Date of Award
Open Access Dissertation
In 1956 Kac studied the Boltzmann equation, 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 an attempt at a solution to the Boltzmann equation, Kac introduced 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 has been abstracted to a property simply called chaos, or Kac’s chaos in classical mechanics. Kac’s chaos has attracted the attention of mathematicians and physicists such as Carlen, Grunbaum, McKean, Mischler, Mouhot, and Sznitman. On the other hand, in ergodic theory, chaos usually refers to the mixing properties of a dynamical system as time tends to infinity. In this thesis, a relationship is derived between classical Kac’s chaos and the notion of mixing. This relationship provides examples of Kac’s chaos built from dynamical systems with certain mixingtype properties. In order to prove this relationship, a famous result of Sznitman is used, which states that the classical form of Kac’s chaos is equivalent to a certain convergence of empirical measures. Further, the quantum version of Kac’s chaos is studied in this thesis. This form of chaos was implicitly introduced by Spohn and explicitly formulated by Gottlieb. The quantum analogue of the result of Sznitman which gives the equivalence of Kac’s chaos to 2-chaoticity and to a certain convergence of empirical measures is proven. Finally, a simple, different proof of a result of Spohn which states that chaos propagates with respect to certain Hamiltonians that define the evolution of the mean field limit for interacting quantum systems is proven.
Musulin, R.(2018). Classical and Quantum Kac’s Chaos. (Doctoral dissertation). Retrieved from https://scholarcommons.sc.edu/etd/4515