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Boltzmann’s reply to the Loschmidt paradox: a commented translation

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Abstract

Boltzmann’s reply to Loschmidt’s reversibility paradox (1877) has baffled many readers, owing to imprecise language and unproven assumptions. Based on a new translation and detailed commentary, it will be shown that this text nevertheless contains the essentials of a correct, insightful interpretation of thermodynamic irreversibility in statistico-mechanical context.

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Notes

  1. Loschmidt 1876, Boltzmann (1877a pp. 116–122).

  2. There already are a nearly complete French translation (Dugas 1959, pp. 188–191) and two complete English translations: Brush (1998, pp. 188–193), Gallavotti (2014, pp. 171–174). Translated extracts and critical comments are found in Klein (1973, pp. 71–74), Brush (1976, pp. 239, 605–607), Sklar (1993, p. 39), von Plato (1994, pp. 87–89), Cercignani (1998, § 5.2), Uffink (2007, § 4.3), Badino (2011, pp. 373–374).

  3. The chief protagonists of this debate are René Dugas, Martin Klein, and Jos Uffink (in favor of radical departure); Jan von Plato, Michel Janssen, and Massimo Badino (in favor of continuity). For references and further discussion, see Darrigol (2018, pp. 542–544).

  4. Loschmidt 1876. Cf. Dugas (1959, pp. 158–184), Darrigol (2018, pp. 181–188). William Thomson anticipated the Loschmidt paradox in 1871 (in private) and in 1874 (in print): see Klein (1970, p. 93), Garber et al. (1995, pp. 178, 187–188, 192–193, 205).

  5. Loschmidt (1876, p. 139).

  6. Today’s physicists know that the Loschmidt return (better: return to the initial microstate after a double velocity-reversal) can be concretely achieved in some systems. See Hahn 1950, Waugh, Rhim, and Pines 1972.

  7. Brush: this theorem.

  8. Brush: we must conceive.

  9. Brush: a certain body.

  10. We thereby include all material points of all bodies that interact with the bodies under consideration, directly or indirectly. Strictly speaking, all bodies of the universe ought to be included, because we cannot build a complex of bodies that has strictly no relation to the other bodies of the universe, we can only imagine it.

  11. Needless to say, for a picture [Anschauung] of the mode of action of natural forces in which this [reversal] is not true (as could happen, for instance, in a dynamic picture), the following considerations would also fail to apply.[The reference to this footnote is missing in the original WB publication; Hasenöhrl reintroduced it in BWA, at a location different from the one assumed in this translation]. [By “dynamic picture,” Boltzmann probably means a view, such as Lesage’s, in which the forces between atoms are traced to motion of intermediate matter].

  12. Brush: the sign of this integral …does not depend on the force law.

  13. Brush omits “rightly” (geradezu).

  14. Brush: and.

  15. Brush omits the “more likely”: will become uniform.

  16. Brush: a long time.

  17. Brush omits this last sentence (after the semicolon).

  18. Brush gives a completely different translation of this sentence. In particular, he renders folgen auf as “result in” instead of “ensue from.” So does Dugas, in French.

  19. Brush: It is only in special cases that…

  20. Brush: if one starts with oxygen and hydrogen mixed in a container…

  21. Brush: there is also an individual very improbably initial condition…

  22. Brush: atom.

  23. Brush omits “nearly”.

  24. Brush: If perhaps this reduction of the second law to the realm of probability makes its application to the entire universe appear dubious, yet the laws of probability theory are confirmed by all experiments carried out in the laboratory.

  25. Boltzmann (1866; 1871, p. 295). Cf. Darrigol (2018, pp. 70–79, 128–133).

  26. Boltzmann (1872, pp. 345–346).

  27. Uffink (2007, § 4.3.1, Point 1) gives this interpretation of the “sophism” and therefore sees a contradiction with Boltzmann’s subsequent admission of initial states for which the entropy law is violated. Without using the word sophism, Sklar (1993, p. 39) correctly remarks that in Boltzmann’s opinion, Loschmidt’s argument does not force us “to posit the existence of specific initial conditions” and that we can instead “take the statistical viewpoint.”

  28. This is properly noted in Uffink (2007, § 4.3.1, Point 2).

  29. Boltzmann had used special cases of Liouville’s theorem since 1868, and he had recently read Watson’s treatise (Watson 1896) in which the theorem was derived and used in broad Hamiltonian context: see Darrigol (2018, pp. 472–475).

  30. Cf. Uffink (2007, § 4.3.1, Point 4).

  31. Boltzmann 1877b. The alternative meaning of “state-distribution” already occurred in Boltzmann’s memoirs of 1871 and 1872.

  32. As is well known, this calculation is done by imagining a reversible transformation connecting the initial state to the end state and integrating the ratio dQ/T.

  33. Boltzmann (1866, 1877a, pp. 122–148). On the latter text, see Darrigol (2021, pp. 199–203).

  34. This confusion is unfortunately reflected in Darrigol (2018, pp. 197, 560). It will be corrected in the paperback edition (Darrigol 2021).

  35. Boltzmann (1877b, p. 166). See Darrigol (2021, pp. 560–563).

  36. In contrast, when we already know that the system has been evolving from a state out of equilibrium, there is only a very small fraction of the microstates compatible with the present macrostate that are compatible with this knowledge, and A1 cannot be true.

  37. See Cercignani 1998, Gallavotti 2014.

  38. See Alder and Wainwright 1960, Orban and Bellemans 1967.

  39. Zermelo (1896, pp. 795–796) strongly attacked A2 in 1896, arguing that probabilities by themselves could say nothing on the evolution of a system. Uffink (2007, § 4.3.1, Points 2 and 3) deplores the lack of justification of both A1 and A2, and his overall judgment of Boltzmann’s reply is unflattering: “One may question whether his considerations of the probability of the initial state hit the nail on the head. Probability theory is equally neutral to the direction of time as is mechanics.”

  40. Boltzmann (1877b, 1881a).

  41. Boltzmann (1872, 1895) (with dice instead of coin), 1897, 1898.

  42. This justification is fragile, because there is no warranty that in actual relaxation experiments all possible initial microstates may occur.

  43. Boltzmann (1868, pp. 95–96; 1881b, pp. 592–593). See Darrigol (2018, pp. 558–560). Boltzmann’s analysis does not apply to systems for which the microcanonical distribution does not yield the observed time-averages. This, the case for systems involving long-range coupling, encountered, for instance, in plasma physics. For an extension of statistical mechanics to such systems, cf. Tsallis 2009.

  44. In such experiments there is a common arrow of time, from the unlifted to the lifted sliding wall. In other words, the temporal asymmetry, which is absent in the complete H curve, is generated by the preparation of the initial state.

  45. Eddington (1929, p. 68).

  46. Boltzmann 1897. Cf. Schrödinger (1951, pp. 193–195).

  47. Boltzmann (1898, p. 637). See Darrigol (2018, pp. 552–553), (also for the related Boltzmann–Schütz argument).

  48. Boltzmann (1895, p. 541). On the British debate, cf. Brown (2009, pp. 182–284). Loschmidt and Boltzmann did not discuss the Boltzmann equation in their first exchange; they did so a few months later (see 2018, pp. 228–231).

  49. The Ehrenfest (1911, pp. 44–45) gave a similar interpretation of some of Boltzmann’s remarks, based on the “concentration curve” (Verdichtungskurve) for the bundle of H-curves starting from the microstates compatible with the given initial distribution. See Sklar (1993, pp. 63–67), Uffink (2007, pp. 60–61).

  50. Boltzmann 1896–1898, vol. 2, p. 112. Cf. Darrigol (2018, p. 443).

  51. These considerations may shed light on the (in)compatibility of Lanford’s theorem with the reversibility of the microdynamics, recently debated in Uffink and Valente (2015) and in Ardourel (2017).

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    Olivier Darrigol

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Darrigol, O. Boltzmann’s reply to the Loschmidt paradox: a commented translation. EPJ H 46, 29 (2021). https://doi.org/10.1140/epjh/s13129-021-00029-2

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