I am very completely satisfied and privileged to place in writing a short foreword for the monograph of Dean Driebe: Fully Chaotic Maps and Broken Time Symmetry. Whatever the technical title this book provides with a problem of elementary significance. To know its which suggests we now need to return to the tragic wrestle that was initiated by the work of the great theoretical physicist Ludwig Boltzmann inside the second half of the nineteenth century. Ludwig Boltzmann tried to emulate in physics what Charles Darwin had executed in biology and to formulate an evolutionary technique by which earlier and future would play completely totally different roles. Boltzmann's work has end in innumerable controversies as a result of the authorized tips of classical mechanics (along with the authorized tips of quan tum mechanics) as traditionally formulated recommend symmetry between earlier and future. As is properly-recognized, Albert Einstein sometimes stated that "Time is an illusion". Definitely, as long as dynamics is said to trajectories fulfill ing the equations of classical mechanics, explaining irreversibility when it involves trajectories appears, as Henri Poincare concluded, as a logical error. After a long wrestle, Boltzmann acknowledged his defeat and launched a probabil ity description via which all microscopic states are imagined to have the similar a priori probability. Irreversibility would then be due to the imperfection of our observations associated solely with the "macroscopic" state described by temperature, pressure and totally different comparable parameters. Irreversibility then appears devoid of any elementary significance. However in the intervening time this place has develop to be untenable.
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