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Poincaré, Poincaré recurrence and the H-theorem: A continued reassessment of Boltzmannian statistical mechanics

Christopher Gregory Weaver

Department of Philosophy, 200 Gregory Hall, 810 South Wright ST MC-468, Urbana, IL 61801, University of Illinois at Urbana-Champaign, USA

Department of Physics, 1110 West Green ST, Urbana, IL 61801, University of Illinois at Urbana-Champaign, USA

Illinois Center for Advanced Studies of the Universe, University of Illinois at Urbana-Champaign, USA

E-mail Address: wgceave9@illinois.edu

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https://doi.org/10.1142/S0217979222300055Cited by:2 (Source: Crossref)

Abstract

In [C. G. Weaver Found. Phys. 51, 1 (2021)], I showed that Boltzmann’s H-theorem does not face a significant threat from the reversibility paradox. I argue that my defense of the H-theorem against that paradox can be used yet again for the purposes of resolving the recurrence paradox without having to endorse heavy-duty statistical assumptions outside of the hypothesis of molecular chaos. As in [C. G. Weaver Found. Phys. 51, 1 (2021)], lessons from the history and foundations of physics reveal precisely how such resolution is achieved.

Keywords:

PACS: 01.65.+g, 01.70.+w, 05.20.Dd, 05.70.−a, 05.70.Ln

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References

1. P. T. Gressman and R. M. Strain, Adv. Math. 227, 2349 (2011). Crossref, Web of Science, Google Scholar
2. D. Albert, Time and Chance (Harvard University Press, Cambridge, MA, 2000). Crossref, Google Scholar
3. M. Audin, Remembering Sofya Kovalevskaya (Springer, London, 2011). Crossref, Google Scholar
4. L. K. Babadzanjanz, Celest. Mech. Dyn. Astron. 56, 427 (1993). Crossref, Web of Science, ADS, Google Scholar
5. L. K. Babadzanjanz, Celest. Mech. 20, 43 (1979). Crossref, Web of Science, ADS, Google Scholar
6. M. Badino, Eur. Phys. J. H 36, 353 (2011). Crossref, Web of Science, Google Scholar
7. J. Barrow-Green, Hist. Math. 37, 164 (2010). Crossref, Web of Science, Google Scholar
8. J. Barrow-Green, The three-body problem, in The Princeton Companion to Mathematics, eds. T. Gowers, J. Barrow-Green and I. Leader (Princeton University Press, Princeton, 2008), pp. 726–728. Google Scholar
9. J. Barrow-Green, “ Jules Henri Poincaré”, in The Princeton Companion to Mathematics, eds. T. Gowers, J. Barrow-Green and I. Leader (Princeton University Press, Princeton, 2008), pp. 785–787. Google Scholar
10. J. Barrow-Green, Poincaré and the Three Body Problem. History of Mathematics, Volume 11 (American Mathematical Society, Providence, 1997). Google Scholar
11. J. Barrow-Green, Arch. History Exact Sci. 48, 107 (1994). Crossref, Web of Science, ADS, Google Scholar
12. G. W. Baxter and J. S. Olafsen, Phys. Rev. Lett. 99, 028001 (2007). Crossref, Web of Science, ADS, Google Scholar
13. L. Boltzmann, Lectures on Gas Theory (Translated by S. G. Brush) Dover, New York, 1964). Google Scholar
14. L. Boltzmann, Wiener Ber. 106, 12 (1897); L. Boltzmann, Wissenschaftliche Abhandlungen von Ludwig Boltzmann, ed. F. Hasenöhrl, Vol. 3 (Barth, Leipzig, 1909), pp. 587–595. Google Scholar
15. L. Boltzmann, ZCW 2, 259 (1897); Ann. Phys. Chem. 296(2), 392 (1897); L. Boltzmann, Wissenschaftliche Abhandlungen von Ludwig Boltzmann, ed. F. Hasenöhrl, Vol. 3 (Barth, Leipzig, 1909), pp. 579–586. Google Scholar
16. L. Boltzmann, ZCW 2, 229 (1896); Ann. Phys. Chem. 293, 773 (1896); L. Boltzmann, Wissenschaftliche Abhandlungen von Ludwig Boltzmann, ed. F. Hasenöhrl, Vol. 3 (Barth, Leipzig, 1909), pp. 567–578. Google Scholar
17. L. Boltzmann, Annalen der Physik und Chemie 291(5), 223 (1895). BWA3, 532–534. Google Scholar
18. L. Boltzmann, Wiener Ber. 76, 373 (1877); L. Boltzmann, Wissenschaftliche Abhandlungen von Ludwig Boltzmann, ed. F. Hasenöhrl, Vol. 2 (Barth, Leipzig, 1909), pp. 164–223. Google Scholar
19. L. Boltzmann, Wiener Ber. 72, 427 (1875); L. Boltzmann, Wissenschaftliche Abhandlungen von Ludwig Boltzmann, ed. F. Hasenöhrl, Vol. 2 (Barth, Leipzig, 1909), pp. 1–30. Google Scholar
20. L. Boltzmann, Wiener Ber. 66, 275 (1872); L. Boltzmann, Wissenschaftliche Abhandlungen von Ludwig Boltzmann, ed. F. Hasenöhrl, Vol. 1 (Barth, Leipzig, 1909), pp. 316–402. Google Scholar
21. J. Boniface, The concept of number from Gauss to Kronecker, in The Shaping of Arithmetic After C. F. Gauss’s Disquisitiones Arithmeticae, eds. C. Goldstein, N. Schappacher and J. Schwermer (Springer, Berlin, 2007), pp. 315–342. Crossref, Google Scholar
22. U. Bottazzini, Stud. Hist. Philos. Mod. Phys. 47, 118 (2014). Crossref, Web of Science, Google Scholar
23. U. Bottazzini, Complex function theory, 1780–1900, in A History of Analysis, eds. H. N. Jahnke (American Mathematical Society, Providence, Rhode Island, 2003), pp. 213–259. Google Scholar
24. H. R. Brown, W. Myrvold and J. Uffink, Stud. Hist. Philos. Mod. Phys. 40, 174 (2009). Crossref, Web of Science, Google Scholar
25. S. G. Brush, The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century Volume 1, Physics and the Atomists (North-Holland, Amsterdam, 1976). Google Scholar
26. S. G. Brush, The Kind of Motion We Call Heat: A History of the Kinetic Theory of Gases in the 19th Century Volume 2 (North Holland, Amsterdam, 1976). Google Scholar
27. S. G. Brush, Arch. Hist. Exact Sci. 12, 1 (1974). Crossref, Web of Science, Google Scholar
28. G. H. Bryan, Nature 52, 29 (1895). Crossref, Google Scholar
29. S. H. Burbury, Nature 51, 78 (1894). Crossref, ADS, Google Scholar
30. S. H. Burbury, Nature 51, 320 (1895). Crossref, ADS, Google Scholar
31. C. Callender, The past histories of molecules, in Probabilities in Physics, eds. C. Beisbart and S. Hartmann (Oxford University Press, New York, 2011), pp. 83–113. Crossref, Google Scholar
32. C. Carathéodory, Gesammelte Mathematische Schriften, Volume 4: Funktionentheorie, reelle Funktionen (C. H. Beck, Munich, 1956) (in German). Google Scholar
33. C. Carathéodory, Sitzungsber. Preuß. Akad. Wiss. Berlin, Math. Phys. Kl. 580 (1919) (in German). Google Scholar
34. T. Carleman, Problèmes Mathématiques dans la Théorie Cinétique des Gaz (Almqvist & Wiksells Boktryckeri AB, Upssala, 1957). Google Scholar
35. T. Carleman, Acta Math. 60, 91 (1933). Crossref, Google Scholar
36. S. Carroll, From Eternity to Here: The Quest for the Ultimate Theory of Time (Dutton, New York, 2010). Google Scholar
37. C. Cercignani, Ludwig Boltzmann: The Man Who Trusted Atoms. (Oxford University Press, New York, 1998). Google Scholar
38. C. Cercignani, The Boltzmann Equation and its Applications (Springer-Verlag, New York, 1988). Crossref, Google Scholar
39. S. Chakraborti et al., arXiv:2109.07742v2 [cond-mat.stat-mech] (2021). Google Scholar
40. R. Cooke, The Mathematics of Sonya Kovalevskaya (Springer-Verlag, New York, 1984). Crossref, Google Scholar
41. E. Culverwell, Nature 50, 617 (1894). Crossref, ADS, Google Scholar
42. O. Darrigol, Eur. Phys. J. H. 46, 29 (2021). Crossref, Web of Science, Google Scholar
43. O. Darrigol, Atoms, Mechanics, and Probability: Ludwig Boltzmann’s Statistico-Mechanical Writings: An Exegesis (Oxford University Press, New York, 2018). Crossref, Google Scholar
44. R. L. Devaney, Am. Math. Monthly 89, 535 (1982). Crossref, Web of Science, Google Scholar
45. F. Diacu, Math. Intell. 18, 66 (1996). Crossref, Web of Science, Google Scholar
46. P. M. C. Dias, Arch. Hist. Exact Sci. 46, 341 (1994). Crossref, Web of Science, Google Scholar
47. Y. Domar, Acta Math. 148, 3 (1982). Crossref, Web of Science, Google Scholar
48. P. Dugac, Arch. Hist. Exact Sci. 10, 41 (1973). Crossref, Web of Science, Google Scholar
49. A. Duncan and M. Janssen, Constructing Quantum Mechanics: Volume 1. The Scaffold 1900–1923 (Oxford University Press, New York, 2019). Crossref, Google Scholar
50. H.-D. Ebbinghaus, Ernst Zermelo: An Approach to His Life and Work, 2nd edn. (Springer, Heidelberg, 2007). Google Scholar
51. A. Einstein, The Collected Papers of Albert Einstein, Volume 2: The Swiss Years: Writings 1900-1909 (Translated by A. Beck), Consultant Peter Havas, Vol. 2 (Princeton University Press, Princeton, NJ, 1989). Google Scholar
52. G. G. Emch and C. Liu, The Logic of Thermostatistical Physics (Springer-Verlag, Berlin, 2002). Crossref, Google Scholar
53. E. Fermi, Thermodynamics (Dover Publications, New York, 1956). Google Scholar
54. R. Frigg and C. Werndl, Monist. 102, 424 (2019). Crossref, Web of Science, Google Scholar
55. R. Frigg and C. Werndl, Entropy: A guide for the perplexed, in Probabilities in Physics, eds. C. Beisbart and S. Hartmann (Oxford University Press, New York, 2011), pp. 115–142. Crossref, Google Scholar
56. E. Garber, S. G. Brush and C. W. F. Everitt (eds.), Maxwell on Heat and Statistical Mechanics: On “Avoiding All Personal Enquiries” of Molecules (MIT Press, Cambridge, MA, 1995). Google Scholar
57. E. Garber, S. G. Brush and C. W. F. Everitt, Kinetic theory and the properties of gases: Maxwell’s work in its Nineteenth-Century context, in Maxwell on Molecules and Gases, eds. E. Garber, S. G. Brush and C. W. F. Everitt (MIT Press, Cambridge, MA, 1986), pp. 1–63. Google Scholar
58. J. W. Gibbs, Elementary Principles in Statistical Mechanics Developed with Especial Reference to the Rational Foundation of Thermodynamics (Dover Publications, New York, 1960). Google Scholar
59. C. Goldstein, Rev. Hist. Math. 17, 211 (2011). Google Scholar
60. C. Goldstein, The Hermitian form of reading the disquisitiones, in The Shaping of Arithmetic After C. F. Gauss’s Disquisitiones Arithmeticae, eds. C. Goldstein, N. Schappacher and J. Schwermer (Springer, Berlin, 2007), pp. 377–410. Crossref, Google Scholar
61. S. Goldstein et al., On the nonequilibrium entropy of large and small systems, in Stochastic Dynamics Out of Equilibrium, eds. G. Giacomin, S. Olla, E. Saada, H. Spohn and G. Stoltz (Springer, Cham, 2017), pp. 581–596. Google Scholar
62. S. Goldstein et al., Ann. Phys. 529, 1600301 (2017). Crossref, Web of Science, Google Scholar
63. S. Goldstein et al., Gibbs and Boltzmann entropy in classical and quantum mechanics, in Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature, eds. V. Allori (World Scientific, Singapore, 2020). Link, Google Scholar
64. S. Goldstein et al., Gibbs and Boltzmann entropy in classical and quantum mechanics, https://arxiv.org/pdf/1903.11870.pdf. (accessed on 11 November 2019). Google Scholar
65. S. Goldstein and J. L. Lebowitz, Physica D 193, 53 (2004). Crossref, Web of Science, ADS, Google Scholar
66. S. Goldstein, Boltzmann’s approach to statistical mechanics, in Chance in Physics, eds. J. Bricmont, D. Dürr, M. C. Galavotti, G. Ghirardi, F. Petruccione and N. Zanghì (Springer-Verlag, Berlin-Heidelberg, 2001), pp. 39–54. Crossref, Google Scholar
67. R. Grant, History of Physical Astronomy: From the Earliest Ages to the Middle of the Nineteenth Century Comprehending a Detailed Account of the Establishment of the Theory of Gravitation by Newton with an Exposition of the Progress of Research on All the Other Subjects of Celestial Physics (Johnson Reprint Corporation, New York and London, 1966). Google Scholar
68. J. Gray, A History of Abstract Algebra: From Algebraic Equations to Modern Algebra (Springer Nature, Switzerland, 2018). Crossref, Google Scholar
69. J. Gray, The Real and the Complex: A History of Analysis in the 19th Century (Springer, New York, 2015). Google Scholar
70. J. Gray, Henri Poincaré: A Scientific Biography (Princeton University Press, Princeton, NJ, 2013). Google Scholar
71. J. Gray, Plato’s Ghost: The Modernist Transformation of Mathematics (Princeton University Press, Princeton, NJ, 2008). Crossref, Google Scholar
72. T. Hawkins, Arch. Hist. Exact Sci. 17, 119 (1977). Crossref, Web of Science, Google Scholar
73. C. Hermite, Euvres de Charles Hermite publiées sous les Auspices de l’Académie des Sciences, ed. É. Picard, Vol. 1–4. (Gauthier-Villars, Paris, 1905). Google Scholar
74. C. Hermite and T. J. Stieltjes, Correspondance d’Hermite et de Stieltjes, eds. B. Baillaud and H. Bourget (Gauthier-Villars, Paris, 1905). Google Scholar
75. K. Huang, Statistical Mechanics, 2nd edn. (John Wiley & Sons, New York, 1987). Google Scholar
76. M. J. Klein, Paul Ehrenfest: Volume 1. The Making of a Theoretical Physicist (North-Holland Publishing Company, Amsterdam, 1970). Google Scholar
77. M. J. Klein, Centaurus 17, 58 (1973) Crossref, ADS, Google Scholar
78. M. Klein, The development of Boltzmann’s statistical ideas, in The Boltzmann Equation: Theory and Applications, eds. E. G. D. Cohen and W. Thirring (Springer Verlag, Vienna, 1973), pp. 53–106. Crossref, Google Scholar
79. A. H. Koblitz, Science, Women and Revolution in Russia (Routledge Publishers, London and New York, 2013). Google Scholar
80. A. H. Koblitz, A Convergence of Lives: Sofia Kovalevskaia, Scientist, Writer, Revolutionary (Birkhäuser, Boston, 1983). Crossref, Google Scholar
81. S. Kopeikin, M. Efroimsky and G. Kaplan, Relativistic Celestial Mechanics of the Solar System (Wiley-VCH, Weinheim, Germany, 2011). Crossref, Google Scholar
82. S. Kovalevskaya, A Russian Childhood (Translated and introduced by B. Stillman) (Springer-Verlag, Berlin, 1978). Google Scholar
83. A. J. Kox, Ann. Sci. 47, 591 (1990). Crossref, Web of Science, Google Scholar
84. A. J. Kox, The correspondence between Boltzmann and H. A. Lorentz, in Ludwig Boltzmann: Internationale Tagung anlässlich des 75. Jahrestages Seines Todes, 5-8 September 1981, Ausgewählte Abhandlungen, eds. R. Sexl and J. Blackmore (Friedr Vieweg & Sohn, Braunschweig/Wiesbaden; Akademische Druck, Graz, 1982), pp. 73–86. Google Scholar
85. H. Kragh, Quantum Generations: A History of Physics in the Twentieth Century (Princeton University Press, Princeton, 1999). Crossref, Google Scholar
86. G. M. Kremer, An Introduction to the Boltzmann Equation and Transport Processes in Gases (Springer-Verlag, Berlin, 2010). Crossref, Google Scholar
87. T. S. Kuhn, Black-Body Theory and the Quantum Discontinuity 1894–1912 (Clarendon Press, Oxford, 1978). Google Scholar
88. N. M. Laurendeau, Statistical Thermodynamics: Fundamentals and Applications (Cambridge University Press, Cambridge, 2005). Crossref, Google Scholar
89. H. Lebesgue, Ann. di Mat. Pura ed Appl. 7, 231 (1902). Crossref, Google Scholar
90. J. L. Lebowitz, Statistical mechanics ensembles and typical behavior of macroscopic systems, in Lecture for the Int. Centre for Theoretical Sciences: Tata Institute for Fundamental Research, 13 July 2021, https://www.icts.res.in/lectures/macroscopicsystems. Google Scholar
91. J. L. Lebowitz, Rev. Mod. Phys. 71, S346 (1999). Crossref, Web of Science, Google Scholar
92. J. L. Lebowitz, Physica A: Stat. Mech. Appl. 194, 1 (1993). Crossref, Web of Science, Google Scholar
93. J. L. Lebowitz, Phys. Today 46, 32 (1993). Crossref, Web of Science, Google Scholar
94. G. W. Leibniz, Philosophical Essays (Translated by R. Ariew and D. Garber) (Hackett Publishing Company, Indianapolis, IN, 1989). Google Scholar
95. G. W. Leibniz, Reflections on the advancement of true metaphysics and particularly on the nature of substance explained by force, in Philosophical Texts, eds. R. Woolhouse and R. S. Francks (Oxford University Press, Oxford, 1998), pp. 139–142. Google Scholar
96. B. Loewer, The mentaculus vision, in Statistical Mechanics and Scientific Explanation: Determinism, Indeterminism and Laws of Nature, eds. V. Allori (World Scientific, Singapore, 2020), pp. 3–29. Link, Google Scholar
97. H. A. Lorentz, Wiener Ber. 95, 115 (1887). Google Scholar
98. J. Loschmidt, Wiener Ber. 73, 128 (1876). Google Scholar
99. J. Losey and R. J. Sadus, J. Phys. Chem. 123, 8268 (2019). Crossref, Web of Science, Google Scholar
100. J. Lützen, The foundation of analysis in the 19th century, in A History of Analysis, ed. H. N. Jahnke (American Mathematical Society, 2003), pp. 155–195. Google Scholar
101. C. Marchal, Regularization of the singularities of Providence, the n-body problem, in Applications of Modern Dynamics to Celestial Mechanics and Astrodynamics, Proc. NATO Advanced Study Institute Held at Cortina d’Ampezzo, Italy, 2–14 August 1981, ed. V. Szebehely (D. Reidel Publishing Company, Dordrecht, 1982), pp. 201–236. Google Scholar
102. T. Maudlin, Br. J. Philos. Sci. 46, 145 (1995). Crossref, Google Scholar
103. J. C. Maxwell, Philos. Mag. 19, 19 (1860); J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, ed. W. D. Niven,Vol. 1 (Cambridge University Press, Cambridge, 1890), pp. 377–391. Google Scholar
104. J. C. Maxwell, Philos. Mag. 20, 21 (1860); J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, ed. W. D. Niven,Vol. 1 (Cambridge University Press, Cambridge, 1890), pp. 392–409. Google Scholar
105. J. C. Maxwell, Philos. Trans. R. Soc. 157, 49 (1867); J. C. Maxwell, The Scientific Papers of James Clerk Maxwell, ed. W. D. Niven, Vol. 2 (Cambridge University Press, Cambridge, 1890), pp. 26–78. Google Scholar
106. G. Mittag-Leffler, Acta Math. 35, 29 (1912). Crossref, Google Scholar
107. P. Nabonnand, Math. Intell. 21, 58 (1999). Crossref, Web of Science, Google Scholar
108. L. W. Nordheim, On the kinetic method in the new statistics and application in the electron theory of conductivity, in Proc. R. Soc. A: Math. Phys. Eng. Sci. 119, 689 (1928). Google Scholar
109. P. Painlevé, Leçons sur la théorie analytique des équations différentielles professes à Stockholm, Sur L’invitation De S. M. Le Roi De Suède Et De Norwége, Vol. 11(2) (Palala Press, 1895–2015). Google Scholar
110. L. Peliti, Statistical Mechanics in a Nutshell (Translated by M. Epstein) (Princeton University Press, Princeton, NJ, 2011). Google Scholar
111. R. Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe (Vintage Books, New York, 2004). Google Scholar
112. H. Poincaré, The Three-Body Problem and the Equations of Dynamics: Poincaré’s Foundational Work on Dynamical Systems Theory (Translated by B. D. Popp) (Springer, Berlin, 2017). Google Scholar
113. H. Poincaré, On the three-body problem and the equations of dynamics, in The Kinetic Theory of Gases: An Anthology of Classic Papers with Historical Commentary (Imperial College Press, London, 2003), pp. 368–376. Google Scholar
114. H. Poincaré, Mechanism and experience, in Kinetic Theory Volume 2: Irreversible Processes, ed. S. G. Brush (Pergamon Press, Oxford, 1966), pp. 203–207. Crossref, Google Scholar
115. H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 3 (Gauthier-Villars, Paris, 1899). Crossref, Google Scholar
116. H. Poincaré, Les Méthodes Nouvelles de la Mécanique Céleste, Vol. 2 (Gauthier-Villars, Paris, 1893). Google Scholar
117. H. Poincaré, Le mécanisme et l’expérience, Rev. Métaphys. Morale 1, 534 (1893). Google Scholar
118. H. Poincaré, Henri. 1892. Les Méthodes Nouvelles de la Mécanique Céleste. Vol. 1 (Gauthier-Paris, 1892). Google Scholar
119. H. Poincaré, Acta Math. 13, 1 (1890); Euvres 7, 262 (1890). Google Scholar
120. H. Poincaré, Sur le problème des trois corps et les équations de la dynamique avec des notes par l’auteur — mémoire couronne du prix de S. M. le Roi Oscar II (1889). Google Scholar
121. B. D. Popp, Translator’s preface, in The Three-Body Problem and the Equations of Dynamics: Poincaré’s Foundational Work on Dynamical Systems Theory (Springer, Berlin, 2017), pp. v–xvii. Google Scholar
122. M. Rågstedt, From Order to Chaos: The Prize Competition in Honour of King Oscar II (Institut Mittag-Leffler The Royal Swedish Academy of Sciences, Auravägen 17, SE 182 60, Djursholm, Sweden, 2022), http://www.mittag-leffler.se/library/henri-poincare. Google Scholar
123. R. E. Robson, T. J. Mehrling and J. Osterhoff, Eur. J. Phys. 38, 065103 (2017). Google Scholar
124. D. E. Rowe, Mathematics in Berlin, 1810–1933, in Mathematics in Berlin, eds. H. G. W. Begehr, H. Koch, J. Kramer, N. Schappacher and E.-J. Thiele (Springer, Basel, AG, 1998), pp. 9–26. Crossref, Google Scholar
125. L. Ruetsche, Interpreting Quantume Theories: The Art of the Possible (Oxford University Press, New York, 2011). Crossref, Google Scholar
126. M. Šuvakov and V. Dmitrašinović, Phys. Rev. Lett. 110, 114301 (2013). Crossref, Web of Science, ADS, Google Scholar
127. D. G. Saari, Collisions, Rings, and Other Newtonian n-Body Problems (American Mathematical Society, Providence, RI, 2005). Crossref, Google Scholar
128. D. G. Saari, A visit to the Newtonian n-body problem via elementary complex variables, Am. Math. Mon. 97, 105 (1990). Crossref, Web of Science, Google Scholar
129. E. Segrè, From Falling Bodies to Radio Waves: Classical Physicists and Their Discoveries (Dover Publications, New York, 1984). Google Scholar
130. C. L. Seigel, Am. Math. Mon. 48, 430 (1941). Crossref, Google Scholar
131. C. L. Siegel and J. K. Moser, Lectures on Celestial Mechanics (Translated by C. I. Kalme) (Springer-Verlag, Berlin, 1995). Google Scholar
132. J. P. Sethna, Statistical Mechanics: Entropy, Order Parameters, and Complexity, 2nd edn. (Oxford University Press, Oxford, 2021). Crossref, Google Scholar
133. H. Spohn, Microscopic time reversibility and the Boltzmann equation, in Chance in Physics, eds. J. Bricmont, D. Dürr, M. C. Galavotti, G. Ghirardi, F. Petruccione and N. Zanghì (Springer-Verlag, Berlin-Heidelberg, 2001), pp. 56–59. Crossref, Google Scholar
134. H. Spohn, Large Scale Dynamics of Interacting Particles (Springer-Verlag, Berlin, 1991). Crossref, Google Scholar
135. G. Holton and G. B. Stephen, Physics, the Human Adventure: From Copernicus to Einstein and Beyond. New Brunswick, NJ: Rutgers University Press. Google Scholar
136. B. Stillman, Introduction, in A Russian Childhood, ed. S. Kovalevskaya (Springer-Verlag, Berlin, 1978), pp. 1–43. Google Scholar
137. W. Thomson, Nature 9, 441 (1874). Crossref, ADS, Google Scholar
138. E. A. Uehling and G. E. Uhlenbeck, Phys. Rev. 43, 552 (1933). Crossref, ADS, Google Scholar
139. J. Uffink, Boltzmann’s work in statistical physics, in The Stanford Encyclopedia of Philosophy, ed. E. N. Zalta (Springer, 2017), https://plato.stanford.edu/archives/spr2017/entries/statphys-Boltzmann/. Google Scholar
140. J. Uffink, Introductory note to 1896a, 1896b, and Boltzmann 1896, 1897, in Collected Works: Volume II, eds. H.-D. Ebbinghaus and A. Kanamori (Springer-Verlag, Berlin, 2013), pp. 188–215. Google Scholar
141. J. Uffink, Compendium of the foundations of classical statistical physics, in Handbook of the Philosophy of Science: Philosophy of Physics Part B, eds. J. Butterfield and J. Earman (Elsevier, Amsterdam, 2007), pp. 923–1074. Crossref, Google Scholar
142. J. Uffink and G. Valente, Time’s arrow and Lanford’s theorem, in Séminaire Poincaré XV Le Temps, (2010), pp. 141–173, https://www.narcis.nl/publication/RecordID/oai:dspace.library.uu.nl:1874%2F202692. Google Scholar
143. B. C. van Fraassen, The Scientific Image (Clarendon Press, Oxford, 1980). Crossref, Google Scholar
144. C. Villani, Entropy production and convergence to equilibrium, in Entropy Methods for the Boltzmann Equation: Lecture Notes in Mathematics, eds. F. Golse and S. Olla (Springer-Verlag, Berlin, 2008), pp. 1–70. Crossref, Google Scholar
145. C. Villani, J. Stat. Phys. 124, 781 (2006). Crossref, Web of Science, ADS, Google Scholar
146. C. Villani, A review of mathematical topics in collisional kinetic theory, in Handbook of Mathematical Fluid Dynamics, eds. S. Friedlander and D. Serre, Vol. 1 (Elsevier Science, Amsterdam, 2002), pp. 71–305. Crossref, Google Scholar
147. F. Verhulst, Henri Poincaré: Impatient Genius (Springer Science, New York, 2012). Crossref, Google Scholar
148. J. von Plato, Creating Modern Probability (Cambridge University Press, Cambridge, 1994). Crossref, Google Scholar
149. Q.-D. Wang, The global solution of the n-body problem, Celest. Mech. Dyn. Astron. 50, 73 (1991); Acta Astro. Sinica 26, 313 (1991). Google Scholar
150. C. G. Weaver, Found. Phys. 51, 1 (2021). Crossref, Web of Science, Google Scholar
151. S. Weinberg, Foundations of Modern Physics (Oxford University Press, New York, 2021). Crossref, Google Scholar
152. K. Weierstrass, Ausgewählte Kapitel aus der Funktionenlehre (Select Chapters from the Theory of Functions). Vorlesung, gehalten in Berlin 1886. Mit der Akademischen Antrittsrede, Berlin 1857, und drei weiteren Originalarbeiten von K. Weierstrass aus den Jahren 1870 bis 1880/86, ed. R. Siegmund-Schultze, B. G. Teubner, Leipzig, 1988) (in German). Google Scholar
153. E. T. Whittaker, A Treatise on the Analytical Dynamics of Particles and Rigid Bodies with Introduction to the Problem of Three Bodies, (Foreword by Sir William McCrea), 4th edn. (Cambridge University Press, Cambridge, 1988). Google Scholar
154. A. Wintner, The Analytical Foundations of Celestial Mechanics (Princeton University Press, Princeton, NJ, 1941). Google Scholar
155. M. Wilson, Physics Avoidance: Essays in Conceptual Strategy (Oxford University Press, Oxford, 2017). Google Scholar
156. E. Zermelo, Ann. Phys. Chem. New Ser. 57, 485 (1896); ZCW 2, 215 (1896). Google Scholar
157. E. Zermelo, Ann. Phys. Chem. New Ser, 59, 793 (1896); ZCW 2, 247 (1896). Google Scholar
158. R. C. Tolman, The Principles of Statistical Mechanics (New York, Dover Publications, 1979). Google Scholar
159. O. Penrose, Foundations of Statistical Mechanics: A Deductive Treatment (Oxford, Pergamon Press, 1970). Google Scholar
160. O. E. Lanford III, “ Time Evolution of Large Classical Systems.” In Dynamical Systems, Theory and Applications: Lecture Notes in Theoretical Physics 38, edited by J. Moser, 1–111. (Berlin, Springer, 1975). Crossref, Google Scholar
161. D. W. Snoke, L. Gangqing and S. M. Girvin, “ The Basis of the Second Law of Thermodynamics in Quantum Field Theory, Ann. Phys. 327, 1825 (2012). Crossref, Web of Science, ADS, Google Scholar
162. P. Ehrenfest and T. Ehrenfest, The Conceptual Foundations of the Statistical Approach in Mechanics. Translated by Michael J. Moravcsik. Mineola: Dover Publications (1990). Google Scholar
163. K. R. Meyer and D. C. Offin, Introduction to Hamiltonian Dynamical Systems and the N-Body Problem. Third Edition (New York, Springer, 2017). Crossref, Google Scholar
164. R. G. Sachs, The Physics of Time Reversal (Chicago, University of Chicago, 1987). Google Scholar
165. O. Stern, “The Method of Molecular Rays”. Nobel Lecture. NobelPrize.org. Nobel Prize Outreach AB 2022. Mon. 25 Jul 2022 (1946), https://www.nobelprize.org/prizes/physics/1943/stern/lecture/. Google Scholar
166. J. North, “ Time in Thermodynamics.” In The Oxford Handbook of the Philosophy of Time, edited by C. Callender (New York, Oxford University Press, 2011). Google Scholar