Statistical Asymmetry: An Entropic Measure of Symmetry

This work introduces formally the concept of statistical asymmetry of a system as an entropic measure of how much it fails to be fully symmetric under a given group of transformations. It is shown that it is able to provide an alternative classification of one-dimensional elementary cellular automata that closely aligns with known others only by measuring symmetry. The behaviour of statistical asymmetry can also be an useful indicator of complex behaviour on two-dimensional discrete processes by following the dynamics of configurations, which is demonstrated in the case of the Geenberg-Hastings model, Conway's game of life, and the random evolution of discrete square matrices.