Irreversibility and spontaneous appearance of coherent behavior in reversible systems

There is empirical evidence that long time numerical simulations of conservative and reversible partial differential equations evolve, as a general rule (exceptions are the integrable models), towards an equilibrium state that is mainly a coherent structure plus small fluctuations inherent in the conservative and reversible character of the original system. The fluctuations account for the energy difference between the initial configuration and the one of the coherent structure. If the energy is not small enough, then the intrinsic fluctuations may destroy the coherent structure. Thus we arrive to the conclusion that a transition arises from a non-coherent state to a coherent structure as we decrease the initial energy below a critical value. This phenomenon has been successfully observed in various numerical simulations. In this article, we stress that this general behavior is also observed in reversible and conservative cellular automata as in the Q2R model. We point out that this conservative and reversible cellular automata is ab initio deterministic and therefore all our numerical computations are not affected by an approximation of any kind.