Appendix 4.5 The Boltzmann equation in the homogeneous isotropic case
- Appendix 5.1 Collision-invariants
- Appendix 5.2 Boltzmann’s inequality and the Maxwellian distribution
- Appendix 5.3 The H-theorem
- Appendix 5.4 The hourglass model
- Appendix 6.1 Likelihood of a distribution
- Appendix 7.1 The canonical distribution for equilibrium states
- Appendix 8.1 The H-theorem for classical polyatomic molecules
- Appendix 8.2 The equipartition problem
- Appendix 9.1 The Stefan–Boltzmann law
- Appendix 9.2 Wien’s law
- Chapter 1
- Chapter 2
- Chapter 3
- Chapter 4
- Chapter 5
- Chapter 6
- Chapter 7
- Chapter 8
- Chapter 9
- Chapter 10
- Chapter 11
- Chapter 12
- Epilogue
- Index