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

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.