  Generalized Gibbs ensembles have been used as powerful tools to describe the
steady state of integrable many-particle quantum systems after a sudden change
of the Hamiltonian. Here we demonstrate numerically, that they can be used for
a much broader class of problems. We consider integrable systems in the
presence of weak perturbations which both break integrability and drive the
system to a state far from equilibrium. Under these conditions, we show that
the steady state and the time-evolution on long time-scales can be accurately
described by a (truncated) generalized Gibbs ensemble with time-dependent
Lagrange parameters, determined from simple rate equations. We compare the
numerically exact time evolutions of density matrices for small systems with a
theory based on block-diagonal density matrices (diagonal ensemble) and a
time-dependent generalized Gibbs ensemble containing only small number of
approximately conserved quantities, using the one-dimensional Heisenberg model
with perturbations described by Lindblad operators as an example.
