Title: A Connection Between Mixing and Kac's Chaos,
Abstract: The Boltzmann equation is an integro-differential equation which describes
the density function of the distribution of the velocities of the molecules of
dilute monoatomic gases under the assumption that the energy is only
transferred via collisions between the molecules. In 1956 Kac studied the
Boltzmann equation and defined a property of the density function that he
called the "Boltzmann property" which describes the behavior of the density
function at a given fixed time as the number of particles tends to infinity.
The Boltzmann property has been studied extensively since then, and now it is
simply called chaos, or Kac's chaos. On the other hand, in ergodic theory,
chaos usually refers to the mixing properties of a dynamical system as time
tends to infinity. A relationship is derived between Kac's chaos and the notion
of mixing.