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Notes on Probabilty and Statistical Mechanics

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David Bohm: Causality and Chance, Letters to Three Women

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Abstract

From the very first letters to Miriam, we find that Bohm is working on probability and statistical mechanics and asking Miriam to use her mathematical expertise in order to help him. In the period up to and including February 1953, there are 14 letters or parts of letters on this topic, most of them quite technical. Then, in 1954, there is a second group of 7 letters, several of them quite long (more than 10 pages).

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Notes

  1. 1.

    (20, 59, pp. 208–209).

  2. 2.

    (21, 64, pp. 221–222).

  3. 3.

    Tolman (1979) in (20, 61, pp. 212–215).

  4. 4.

    (20, 61, pp. 212–215).

  5. 5.

    (21, 64, pp. 221–222).

  6. 6.

    (21, 64, pp. 221–224).

  7. 7.

    (22, 67, pp. 240–242).

  8. 8.

    (22, 68, p. 246).

  9. 9.

    (23, 75, pp. 259–261).

  10. 10.

    (23, 75, p. 260).

  11. 11.

    In the incomplete letter (23, 76, pp. 262–263).

  12. 12.

    (23, 78, pp. 265–270).

  13. 13.

    (24, 83, pp. 277–281).

  14. 14.

    (24, 83, p. 279).

  15. 15.

    (24, 84, pp. 281–284).

  16. 16.

    See Figs. 9.1, 9.2 and 9.3 for the original copies of this letter.

  17. 17.

    (24, 86, pp. 284–285).

  18. 18.

    Actually it was not Weyl, but the mathematician E. Borel, who in 1909 first looked at the so-called “normal” numbers, beween 0 and 1, which would give rise to a uniform distribution. See Niven (2005), Chap. VIII.

  19. 19.

    (24, 87, pp. 286–289).

  20. 20.

    See for example Stewart (1990), p. 112, where this model is used with \(K=10\), so that the change from one angle to the next in the sequence is given by shifting the decimal point to the right, and then dropping the whole number before the decimal point. Stewart’s book is a useful introduction to Chaos Theory.

  21. 21.

    Note that this assumes the “essential randomness” of standard quantum mechanics. Bohm’s causal interpretation would assume only a difference of degree between the two cases.

  22. 22.

    For a nice introduction see Bennett (1999). For Bohm’s model as a pseudo random number generator for large K, see Broer and Takens (2011), p. 99 (Transition from chaotic to stochastic).

  23. 23.

    See also Bohm and Schützer (1955), Sect. 7.

  24. 24.

    (27, 104, pp. 344–345).

  25. 25.

    See Dürr and Teufel (2009), Sect. 4.3.

  26. 26.

    For the Strong Law see (20, 58, p. 205), (21, 64, pp. 223–224) and Bohm thanking Miriam for her help on this topic in (23, 77, p. 264). For Bohm’s dismissal of Lesbesgue integration see (23, 75, pp. 259–260).

  27. 27.

    (23, 77, p. 263).

  28. 28.

    (25, 89, p. 295).

  29. 29.

    (27, 104, pp. 344–345).

  30. 30.

    (27, 104, pp. 344–345).

  31. 31.

    (26, 96, pp. 317–318).

  32. 32.

    (18, 42, pp. 161–163).

  33. 33.

    (30, 114, pp. 378–386), (30, 115, pp. 391–393), (31, 117, pp. 397–403), (31, 119, pp. 406–407), (31, 120, pp. 408–414), (32, 121, pp. 417–423) and (32, 122, pp. 423–427).

  34. 34.

    (28, 106, p. 351).

  35. 35.

    Bunge (1951).

  36. 36.

    See Bunge (2016), p. 90.

  37. 37.

    (18, 43, pp. 163–165) and (18, 46, pp. 170–173).

  38. 38.

    This is not true, a tradition of Marxist work on biology was developed in the USSR and suppressed under Stalin. See, for example, the article written by the American geneticist and Nobel prize winner, H.J. Muller, Lenin on Genetics, reproduced in Graham (1971), pp. 451–472. Muller supported the Soviet Union to the extent of moving to Russia in 1933. Stalin’s opposition to him and the growing support for Lysenko forced him to leave by 1937.

  39. 39.

    Marx and Engels (1988), p. 498. The example of “Darwin, in his epoch-making work” is on p. 501.

  40. 40.

    In Wilkins (1986), VI, Bohm says “I thought Schönberg had a deeper view of these things than most of the left-wing people. In a way, he helped to show me that I had been approaching the thing in a narrow way, by just looking at causality, without bringing in the opposite side of chance.”

  41. 41.

    (31, 117, pp. 402–403).

  42. 42.

    Ibid.

  43. 43.

    (31, 119, p. 407).

  44. 44.

    Bohm (1957), p. 2.

  45. 45.

    See (30, 114, p. 379).

  46. 46.

    (31, 119, p. 407).

  47. 47.

    (32, 121, p. 418).

  48. 48.

    Bohm and Schützer (1955), Sect. 6.

  49. 49.

    As in (21, 64, pp. 221–222) above.

  50. 50.

    Bohm and Schützer (1955), Sect. 10.

  51. 51.

    (31, 120, pp. 414–415) and (32, 122, pp. 425–427).

  52. 52.

    (31, 120, pp. 408–413).

  53. 53.

    Bohm uses radian measure, so \(360^\circ \) is \(2\pi \)radians.

  54. 54.

    To be more precise, one can always predict forward in time by using \(\theta _{n+1} = K \theta _n\), but one cannot predict backwards or retrodict.

  55. 55.

    Or “chaos”, as in the above discussion relating to (24, 84, pp. 281–284).

  56. 56.

    Bohm refers to a key theorem in statistics, the “law of large numbers”.

  57. 57.

    As, for example, the statistical mechanics of an ideal gas give rise to thermodynamical laws.

  58. 58.

    See (30, 114, pp. 380–382) for examples.

  59. 59.

    (30, 114, pp. 378–381).

  60. 60.

    Bohm and Schützer (1955), Sects. 3–5.

  61. 61.

    For a more thorough history, see Plato (1994). Von Plato (p. 231) notes the extraordinary support for Kolmogorov’s approach and the dismissal of von Mises by none other than J.L. Doob, who Miriam had dealings with, as noted above. Doob wrote that Kolmogorov “transformed the character of the calculus of probabilities, moving it into mathematics from its previous state as a collection of calculations inspired by a vague non-mathematical context.”

  62. 62.

    In which, as noted above, the ratio of the number of times a particular event occurs to the total number of events is used, taking larger and larger totals, in order to approximate a probability.

  63. 63.

    (30, 114, pp. 382–386), (30, 115, pp. 391–392), (31, 117, pp. 398–402), (31, 119, p. 407), (32, 122, pp. 424–427), as well as in Bohm and Schützer (1955), Sect. 9.

  64. 64.

    (31, 119, p. 407).

  65. 65.

    In (20, 61, pp. 213–215).

  66. 66.

    Bohm and Schützer (1955), pp. 1030–1031.

  67. 67.

    Bohm and Schützer (1955), pp. 1018–1020.

  68. 68.

    Lebowitz (1993).

  69. 69.

    Dürr and Teufel (2009), Sect. 4.1.

  70. 70.

    Frigg (2011).

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    Chris Talbot

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Talbot, C. (2017). Notes on Probabilty and Statistical Mechanics. In: Talbot, C. (eds) David Bohm: Causality and Chance, Letters to Three Women. Springer, Cham. https://doi.org/10.1007/978-3-319-55492-1_9

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