Quantum Interacting Particle Systems: Lecture Notes of the Volterra-Cirm International School, Trento, Italy, 23-29 September 2000 (Qp-Pq, Quantum Probability and White Noise Analysis, V. 14)

The dynamics of infinite classical lattice systems has been considered and has led to the study of the properties of ergodicity and convergence to equilibrium of a new class of Markov semigroups. Quantum analogues of these semigroups have also been considered. However, the problem of deriving these Markovian semigroups and, what is much more interesting, the associated stochastic flows, as limits of Hamiltonian systems, rather than postulating their form on a phenomenological basis, is essentially open both in the classical case and in the quantum case. This work conjectures that, by coupling a quantum spin system in finite volume to a quantum field via a suitable interaction, applying the stochastic golden rule and taking the thermodynamic limit, one may obtain a class of quantum flows which, when restricted to an appropriate Abelian subalgebra, gives rise to the classical interacting particle systems studied in classical statistical mechanics. In the first chapter of the book, it is proved that this conjecture is true and that the class of quantum-dynamical semigroups arising from the stochastic limit, in the weak coupling regime, has a rich structure which allows one in some cases to write down explicitly their invariant or equilibrium distributions. Chapter 2 discusses simple and effective methods to analyze qualitatively the behaviour of quantum Markov semigroups. The general methods discussed in the first two chapters are mainly effective in finite volume. New ideas and techniques which are specific to certain classes of generators need to be developed for use in infinite volume; this is the subject of the last chapter.