I am very pleased and privileged to write a short foreword for the
monograph of Dean Driebe: Fully Chaotic Maps and Broken Time Symmetry.
Despite the technical title this book deals with a problem of
fundamental importance. To appreciate its meaning we have to go back to
the tragic struggle that was initiated by the work of the great
theoretical physicist Ludwig Boltzmann in the second half of the 19th
century. Ludwig Boltzmann tried to emulate in physics what Charles
Darwin had done in biology and to formulate an evolutionary approach in
which past and future would play different roles. Boltzmann's work has
lead to innumerable controversies as the laws of classical mechanics (as
well as the laws of quan- tum mechanics) as traditionally formulated
imply symmetry between past and future. As is well known, Albert
Einstein often stated that "Time is an illusion". Indeed, as long as
dynamics is associated with trajectories satisfy- ing the equations of
classical mechanics, explaining irreversibility in terms of trajectories
appears, as Henri Poincare concluded, as a logical error. After a long
struggle, Boltzmann acknowledged his defeat and introduced a probabil-
ity description in which all microscopic states are supposed to have the
same a priori probability. Irreversibility would then be due to the
imperfection of our observations associated only with the "macroscopic"
state described by temperature, pressure and other similar parameters.
Irreversibility then appears devoid of any fundamental significance.
However today this position has become untenable.
