  We study the non-equilibrium dynamics of driven spin lattices in the presence
of decoherence caused by either laser phase noise or strong decay. In the first
case, we discriminate between correlated and uncorrelated noise and explore
their effect on the mean density of Rydberg states and the full counting
statistics (FCS). We find that while the mean density is almost identical in
both cases, the FCS differ considerably. The main method employed is the
Langevin equation (LE) but for the sake of efficiency in certain regimes, we
use a Markovian master equation and Monte Carlo rate equations, respectively.
In the second case, we consider dissipative systems with more general power-law
interactions. We determine the phase diagram in the steady state and analyse
its generation dynamics using Monte Carlo rate equations. In contrast to
nearest-neighbour models, there is no transition to long-range-ordered phases
for realistic interactions and resonant driving. Yet, for finite laser
detunings, we show that Rydberg lattices can undergo a dissipative phase
transition to a long-range-ordered antiferromagnetic (AF) phase. We identify
the advantages of Monte Carlo rate equations over mean field (MF) predictions.
