Propagation of Chaos for Kac-like Particle Systems

Doctoral thesis, 2015

This thesis concerns various aspects of the Kac model. The Kac model is a Markov jump process for a particle system where the total kinetic energy of the system is conserved. This particle model is connected to a limiting equation, describing the evolution of a one-particle density, when the numbers of particles tends to infinity. To rigourously derive the limiting equation, Kac proved that propagation of chaos holds for his model. Vaguely speaking, here chaos or chaotic means that the two-particle density can be written as a product of two one-particle densities when the numbers of particles tends to infinity. Propagation of chaos means that this property is propagated in time. The thesis contains three papers. The thermosttated Kac model is particle model where the jumps are modeled as in the Kac model. In additions to the jumps, the particles are accelerated between the jumps under the presence of a uniform force with a thermostat, acting equally on all particles. The thermostat ensures that no extra energy is supplied into the system by the force. In paper I we show that propagation of chaos holds for the thermosttated Kac model. In paper II we study a modified Kac model where the expression for the kinetic energy of a particle is replaced by an arbitrary energy function. This includes a one-dimensional Kac model for relativistic particles. We show that uniform distribution on the manifold defined by the conservation of total energy is chaotic. We also show that propagation of chaos holds for these modified Kac models. The BGK equation is an approximation to the Boltzmann equation by a relaxation term. In paper III we study a particle model involving jumps and exchanges between particles. We show that this particle model is connected to the one-dimensional spatially homogeneous BGK equation when the numbers of particles tends to infinity.