Statistical properties of spatiotemporal dynamical systems

In this paper we consider spatiotemporal dynamical systems modeled by coupled, or uncoupled but noise-driven, map lattices. In particular, we examine reports in the literature where it is found that the distribution of certain mean-field quantities violates the law of large numbers (hence nonstatistical) but not the central-limit theorem. Our results show that the origin of such nonstatistical behavior is due to the statistical dependence between random variables at different lattice sites, thus rendering nonapplicable to such situations the law of large numbers and the central-limit theorem. Additional issues explored include the discussion of a special class of systems where nonstatistical behavior is not observed and the physical motivation for considering uncoupled but noise-driven map lattices.