- 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
The Boltzmann equation
The Boltzmann equation
- Chapter:
- (p.86) 4 The Boltzmann equation
- Source:
- Ludwig Boltzmann
- Author(s):
CARLO CERCIGNANI
- Publisher:
- Oxford University Press
The problem of irreversibility came to the forefront in kinetic theory with Ludwig Boltzmann. In 1872, Boltzmann not only derived the equation that bears his name, but also introduced a definition of entropy in terms of the distribution function of the molecular velocities. By 1868, Boltzmann had already extended James Clerk Maxwell's distribution to the case where the molecules are in equilibrium in a force field with potential, including the case of polyatomic molecules. Boltzmann interprets Maxwell's distribution function in two different ways: the first way is based on the fraction of a sufficiently long time interval, during which the velocity of a specific molecule has values within a certain volume element in velocity space; whereas the second way is based on the fraction of molecules which, at a given instant, have a velocity in the said volume element. This chapter discusses the Boltzmann equation, irreversibility and kinetic theory, and Boltzmann's 1872 paper on the thermal equilibrium of gas molecules.
Keywords: entropy, James Clerk Maxwell, Boltzmann equation, irreversibility, kinetic theory, molecules, thermal equilibrium, distribution function, molecular velocities
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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