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### Summary

### Summary

Ludwig Boltzmann arguably played the key role in establishing that submicroscopic structures underlie the ordinary world. He had a tremendous impact on late 19th-century and early 20th-century physics, and he anticipated many contemporary ideas, including Kuhn's theory of scientific revolutions and recent theories of knowledge based on Darwinian principles. This book is the first accessible biography of this important figure. Without relying on equations, it provides a deep look at the full range of his scientific and philosophical ideas, discussing both their original context and their relevance today. The book also gives a concise portrait of Boltzmann's life, which, despite his successes, ended tragically in suicide. Drawing on recent research related to some of Boltzmann's more controversial ideas, this book offers fascinating insights into the birth of modern physics.

### Author Notes

Carlo Cercignani is a Professor of Theoretical Mechanics at Politecnico di Milano.

### Table of Contents

Figure acknowledgements | p. xvii |

Introduction | p. 1 |

1 A short biography of Ludwig Boltzmann | p. 5 |

1.1 Youth and happy years | p. 5 |

1.2 The crisis | p. 20 |

1.3 Restlessness | p. 22 |

1.4 Scientific debates and travels | p. 26 |

1.5 The tragic fate of a great scientist | p. 34 |

1.6 Boltzmann as a teacher | p. 37 |

1.7 Boltzmann and inventions | p. 38 |

1.8 Ludwig Boltzmann and his times | p. 39 |

1.9 A poem by Ludwig Boltzmann | p. 46 |

1.10 Boltzmann's personality | p. 48 |

2 Physics before Boltzmann | p. 50 |

2.1 From Galileo and Newton to the early atomic theories | p. 50 |

2.2 The first connections between heat and mechanical energy | p. 56 |

2.3 The springtime of thermodynamics | p. 59 |

2.4 Electricity and magnetism | p. 65 |

3 Kinetic theory before Boltzmann | p. 71 |

3.1 Early kinetic theories | p. 71 |

3.2 The beginnings of modern kinetic theory and the problem of justifying the Second Law | p. 80 |

4 The Boltzmann equation | p. 86 |

4.1 Irreversibility and kinetic theory | p. 86 |

4.2 The great paper of 1872 | p. 88 |

4.3 A critique of Boltzmann's approach | p. 93 |

5 Time irreversibility and the H-theorem | p. 96 |

5.1 Introduction | p. 96 |

5.2 Loschmidt's paradox | p. 97 |

5.3 Poincare's recurrence and Zermelo's paradox | p. 100 |

5.4 The physical and mathematical resolution of the paradoxes | p. 102 |

5.5 Timers arrow and the expanding universe | p. 109 |

5.6 Is irreversibility objective or subjective? | p. 112 |

5.7 Concluding remarks | p. 118 |

6 Boltzmann's relation and the statistical interpretation of entropy | p. 120 |

6.1 The probabilistic interpretation of thermodynamics | p. 120 |

6.2 Explicit use of probability for a gas with discrete energies | p. 121 |

6.3 Energy is continuous | p. 125 |

6.4 The so-called H-curve | p. 129 |

7 Boltzmann, Gibbs, and equilibrium statistical mechanics | p. 134 |

7.1 Introduction | p. 134 |

7.2 A great American scientist of the nineteenth century: J.W. Gibbs | p. 135 |

7.3 Why is statistical mechanics usually attributed to Gibbs and not to Boltzmann? | p. 140 |

7.4 Gibbs's treatise | p. 142 |

7.5 French scientists on statistical mechanics | p. 145 |

7.6 The problem of trend to equilibrium and ergodic theory | p. 146 |

7.7 Planck and statistical mechanics | p. 150 |

8 The problem of polyatomic molecules | p. 153 |

8.1 The problem of specific heats | p. 153 |

8.2 The H-theorem for polyatomic molecules | p. 154 |

8.3 Specific heats again | p. 155 |

8.4 Boltzmann's ideas on specific heats, and twentieth century contributions | p. 157 |

9 Boltzmann's contributions to other branches of physics | p. 160 |

9.1 Boltzmann's testing of Maxwell's theory of electromagnetism | p. 160 |

9.2 Boltzmann lays the foundations of hereditary mechanics | p. 161 |

9.3 Back to electromagnetism | p. 162 |

9.4 A true pearl of theoretical physics | p. 163 |

9.5 Mathematics and foundations of mechanics | p. 164 |

10 Boltzmann as a philosopher | p. 170 |

10.1 A realist, but not a naive one | p. 170 |

10.2 Laws of thought and scientific concepts | p. 177 |

10.3 Ethics, aesthetics, religion | p. 181 |

10.4 Philosophy of science | p. 184 |

10.5 Boltzmann's views on scientific revolutions | p. 189 |

10.6 Boltzmann's education in philosophy | p. 191 |

10.7 Did Boltzmann abandon realism? | p. 192 |

11 Boltzmann and his contemporaries | p. 198 |

11.1 The contacts between Boltzmann and his colleagues | p. 198 |

11.2 Maxwell | p. 198 |

11.3 Lorentz | p. 200 |

11.4 Boltzmann and the energetists | p. 202 |

11.5 Planck | p. 210 |

11.6 Students and younger colleagues | p. 211 |

12 The influence of Boltzmann's ideas on the science and technology of the twentieth century | p. 214 |

12.1 Brownian motion | p. 214 |

12.2 Enter Einstein | p. 215 |

12.3 Black-body radiation | p. 217 |

12.4 Einstein again | p. 220 |

12.5 The role of Boltzmann's ideas during the twentieth century | p. 223 |

Epilogue | p. 226 |

Chronology | p. 227 |

"A German professor's journey into Eldorado" | p. 231 |

Appendices | p. 251 |

A 3.1 Calculation of pressure in a rarefied gas | p. 251 |

A 4.1 The Liouville equation | p. 255 |

A 4.2 Calculation of the effect of collisions of one particle with another | p. 259 |

A 4.3 The BBGKY hierarchy | p. 261 |

A 4.4 The Boltzmann hierarchy and its relation to the Boltzmann equation | p. 264 |

A 4.5 The Boltzmann equation in the homogeneous isotropic case | p. 266 |

A 5.1 Collision-invariants | p. 267 |

A 5.2 Boltzmann's inequality and the Maxwell distribution | p. 271 |

A 5.3 The H-theorem | p. 273 |

A 5.4 The hourglass model | p. 277 |

A 6.1 Likelihood of a distribution | p. 280 |

A 7.1 The canonical distribution for equilibrium states | p. 283 |

A 8.1 The H-theorem for classical polyatomic molecules | p. 287 |

A 8.2 The equipartition problem | p. 291 |

A 9.1 The Stefan-Boltzmann law | p. 294 |

A 9.2 Wien's law | p. 295 |

References | p. 297 |

Index | p. 319 |