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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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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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  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.


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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).

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