Quantifying Self-Organization and Coherent Structures with Statistical Complexity

Despite broad interest in self-organizing systems, there are few quantitative criteria for self-organization which can be applied to dynamical models, let alone experimental data. The existing criteria all give counter-intuitive results in important cases. A resolution is offered by a recently-proposed criterion, namely an internally-generated increase in the statistical complexity, the amount of information required for optimal prediction of the system's dynamics. This complexity can be precisely defined for spatially-extended dynamical systems, using the probabilistic ideas of mutual information and minimal sufficient statistics. The definition also leads to a general method for predicting such systems. Examining the variation in the statistical complexity over space and time provides a way of automatically identifying the coherent structures generated by the system. Results on two important classes of cellular automata (CA) --- cyclic CA, which model excitable media, and sandpile CA, which are prototypes of self-organized criticality --- illustrate the general ideas.