  In this review, we discuss the decoherence and thermalization of a quantum
spin system interacting with a spin bath environment, by numerically solving
the time-dependent Schr\"{o}dinger equation of the whole system. The effects of
the topologic structure and the initial state of the environment on the
decoherence of the two-spin and many-spin system are discussed. The role of
different spin-spin coupling is considered. We show under which conditions the
environment drives the reduced density matrix of the system to a fully
decoherent state, and how the diagonal elements of the reduced density matrix
approach those expected for the system in the microcanonical or canonical
ensemble, depending on the character of the additional integrals of motion. Our
demonstration does not rely on time-averaging of observables nor does it assume
that the coupling between system and bath is weak. Our findings show that the
microcanonical distribution (in each eigenenergy subspace) and canonical
ensemble are states that may result from pure quantum dynamics, suggesting that
quantum mechanics may be regarded as the foundation of quantum statistical
mechanics. Furthermore, our numerical results show that a fully decoherent
quantum system prefers to stay in an equilibrium state with a maximum entropy,
indicating the validity of the second law of thermodynamics in the decoherence
process.
