  When Rydberg states are excited in a dense atomic gas the mean number of
excited atoms reaches a stationary value after an initial transient period. We
shed light on the origin of this steady state that emerges from a purely
coherent evolution of a closed system. To this end we consider a
one-dimensional ring lattice, and employ the perfect blockade model, i.e. the
simultaneous excitation of Rydberg atoms occupying neighboring sites is
forbidden. We derive an equation of motion which governs the system's evolution
in excitation number space. This equation possesses a steady state which is
strongly localized. Our findings show that this state is to a good accuracy
given by the density matrix of the microcanonical ensemble where the
corresponding microstates are the zero energy eigenstates of the interaction
Hamiltonian. We analyze the statistics of the Rydberg atom number count
providing expressions for the number of excited Rydberg atoms and the Mandel
Q-parameter in equilibrium.
