Minimal irreversible quantum mechanics. The decay of unstable states
  Brownian motion is modelled by a harmonic oscillator (Brownian particle) interacting with a continuous set of uncoupled harmonic oscillators. The interaction is linear in the coordinates and the momenta. The model has an analytical solution that is used to study the time evolution of the reduced density operator. It is derived in a closed form, in the one-particle sector of the model. The irreversible behavior of the Brownian particle is described by a reduced density matrix. 
Diego G. Arbo, Mario A. Castagnino, Fabian H. Gaioli, Sergio Iguri