On reduction of the wave-packet, decoherence, irreversibility and the second law of thermodynamics

We prove a quantum version of the second law of thermodynamics: the (quantum) Boltzmann entropy increases if the initial (zero time) density matrix decoheres, a condition generally satisfied in Nature. It is illustrated by a model of wave-packet reduction, the Coleman-Hepp model, along the framework introduced by Sewell in his approach to the quantum measurement problem. Further models illustrate the monotonic-versus-non-monotonic behavior of the quantum Boltzmann entropy in time. As a last closely related topic, decoherence, which was shown by Narnhofer and Thirring to enforce macroscopic purity in the case of quantum K systems, is analysed within a different class of quantum chaotic systems, viz. the quantum Anosov models as defined by Emch, Narnhofer, Sewell and Thirring. A review of the concept of quantum Boltzmann entropy, as well as of some of the rigorous approaches to the quantum measurement problem within the framework of Schr\"{o}dinger dynamics, is given, together with an overview of the C* algebra approach, which encompasses the relevant notions and definitions in a comprehensive way.