The Kac model: variations on a theme

Abstract

The Kac master equation provides a simple framework to understand systems of particles that interact through pairwise collisions. This article is a short review of results, chief among them is the approach to equilibrium for a gas of particles that undergo energy and momentum preserving collisions, as well as results on the entropy and information decay for a one-dimensional Kac system coupled to a reservoir. The principles underlying the Kac master equation can be extended to a Quantum Master Equation (QME) where the time evolution acts on density matrices and is a completely positive trace-preserving map. There is a rich set of equilibrium states and there is also a notion of propagation of chaos that leads to the Quantum Kac–Boltzmann equation. Likewise, the gap of the generator of the QME can be computed in certain special cases.