  We analyze the dynamics of N interacting spins (quantum register)
collectively coupled to a thermal environment. Each spin experiences the same
environment interaction, consisting of an energy conserving and an energy
exchange part.
  We find the decay rates of the reduced density matrix elements in the energy
basis. We show that if the spins do not interact among each other, then the
fastest decay rates of off-diagonal matrix elements induced by the energy
conserving interaction is of order N^2, while that one induced by the energy
exchange interaction is of the order N only. Moreover, the diagonal matrix
elements approach their limiting values at a rate independent of N.
  For a general spin system the decay rates depend in a rather complicated (but
explicit) way on the size N and the interaction between the spins.
  Our method is based on a dynamical quantum resonance theory valid for small,
fixed values of the couplings. We do not make Markov-, Born- or weak coupling
(van Hove) approximations.
