focused on the possible realization of statistical mechanics for reversible cellular automata. His studies concerned explicitly all class of rule in the one dimensional case, the Q2R being only a special case. However, the Q2R (90R in his terminology), is the analogue of an ideal gas of particles with speeds +1 or -1, a system that cannot reach equilibrium in practice. But, it is ergodic only in thermodynamical equilibrium. More recently, in Ref. [@goles], one of us (SR) has studied numerically the irreversible behavior and the existence of a spontaneous transition from a non-coherent state to a coherent state.

The present article is organized as follows, the Q2R model, as well as its main features and findings are presented in section \[model\]. This section is sub-divided as follows: we briefly report the numerical studies of Ref. [@goles] in sub-section \[longtime\]; the phase space properties, in particular some results on the distribution of periods of the dynamics is reported in subsection \[PhaseSpace\]; and, finally, the scope of the paper is presented in sub section \[scope\]. Section \[masterEqn\], introduces the notion of a master equation for the statistical description of the dynamics. Finally, in Section \[Results\] we provide some precise examples, where 