Mathematical problems concerning the Kac model
Tossounian, Hagop B.
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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  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 . 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 , , and  joint work with my collaborators Federico Bonetto, Michael Loss, and Ranjini Vaidyanathan. The work in  extends the work in  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  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  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  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  is also given.