Superselection sectors, pure states, Bose and Fermi distributions by completely positive quantum-motions

The connection of the operators V, building up the Kossakowski-Lindblad generator, with the asymptotic states of the corresponding completely positive quantum-maps is discussed. Maps leading to decolierence are constructed, the importance of zero-modes in the absolute value root V+V of V for the generation of pure states from arbitrary mixed states is illustrated. The universal role of equipartite states appears when unitary V are chosen. The 'damped oscillator model' is generalized to yield Bose and Fermi distributions as asymptotic states for systems described by a Hamiltonian and other constants of motion. Calculations are performed in finite dimensional Hilbert spaces.