## Mathematical problems concerning the Kac model

##### Abstract

This thesis deals with the Kac model in kinetic theory. Kac’s model is a linear, space homogeneous, n-particle model created by Mark Kac in 1956 in [14] in an attempt to give a derivation of Boltzmann’s equation. The marginals of a distribution under Kac’s evolution are connected to a simple Boltzmann-type equation via the mechanism of “propagation of chaos” given by Kac in [14]. Kac evolution preserves total (kinetic) energy and is ergodic, having the uniform distribution on the constant energy sphere as its equilibrium. The generator of the associated Markov process has a spectral gap that is bounded away from zero uniformly in n. A central question in the field is the speed of approach to equilibrium (or rate of equilibration). The thesis gives the results of the papers [25], [3], and [26] joint work with my collaborators Federico Bonetto, Michael Loss, and Ranjini Vaidyanathan. The work in [25] extends the work in [2] by studying the rate of approach to equilibrium when a fraction α = m/n of the particles interact with a “strong” thermostat. Results in the spectral gap and the (negative of) relative entropy metric are obtained. The work in [3] shows, using both the L2 metric and the Fourier-based metric d2 , that the evolution of the system interacting with the ideal infinite particle thermostat used in the model in [2] can be approximated by the evolution of the same system interacting with a large but finite reservoir. This approximation does not deteriorate with time, and it improves as the number of reservoir particles increases.
The work in [26] studies the Kac evolution in the absence of thermostats and reservoirs using
the metric d2. It finds an upper bound to the approach to equilibrium, and constructs a family
of initial states that for time t0 independent of n shows practically no approach to equilibrium
in d2. An independent propagation of chaos result for the model in [25] is also given.