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Looking at the Arrow of Time and Loschmidt’s Paradox Through the Magnifying Glass of Mathematical-Billiard

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Abstract

The contrast between the past-future symmetry of mechanical theories and the time-arrow observed in the behaviour of real complex systems doesn’t have nowadays a fully satisfactory explanation. If one confides in the Laplace-dream that everything be exactly and completely describable by the known mechanical differential equations, the whole experimental evidence of the irreversibility of real complex processes can only be interpreted as an illusion due to the limits of human brain and shortness of human history. In this work it is surmised that in the description of real events it would be more reasonable to renounce exactness and completeness of mechanical differential equations, assuming that also further effects exist in nature, governed by different kinds of rules, in spite of being so weak to be directly unobservable in single motions. This surmise can explain not only the time-arrow, but also why, in particular cases, it can happen that approximate and/or statistical models represent an improvement of mechanical theories instead of an approximation: this happened for Boltzmann gas-model and also for the famous work of Max Planck on blackbody-radiation. And it also appears as a more promising “working hypothesis”, stimulating and guiding us to learn more about limits and origin of the basic equations, and also about the nature of chance and the meaning of probability, which is nowadays not clear in spite of the fundamental role it plays in physics. Particularly that kind of probability which gives the connection between quantum–mechanical differential equations and observable events.

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Notes

  1. As a meaning of “professional deformation” we suggest the “Law of Maslow Hammer”: “I suppose it is tempting, if the only tool you have is a hammer, to treat everything as if it were a nail”. And differential equations are indeed the main tool we have to understand nature.

  2. I mean here not the simple concept of approximation or incompleteness that can be “cured” by reducing computational errors, or adding new variables, but the possible presence in nature of effects not describable within the known analytical and canonical formalism.

  3. You can put into evidence the “moral” character of the laws of thermodynamics writing them as follows:

    1st law: Remember, man, that you will never realize a process having the result of bringing a weight to a higher place without the sacrifice of something else.

    2nd law: Remember, man, that you will never realize a process having the result of bringing a weight to a higher place with the only consequence that something becomes colder.

  4. Speaking about initial and final conditions is not problematic if the motion is exactly periodic and you take a full period for the time elapsed from initial to final condition. In this case, a finite time interval contains the full evolution of the trajectory, i.e. both future and past of any point.

  5. Also in the presence of small perturbations of many other kinds (e.g. introducing weak numerical inaccuracies), it will happen that motions of type 1 will mostly remain of type 1, while motions of type 2 will mostly transform into motions of type 1.

  6. With “independent perturbation” we always mean perturbations generated according to rules that are different from dynamic-equations and are not retraced backwards by Loschmidt’s inversion.

  7. If you like a bit of humour, you could interpret these considerations, saying that breaking Loschmidt’s time-symmetry is, for the universe, a necessary condition to tolerate the existence of mathematicians. Anyway we note that this is a completely different kind of argument than “Boltzmann-brain paradox”, except that both appear to take into account the existence of human intelligence.

References

  1. Halliwell, J.J., Pérez-Mercader, J., Zurek, W.H.: Physical Origins of Time Asymmetry. Cambridge University Press, Cambridge (1996)

    Google Scholar 

  2. Zeh, H.D.: The Physical Basis of the Direction of Time. The Frontier Collection. Springer, New York (2007)

    MATH  Google Scholar 

  3. Lyapunov, A. M.: Ph.D. Thesis (in Russian); Stability of Motion. St. Petersburg University, Russia (1892)

  4. Prigogine, I., Stengers, I.: The End of Certainty. Simon and Shuster (1997)

  5. Tegmark, M.: The Mathematical Universe, Foundations of Physics, vol. 38. Springer, New York (2008)

    MATH  Google Scholar 

  6. Amelino-Camelia, G.: Oltre l’orizzonte. Codice edizioni, Torino (2017). ISBN 978-88-7578-720-2

    Google Scholar 

  7. Smolin, L.: Time Reborn. Mariner Books, New York (2014)

    Google Scholar 

  8. Smolin, L.: A naturalist account of the limited, and hence reasonable, effectiveness of mathematics in physics, (2015) arXiv: 1506.03733

  9. Canales, J.: The Physicist and the Philosopher. Princeton University Press, Princeton (2015)

    Book  Google Scholar 

  10. Kendall, M.G., Stuart, A.: The Advanced Theory of Statistic. Charles Griffin & Company Limited, London (1958)

    Google Scholar 

  11. Fisher, R.A.: Statistical Methods and Scientific Inference. Oliver and Boyd, Edinburgh (1956)

    MATH  Google Scholar 

  12. Jaynes, E.T.: Prior probabilities. IEEE Trans. Syst. Sci. Cybern. 4(3), 227–241 (1968)

    Article  Google Scholar 

  13. Ter Haar, D.: Elements of Statistical Mechanicsm, Holt. Rinehart and Winston, New York (1960)

    Google Scholar 

  14. Jaynes, E.T.: Information Theory and Statistical Mechanics. Brandeis Lectures: Statistical Physics, vol. 3. W. A. Benjamin Inc., New York (1963)

    Google Scholar 

  15. Prigogine, I.: Time Structure and Fluctuations, Nobel Lecture (1977)

  16. Kondepudi, D., Prigogine, I.: Modern Thermodynamics, from Heat Engines to Dissipative Structures. Wiley, New York (1998)

    MATH  Google Scholar 

  17. Planck, M.: Treatise on Thermodynamics. Dover Publications, New York (1926)

    Google Scholar 

  18. Bohm, D.: Causality and Chance in Modern Physics. University of Pennsylvania Press, Philadelphia (1957)

    Book  Google Scholar 

  19. Santos, E.: Stochastic electrodynamics and the interpretation of nonrelativistic quantum mechanics. (2014) arXiv:1205.0916v2[quant-ph]

  20. Popper, K.: Conjectures and Refutations. The Growth of scientific Knowledge. Basic Books, New York (1962)

    Google Scholar 

  21. Cortes, M., Gomes, H., Smolin, L.: Time asymmetric extensions of general relativity. Phys. Rev. D 92, 043502 (2015)

    ADS  MathSciNet  Article  Google Scholar 

  22. Landau, E.D., Lifshitz, E.M.: Quantum Mechanics. Pergamon Press, London (1958)

    MATH  Google Scholar 

  23. Gouëzel, S., Lanneau, E.: Un théorème de Kerckhoff, Masur et Smillie: Unique ergodicité sur le surfaces plates. Śémin. Congr. 20, 113–145. (2010) arXiv:math/0611827

  24. Reichenbach, H.: The Direction of Time. In: Maria Reichenbach (ed.) University of California Press (1971)

  25. Yukalov, V.I.: Irreversibility of time for quasi-isolated systems. Phys. Lett. A 308, 313 (2003)

    ADS  MathSciNet  Article  Google Scholar 

  26. Turing, A.: On computable numbers with an application to Entscheidungsproblem. Proc. Lond. Math. Soc. 42, 230–265 (1936)

    MathSciNet  MATH  Google Scholar 

  27. Petrosky, T., Prigogine, I.: Poincaré resonances and the limits of trajectory dynamics. Proc. Natl. Acad. Sci. U.S.A. 90, 9393–9397 (1993)

    ADS  Article  Google Scholar 

  28. Cubitt T.S, Perez-Garcia, D., Wolf, M. M.: Undecidability of the spectral gap. Nature 528, 207–211. (2015) arXiv: 1502.04135v3 (2018)

  29. Abbott, D.: The reasonable ineffectiveness of mathematics In: Proceedings of the IEEE, vol. 101, no. 10, October (2013)

  30. Welton, T.A.: Some observable effects of the quantum-mechanical fluctuations of the electromagnetic field. Phys. Rev. 74, 1157 (1948)

    ADS  Article  Google Scholar 

  31. Amelino-Camelia, G., Ellis, J., Mayromatos, N.E., Nanopoulos, D.V.: Distance measurement and wave dispersion in a liouville-string approach to quantum gravity. Int. J. Mod. Phys. A 12(03):607–623 (1996). arXiv:hep-th/9605211v129

    ADS  MathSciNet  Article  Google Scholar 

  32. Lucretius, T.: De Rerum Natura (1st century BC)

  33. Heisenberg, W.: The physical Principles of the Quantum Theory. Dover Publications, New York (1930)

    MATH  Google Scholar 

  34. Penrose, R.: Consciousness and the Universe. Cosmology Science Publisher. IBSN 10: 10938024435 (2015)

  35. Borges, J.L.: The writing of the god. In: Aleph and Other Stories (ed.) Plume 1979 (1949)

  36. Connes A., Rovelli, C.: Von Neumann Algebra Automorphisms and time-thermodynamics relation in general covariant quantum Theories. (2008). arXiv: gr-qc/9406019,v1

  37. Rovelli, C.: L’ordine del tempo, Adelphi (2017)

  38. Cortes, M., Smolin, L.: The universe as a process of unique events. Phys. Rev. D 90, 084007 (2014)

    ADS  Article  Google Scholar 

  39. ‘t Hooft, G.: The cellular automaton interpretation of quantum mechanics. (2015) arXiv:1495.1548v3 [quant-ph]

  40. ‘t Hooft, G.: Time, the arrow of time, and quantum mechanics. (2018). arXiv:1804.01383v1[quant-ph]

  41. Harsham, N.L.: Symmetry, structure, and emergent subsystems. (2018). arXiv:1801.08755v1[quant-ph]26Jan2018

  42. Coceva, C., Stefanon, M.: Experimental aspects of the statistical theory of nuclear spectra fluctuations. Nucl. Phys. A 315, 1–20 (1979)

    ADS  Article  Google Scholar 

  43. Stefanon, M.: Description of a statistical method of analysis of neutron resonance data. Nucl. Instrum. Methods 174(1980), 243–252 (1980)

    ADS  Article  Google Scholar 

  44. Ullmo, D.: Bohigas-Giannoni-Schmit conjecture. Scholarpedia 11, 31721 (2016)

    ADS  Article  Google Scholar 

  45. Dolev, Y.: Physics’ silence on time. Eur. J. Philos. Sci. 8(3), 455–469 (2018)

    MathSciNet  Article  Google Scholar 

  46. Smolin, L.: A real ensemble interpretation of quantum mechanics. Found. Phys. (2012). https://doi.org/10.1007/s10701-012-9666-4

    MathSciNet  Article  MATH  Google Scholar 

  47. Monod J.: Chance and Necessity. Mass Market Paperback (1972)

  48. Coveney, P., Highfield, R.: The Arrow of Time, Fawcett Columbine Book, First Paperback Edition (1992)

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Stefanon, M. Looking at the Arrow of Time and Loschmidt’s Paradox Through the Magnifying Glass of Mathematical-Billiard. Found Phys 49, 1231–1251 (2019). https://doi.org/10.1007/s10701-019-00295-7

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Keywords

  • Time arrow
  • Irreversibility
  • Probability
  • Statistical inference
  • Determinism