- 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.294) Appendix 9.1 The Stefan–Boltzmann law

# (p.294) Appendix 9.1 The Stefan–Boltzmann law

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

As indicated in the main text, Boltzmann’s proof of the law suggested by Stefan on the basis of rather inaccurate experimental data rests on the concept of radiation pressure. Let us imagine an enclosure closed by a (slowly) movable piston with a reflecting surface. Electromagnetic waves exert a pressure on the piston which, as indicated in the text, is *p* = *e*/3. This pressure is due to the momentum which the electromagnetic field carries with it, according to Maxwell’s equations. Since this momentum density has a magnitude *g* = *e/c*, where *c* is the speed of light, the pressure can be computed as was done in Chapter 3 for a gas; the only difference is that the speed of the waves is constant and equal to *c*. The fact that we obtain *p* = *e*/3, rather than *p* = 2*e*/3 as in the kinetic theory of gases, is due to the fact that *e* = *gc* rather than *e* = *gc*/2 (light cannot be treated as a non-relativistic particle).

We can then write, if *V* is the volume of the enclosure and *S* its entropy:

*T*and

*V*:

*S*/∂

*V*with respect to

*T*to the derivative of ∂

*S*/∂

*T*with respect to

*V*, thus obtaining

*σ*is an integration constant. An easy argument also gives

*S*= (4/3)σ

*VT*

^{3}.

- 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